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1
Abdenova, Gaukhar, Karlygash Bazikova, and Zhangul Kenzhegalym. "A parabolic model in the form of space states of the dynamics of savings." Analysis and data processing systems, no. 2 (June 18, 2021): 7–18. http://dx.doi.org/10.17212/2782-2001-2021-2-7-18.
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An important place in the theory of partial differential equations and its applications is occupied by the heat equation, a representative of the class of the so-called parabolic equations. It is known that to check the correctness of a mathematical model based on a parabolic equation, the existence of its solution is very important since a mathematical model is not always adequate to a specific phenomenon and the existence of a solution to a corresponding mathematical problem does not follow from the existence of a solution to a real applied problem. Therefore, methods for solving partial differential equations, both analytical and numerical, are always relevant. Nowadays, a computational experiment has become a powerful tool for theoretical research. It is carried out over a mathematical model of the object under study, but at the same time, other parameters are calculated using one of the parameters of the model and conclusions are drawn about the properties of the object or phenomenon under study. The problem of passive parametric identification of systems with distributed parameters for resource accumulation dynamics of many households using a stochastic distributed model in the form of a state space with regard to the white noise of the dynamics model of the object under study and the white noise of the model of a linear-type measuring system is considered in the paper. The use of the finite difference method allowed us to reduce the solution of partial differential equations of a parabolic type to the solution of a system of linear finite difference and algebraic equations represented by models in the form of a state space. It was also proposed to use a filtering algorithm based on the Kalman scheme for reliable estimation of the object behavior. Calculations were carried out using the Matlab mathematical system based on statistical data for five years, taken from the site “Agency for Strategic Planning and Reforms of the Republic of Kazakhstan Bureau of National Statistics”. Estimation of the coefficients of the equations for the household resource accumulation in the form of a state space using this technique is sufficiently universal and can be applied in other fields of science and technology.
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Friedly, J. C. "Transient Response of a Coupled Conduction and Convection Heat Transfer Problem." Journal of Heat Transfer 107, no. 1 (1985): 57–62. http://dx.doi.org/10.1115/1.3247402.
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Systems of dynamic models involving the coupling of both conduction and convection offer significant theoretical challenges because of the interaction between parabolic and hyperbolic types of responses. Recent results of state space theory for coupled partial differential equation models are applied to conjugate heat transfer problems in an attempt to understand this interaction. Definition of a matrix of Green’s functions for such problems permits the transient responses to be resolved directly in terms of the operators’ spectral properties when they can be obtained. Application of the theory to a simple conjugate heat transfer problem is worked out in detail. The model consists of the transient energy storage or retrieval in a stationary, single dimensioned matrix through which an energy transport fluid flows. Even though the partial differential operator is nonself-adjoint, it is shown how its spectral properties can be obtained and used in the general solution. Computations are presented on the effect of parameters on the spectral properties and the nature of the solution. Comparison is made with several readily solvable limiting cases of the equations.
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Phan, Tuấn Đình, and Ha Ngoc Hoang. "NONLINEAR CONTROL OF TEMPERATURE PROFILE OF UNSTABLE HEAT CONDUCTION SYSTEMS: A PORT HAMILTONIAN APPROACH." Journal of Computer Science and Cybernetics 32, no. 1 (2016): 61–74. http://dx.doi.org/10.15625/1813-9663/32/1/6401.
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This paper focuses on boundary control of distributed parameter systems (also called infinite dimensional systems). More precisely, a passivity based approach for the stabilization of temperature profile inside a well-insulated bar with heat conduction in a one-dimensional described by parabolic partial differential equations (PDEs) is developed. This approach is motivated by an appropriate model reduction schema using the finite difference approximation method. On this basis, it allows to discretize and then, write the original parabolic PDEs into a Port Hamiltonian (PH) representation. From this, the boundary control input is therefore synthesized using passive tools to stabilize the temperature at a desired reference profile asymptotically. The infinite dimensional nature of the original distributed parameter system in the PH framework is also discussed. Numerical simulations illustrate the application of the developments.
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Straughan, B. "Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2157 (2013): 20130187. http://dx.doi.org/10.1098/rspa.2013.0187.
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There is increasing interest in convection in local thermal non-equilibrium (LTNE) porous media. This is where the solid skeleton and the fluid may have different temperatures. There is also increasing interest in thermal wave motion, especially at the microscale and nanoscale, and particularly in solids. Much of this work has been based on the famous model proposed by Carlo Cattaneo in 1948. In this paper, we develop a model for thermal convection in a fluid-saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. However, we base our thermodynamics for the fluid on Fourier's law of heat conduction, whereas we allow the solid skeleton to transfer heat by means of the Cattaneo heat flux theory. This leads to a novel system of partial differential equations involving Darcy's law, a parabolic fluid temperature equation and effectively a hyperbolic solid skeleton temperature equation. This system leads to novel physics, and oscillatory convection is found, whereas for the standard LTNE Darcy model, this does not exist. We are also able to derive a rigorous nonlinear global stability theory, unlike work in thermal convection in other second sound systems in porous media.
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Eremin, A. V., and K. V. Gubareva. "Analytical solution to the problem of heat transfer using boundary conditions of the third kind." Vestnik IGEU, no. 6 (2019): 67–74. http://dx.doi.org/10.17588/2072-2672.2019.6.067-074.
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Non-stationary heat transmission within solid bodies is described using parabolic and hyperbolic equations. Currently, numerical methods for studying the processes of heat and mass transfer in the flows of liquids and gases have disseminated. Modern programs allow the automatic construction of computational grids, solutions to the systems of equations and offer a wide range of tools for analysis. Approximate analytical solutions have significant advantages compared to numerical ones. In particular, the solutions obtained in an analytical form allow performing parametric analysis of the system under study, configuration and programming of measurement devices, etc. Based on the joint use of additional desired function and additional boundary conditions in the integral method of heat balance, a method of mathematical modeling for the heat transfer process in a plate under symmetric boundary conditions of the third kind has been developed. Using the heat flux density as a new desired function, the method for solving heat conduction problems with boundary conditions of the third kind has been proposed. Finding a solution to the partial differential equation with respect to the temperature function presents integrating an ordinary differential equation with respect to the heat flux density on the surface of the studied zone. It has been shown that isotherms appear on the surface of the plate with a certain initial velocity which depends on the heat transfer intensity. The calculation results have been compared to the exact solution. The presented method can be used in determining the heat flux density of buildings and heating devices, finding heat losses during convective heat transfer and designing heat transfer equipment. The results can be applied to increase the validity and reliability of the calculation of actual losses and balance of thermal energy. The method reliability, validity and a high degree of approximation with about 3% inaccuracy have been demonstrated. The accuracy of the solution depends on the number of approximations performed and is determined by the degree of the approximating polynomial.
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Cruz-Quintero, Eduardo, and Francisco Jurado. "Boundary Control for a Certain Class of Reaction-Advection-Diffusion System." Mathematics 8, no. 11 (2020): 1854. http://dx.doi.org/10.3390/math8111854.
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There are physical phenomena, involving diffusion and structural vibrations, modeled by partial differential equations (PDEs) whose solution reflects their spatial distribution. Systems whose dynamics evolve on an infinite-dimensional Hilbert space, i.e., infinite-dimensional systems, are modeled by PDEs. The aim when designing a controller for infinite-dimensional systems is similar to that for finite-dimensional systems, i.e., the control system must be stable. Another common goal is to design the controller in such a way that the response of the system does not be affected by external disturbances. The controller design for finite-dimensional systems is not an easy task, so, the controller design for infinite-dimensional systems is even more challenging. The backstepping control approach is a dominant methodology for boundary feedback design. In this work, we try with the backstepping design for the boundary control of a reaction-advection-diffusion (R-A-D) equation, namely, a type parabolic PDE, but with constant coefficients and Neumann boundary conditions, with actuation in one of these latter. The heat equation with Neumann boundary conditions is considered as the target system. Dynamics of the open- and closed-loop solution of the PDE system are validated via numerical simulation. The MATLAB®-based numerical algorithm related with the implementation of the control scheme is here included.
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JOSHI, DHRUTI B., and A. K. DESAI. "Reduction of Certain Type of Parabolic Partial Differential Equations to Heat Equation." Journal of Ultra Scientist of Physical Sciences Section A 30, no. 9 (2018): 353–61. http://dx.doi.org/10.22147/jusps-a/300901.
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Jin, Sixian, and Henry Schellhorn. "Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations." International Journal of Stochastic Analysis 2017 (February 27, 2017): 1–12. http://dx.doi.org/10.1155/2017/2876961.
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We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.
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KUMAR, B. V. RATHISH, and MANI MEHRA. "A WAVELET-TAYLOR GALERKIN METHOD FOR PARABOLIC AND HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS." International Journal of Computational Methods 02, no. 01 (2005): 75–97. http://dx.doi.org/10.1142/s0219876205000375.
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In this study a set of new space and time accurate numerical methods based on different time marching schemes such as Euler, leap-frog and Crank-Nicolson for partial differential equations of the form [Formula: see text], where ℒ is linear differential operator and [Formula: see text] is a nonlinear function, are proposed. To produce accurate temporal differencing, the method employs forward/backward time Taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation. This yields a generalized time discretized scheme which is approximated in space by Galerkin method. The compactly supported orthogonal wavelet bases developed by Daubechies are used in Galerkin scheme. This new wavelet-Taylor Galerkin approach is successively applied to heat equation, convection equation and inviscid Burgers' equation.
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Kafle, J., L. P. Bagale, and D. J. K. C. "Numerical Solution of Parabolic Partial Differential Equation by Using Finite Difference Method." Journal of Nepal Physical Society 6, no. 2 (2020): 57–65. http://dx.doi.org/10.3126/jnphyssoc.v6i2.34858.
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In the real world, many physical problems like heat equation, wave equation, Laplace equation and Poisson equation are modeled by partial differential equations (PDEs). A PDE of the form ut = α uxx, (α > 0) where x and t are independent variables and u is a dependent variable; is a one-dimensional heat equation. This is an example of a prototypical parabolic equation. The heat equation has analytic solution in regular shape domain. If the domain has irregular shape, computing analytic solution of such equations is difficult. In this case, we can use numerical methods to compute the solution of such PDEs. Finite difference method is one of the numerical methods that is used to compute the solutions of PDEs by discretizing the given domain into finite number of regions. Here, we derived the Forward Time Central Space Scheme (FTCSS) for this heat equation. We also computed its numerical solution by using FTCSS. We compared the analytic solution and numerical solution for different homogeneous materials (for different values of diffusivity α). There is instantaneous heat transfer and heat loss for the materials with higher diffusivity (α) as compared to the materials of lower diffusivity. Finally, we compared simulation results of different non-homogeneous materials.
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Cywiak-Códova, D., G. Gutiérrez-Juárez, and And M. Cywiak-Garbarcewicz. "Spectral generalized function method for solving homogeneous partial differential equations with constant coefficients." Revista Mexicana de Física E 17, no. 1 Jan-Jun (2020): 11. http://dx.doi.org/10.31349/revmexfise.17.11.
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A method based on a generalized function in Fourier space gives analytical solutions to homogeneous partial differential equations with constant coefficients of any order in any number of dimensions. The method exploits well-known properties of the Dirac delta, reducing the differential mathematical problem into the factorization of an algebraic expression that finally has to be integrated. In particular, the method was utilized to solve the most general homogeneous second order partial differential equation in Cartesian coordinates, finding a general solution for non-parabolic partial differential equations, which can be seen as a generalization of d'Alambert solution. We found that the traditional classification, i.e., parabolic, hyperbolic and elliptic, is not necessary reducing the classification to only parabolic and non-parabolic cases. We put special attention for parabolic partial differential equations, analyzing the general 1D homogeneous solution of the Photoacoustic and Photothermal equations in the frequency and time domain. Finally, we also used the method to solve Helmholtz equation in cylindrical coordinates, showing that it can be used in other coordinates systems.
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Kravchenko, Vladislav V., Josafath A. Otero, and Sergii M. Torba. "Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients." Advances in Mathematical Physics 2017 (2017): 1–5. http://dx.doi.org/10.1155/2017/2947275.
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A complete family of solutions for the one-dimensional reaction-diffusion equation, uxx(x,t)-q(x)u(x,t)=ut(x,t), with a coefficient q depending on x is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper, the Dirichlet boundary conditions are considered; however, the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.
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K.C., Durga Jang, and Ganesh Bahadur Basnet. "Discrete Maximum Principle in One-Dimensional Heat Equation." Journal of Advanced College of Engineering and Management 2 (November 29, 2016): 5. http://dx.doi.org/10.3126/jacem.v2i0.16093.
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<p>The maximum principle plays key role in the theory and application of a wide class of real linear partial differential equations. In this paper, we introduce ‘Maximum principle and its discrete version’ for the study of second-order parabolic equations, especially for the one-dimensional heat equation. We also give a short introduction of formation of grid as well as finite difference schemes and a short prove of the ‘Discrete Maximum principle’ by using different schemes of heat equation.</p><p><strong>Journal of Advanced College of Engineering and Management</strong>, Vol. 2, 2016, Page: 5-10</p>
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D’Acunto, Berardino, Luigi Frunzo, Vincenzo Luongo, and Maria Rosaria Mattei. "Modeling Heavy Metal Sorption and Interaction in a Multispecies Biofilm." Mathematics 7, no. 9 (2019): 781. http://dx.doi.org/10.3390/math7090781.
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A mathematical model able to simulate the physical, chemical and biological interactions prevailing in multispecies biofilms in the presence of a toxic heavy metal is presented. The free boundary value problem related to biofilm growth and evolution is governed by a nonlinear ordinary differential equation. The problem requires the integration of a system of nonlinear hyperbolic partial differential equations describing the biofilm components evolution, and a systems of semilinear parabolic partial differential equations accounting for substrates diffusion and reaction within the biofilm. In addition, a semilinear parabolic partial differential equation is introduced to describe heavy metal diffusion and sorption. The biosoption process modeling is completed by the definition and integration of other two systems of nonlinear hyperbolic partial differential equations describing the free and occupied binding sites evolution, respectively. Numerical simulations of the heterotrophic-autotrophic interaction occurring in biofilm reactors devoted to wastewater treatment are presented. The high biosorption ability of bacteria living in a mature biofilm is highlighted, as well as the toxicity effect of heavy metals on autotrophic bacteria, whose growth directly affects the nitrification performance of bioreactors.
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Meng, Xiangqian, and Erkan Nane. "Space-time fractional stochastic partial differential equations with Lévy noise." Fractional Calculus and Applied Analysis 23, no. 1 (2020): 224–49. http://dx.doi.org/10.1515/fca-2020-0009.
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AbstractWe consider non-linear time-fractional stochastic heat type equation$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg] \end{array} $$and$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h)\stackrel{\cdot}{N }(t,x,h)\bigg] \end{array} $$in (d + 1) dimensions, where α ∈ (0, 2] and d < min{2, β−1}α, ν > 0, $\begin{array}{} \partial^\beta_t \end{array} $ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, $\begin{array}{} I^{1-\beta}_t \end{array} $ is the fractional integral operator, N(t, x) are Poisson random measure with Ñ(t, x) being the compensated Poisson random measure. σ : ℝ → ℝ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in [16, 33]. Under the linear growth of σ, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when σ grows faster than linear.
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Miller, Andrew, Jan Petrich, and Shashi Phoha. "Advanced Image Analysis for Learning Underlying Partial Differential Equations for Anomaly Identification." Journal of Imaging Science and Technology 64, no. 2 (2020): 20510–1. http://dx.doi.org/10.2352/j.imagingsci.technol.2020.64.2.020510.
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Abstract In this article, the authors adapt and utilize data-driven advanced image processing and machine learning techniques to identify the underlying dynamics and the model parameters for dynamic processes driven by partial differential equations (PDEs). Potential applications include non-destructive inspection for material crack detection using thermal imaging as well as real-time anomaly detection for process monitoring of three-dimensional printing applications. A neural network (NN) architecture is established that offers sufficient flexibility for spatial and temporal derivatives to capture the physical dependencies inherent in the process. Predictive capabilities are then established by propagating the process forward in time using the acquired model structure as well as individual parameter values. Moreover, deviations in the predicted values can be monitored in real time to detect potential process anomalies or perturbations. For concept development and validation, this article utilizes well-understood PDEs such as the homogeneous heat diffusion equation. Time series data governed by the heat equation representing a parabolic PDE is generated using high-fidelity simulations in order to construct the heat profile. Model structure and parameter identification are realized through a shallow residual convolutional NN. The learned model structure and associated parameters resemble a spatial convolution filter, which can be applied to the current heat profile to predict the diffusion behavior forward in time.
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Domoshnitsky, Alexander. "About Asymptotic and Oscillation Properties of the Dirichlet Problem for Delay Partial Differential Equations." gmj 10, no. 3 (2003): 495–502. http://dx.doi.org/10.1515/gmj.2003.495.
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Abstract In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet boundary value problem for the heat equation: the maximum principle is not valid, unbounded solutions appear while all solutions of the classical Dirichlet problem tend to zero at infinity, for “narrow enough zones” all solutions oscillate instead of being positive. We establish that the Dirichlet problem for the wave equation with delay can possess unbounded solutions. We estimate zones of positivity of solutions for hyperbolic equations.
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Watson, N. A. "Growth of solutions of weakly coupled parabolic systems and Laplace's equation." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 3 (1986): 391–403. http://dx.doi.org/10.1017/s1446788700033851.
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AbstractLet ui(x, t) be the ith component of a nonnegative solution of a weakly coupled system of second-order, linear, parabolic partial differential equations, for x ∈ Rn and 0 < t < T. We obtain lower estimates, near t = 0, for the Lebesgue measure of the set of x for which exceeds 1. Related results for Poisson integrals on a half-space are also described, some applications are given, and interesting comparisons emerge.
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Abdullayev, Akmaljon, Kholsaid Kholturayev, and Nigora Safarbayeva. "Exact method to solve of linear heat transfer problems." E3S Web of Conferences 264 (2021): 02059. http://dx.doi.org/10.1051/e3sconf/202126402059.
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When approximating multidimensional partial differential equations, the values of the grid functions from neighboring layers are taken from the previous time layer or approximation. As a result, along with the approximation discrepancy, an additional discrepancy of the numerical solution is formed. To reduce this discrepancy when solving a stationary elliptic equation, parabolization is carried out, and the resulting equation is solved by the method of successive approximations. This discrepancy is eliminated in the approximate analytical method proposed below for solving two-dimensional equations of parabolic and elliptic types, and an exact solution of the system of finite difference equations for a fixed time is obtained. To solve problems with a boundary condition of the first kind on the first coordinate and arbitrary combinations of the first, second and third kinds of boundary conditions on the second coordinate, it is proposed to use the method of straight lines on the first coordinate and ordinary sweep method on the second coordinate. Approximating the equations on the first coordinate, a matrix equation is built relative to the grid functions. Using eigenvalues and vectors of the three-diagonal transition matrix, linear combinations of grid functions are compiled, where the coefficients are the elements of the eigenvectors of the three-diagonal transition matrix. Boundary conditions, and for a parabolic equation, initial conditions are formed for a given combination of grid functions. The resulting one-dimensional differential-difference equations are solved by the ordinary sweep method. From the resulting solution, proceed to the initial grid functions. The method provides a second order of approximation accuracy on coordinates. And the approximation accuracy in time when solving the parabolic equation can be increased to the second order using the central difference in time. The method is used to solve heat transfer problems when the boundary conditions are expressed by smooth and discontinuous functions of a stationary and non-stationary nature, and the right-hand side of the equation represents a moving source or outflow of heat.
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Zhang, Xue Yan, Qian Li, Da Quan Gu, and Tai Ping Hou. "Methods of Moving Boundary Based on Artificial Boundary in Heat Conduction Direct Problem." Advanced Materials Research 490-495 (March 2012): 2282–85. http://dx.doi.org/10.4028/www.scientific.net/amr.490-495.2282.
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The initial boundary value problem for parabolic equation with Neumann boundary condition is a kind of classical problem in partial differential equations. In this paper we use the artificial boundary to solve the moving boundary problem. Potential theory and difference method are discussed. Numerical results are given to support the proposed schemes and to give the compare of the two methods.
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Johnpillai, Andrew G., Fazal M. Mahomed, and Saeid Abbasbandy. "Fundamental Solution via Invariant Approach for a Brain Tumor Model and its Extensions." Zeitschrift für Naturforschung A 69, no. 12 (2014): 725–32. http://dx.doi.org/10.5560/zna.2014-0064.
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AbstractWe firstly show how one can use the invariant criteria for a scalar linear (1+1) parabolic partial differential equations to perform reduction under equivalence transformations to the first Lie canonical form for a class of brain tumor models. Fundamental solution for the underlying class of models via these transformations is thereby found by making use of the well-known fundamental solution of the classical heat equation. The closed-form solution of the Cauchy initial value problem of the model equations is then obtained as well. We also demonstrate the utility of the invariant method for the extended form of the class of brain tumor models and find in a simple and elegant way the possible forms of the arbitrary functions appearing in the extended class of partial differential equations. We also derive the equivalence transformations which completely classify the underlying extended class of partial differential equations into the Lie canonical forms. Examples are provided as illustration of the results.
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Cannarsa, P., P. Martinez, and J. Vancostenoble. "The cost of controlling strongly degenerate parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 2. http://dx.doi.org/10.1051/cocv/2018007.
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We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).
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ESTEP, DONALD J., and ROY D. WILLIAMS. "ACCURATE PARALLEL INTEGRATION OF LARGE SPARSE SYSTEMS OF DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 06, no. 04 (1996): 535–68. http://dx.doi.org/10.1142/s0218202596000213.
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We describe a MIMD parallel code to solve a general class of ordinary differential equations, with particular emphasis on the large, sparse systems arising from space discretization of systems of parabolic partial differential equations. The main goals of this work are sharp bounds on the accuracy of the computed solution and flexibility of the software. We discuss the sources of error in solving differential equations, and the resulting constraints on time steps. We also discuss the theory of a posteriori error analysis for the Galerkin finite element methods, and its implementation in error control and estimation. The software is designed in a matrix-free fashion, which enables the solver to effectively tackle large sparse systems with minimal memory consumption and an easy and natural transition to MIMD (distributed memory) parallelism. In addition, there is no need for the choice of a particular representation of a sparse matrix. All memory is dynamically allocated, with a new expandable array object used for archiving results. The implicit solution of the discrete equations is carried out by replaceable modules: the nonlinear solver module may be a full Newton scheme or a quasi-Newton; these in turn are implemented with a linear solver, for which we have used both a direct solver and QMR, an iterative (Krylov space) method. Three computations are presented: the Lorenz system, which has dimension three and the discretized versions of the (partial-differential) bistable equation in one and two dimensions. The Lorenz system demonstrates the quality of the error estimation. The discretized bistable examples provide large sparse systems, and our precise error estimation shows, contrary to standard error estimates, that reliable computation is possible for large times.
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Uskov, Vladimir I. "Solution of a problem for a system of third order partial differential equations." Russian Universities Reports. Mathematics, no. 133 (2021): 68–76. http://dx.doi.org/10.20310/2686-9667-2021-26-133-68-76.
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An initial-boundary value problem for a system of third-order partial differential equations is considered. Equations and systems of equations with the highest mixed third derivative describe heat exchange in the soil complicated by the movement of soil moisture, quasi-stationary processes in a two-component semiconductor plasma, etc. The system is reduced to a differential equation with a degenerate operator at the highest derivative with respect to the distinguished variable in a Banach space. This operator has the property of having 0 as a normal eigenvalue, which makes it possible to split the original equations into an equation in subspaces. The conditions are obtained under which a unique solution to the problem exists; the analytical formula is found.
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ARTAMKIN, I. V. "COLORED GRAPHS, BURGERS EQUATION AND HESSIAN CONJECTURE." International Journal of Mathematics 18, no. 07 (2007): 797–808. http://dx.doi.org/10.1142/s0129167x0700431x.
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We prove that generating series for colored modular graphs satisfy some systems of partial differential equations generalizing Burgers or heat equations. The solution is obtained by genus expansion of the generating function. The initial term of this expansion is the corresponding generating function for trees. For this term the system of differential equations is equivalent to the inversion problem for the gradient mapping defined by the initial condition. This enables to state the Jacobian conjecture in the language of generating functions. The use of generating functions provides rather short and natural proofs of resent results of Zhao and of the well-known Bass–Connell–Wright tree inversion formula.
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Tuckerman, Laurette S. "Laplacian Preconditioning for the Inverse Arnoldi Method." Communications in Computational Physics 18, no. 5 (2015): 1336–51. http://dx.doi.org/10.4208/cicp.281114.290615a.
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AbstractMany physical processes are described by elliptic or parabolic partial differential equations. For linear stability problems associated with such equations, the inverse Laplacian provides a very effective preconditioner. In addition, it is also readily available in most scientific calculations in the form of a Poisson solver or an implicit diffusive timestep. We incorporate Laplacian preconditioning into the inverse Arnoldi method, using BiCGSTAB to solve the large linear systems. Two successful implementations are described: spherical Couette flow described by the Navier-Stokes equations and Bose-Einstein condensation described by the nonlinear Schrödinger equation.
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Dubljevic, Stevan. "Model predictive control of diffusion-reaction processes." Chemical Industry and Chemical Engineering Quarterly 11, no. 1 (2005): 10–18. http://dx.doi.org/10.2298/ciceq0501010d.
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Parabolic partial differential equations naturally arise as an adequate representation of a large class of spatially distributed systems, such as diffusion-reaction processes, where the interplay between diffusive and reaction forces introduces complexity in the characterization of the system, for the purpose of process parameter identification and subsequent control. In this work we introduce a model predictive control (MPC) framework for the control of input and state constrained parabolic partial differential equation (PDEs) systems. Model predictive control (MPC) is one of the most popular control formulations among chemical engineers, manly due to its ability to account for the actuator (input) constraints that inevitably exist due to finite actuator power and its ability to handle state constraints within an optimal control setting. In controller synthesis, the initially parabolic partial differential equation of the diffusion reaction type is transformed by the Galerkin method into a system of ordinary differential equations (ODEs) that capture the dominant dynamics of the PDE system. Systems obtained in such a way (ODEs) are used as the basis for the synthesis of the MPC controller that explicitly accounts for the input and state constraints. Namely, the modified MPC formulation includes a penalty term that is directly added to the objective function and through the appropriate structure of the controller state constraints accounts for the infinite dimensional nature of the state of the PDE system. The MPC controller design method is successively applied to control of the diffusion-reaction process described by linear parabolic PDE, by demonstrating stabilization of the non-dimensional temperature profile around a spatially uniform unstable steady-state under satisfaction of the input (actuator) constraints and allowable non-dimensional temperature (state) constraints.
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Campo, Antonio, Abraham J. Salazar, Diego J. Celentano, and Marcos Raydan. "Accurate analytical/numerical solution of the heat conduction equation." International Journal of Numerical Methods for Heat & Fluid Flow 24, no. 7 (2014): 1519–36. http://dx.doi.org/10.1108/hff-01-2013-0030.
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Purpose – The purpose of this paper is to address a novel method for solving parabolic partial differential equations (PDEs) in general, wherein the heat conduction equation constitutes an important particular case. The new method, appropriately named the Improved Transversal Method of Lines (ITMOL), is inspired in the Transversal Method of Lines (TMOL), with strong insight from the method of separation of variables. Design/methodology/approach – The essence of ITMOL revolves around an exponential variation of the dependent variable in the parabolic PDE for the evaluation of the time derivative. As will be demonstrated later, this key step is responsible for improving the accuracy of ITMOL over its predecessor TMOL. Throughout the paper, the theoretical properties of ITMOL, such as consistency, stability, convergence and accuracy are analyzed in depth. In addition, ITMOL has proven to be unconditionally stable in the Fourier sense. Findings – In a case study, the 1-D heat conduction equation for a large plate with symmetric Dirichlet boundary conditions is transformed into a nonlinear ordinary differential equation by means of ITMOL. The numerical solution of the resulting differential equation is straightforward and brings forth a nearly zero truncation error over the entire time domain, which is practically nonexistent. Originality/value – Accurate levels of the analytical/numerical solution of the 1-D heat conduction equation by ITMOL are easily established in the entire time domain.
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ZAIN, F. MUHAMMAD, M. GARDA KHADAFI, and P. H. GUNAWAN. "ANALISIS KONVERGENSI METODE BEDA HINGGA DALAM MENGHAMPIRI PERSAMAAN DIFUSI." E-Jurnal Matematika 7, no. 1 (2018): 1. http://dx.doi.org/10.24843/mtk.2018.v07.i01.p176.
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The diffusion equation or known as heat equation is a parabolic and linear type of partial differential equation. One of the numerical method to approximate the solution of diffusion equations is Finite Difference Method (FDM). In this study, the analysis of numerical convergence of FDM to the solution of diffusion equation is discussed. The analytical solution of diffusion equation is given by the separation of variables approach. Here, the result show the convergence of rate the numerical method is approximately approach 2. This result is in a good agreement with the spatial error from Taylor expansion of spatial second derivative.
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Jentzen, Arnulf, and Peter E. Kloeden. "Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2102 (2008): 649–67. http://dx.doi.org/10.1098/rspa.2008.0325.
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We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler scheme, and show that it converges (in the strong sense) faster than the classical numerical schemes, such as the linear-implicit Euler scheme or the Crank–Nicholson scheme, for this equation with the general noise. In particular, we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical schemes using only basic increments of the noise process reported previously. By contrast, our scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.
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Perev, Kamen. "SYSTEM CHARACTERISTICS OF DISTRIBUTED PARAMETER SYSTEMS." Proceedings of the Technical University of Sofia 70, no. 3 (2020): 34–44. http://dx.doi.org/10.47978/tus.2020.70.03.018.
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The paper considers the problem of distributed parameter systems modeling. The basic model types are presented, depending on the partial differential equation, which determines the physical processes dynamics. The similarities and the differences with the models described in terms of ordinary differential equations are discussed. A special attention is paid to the problem of heat flow in a rod. The problem set up is demonstrated and the methods of its solution are discussed. The main characteristics from a system point of view are presented, namely the Green function and the transfer function. Different special cases for these characteristics are discussed, depending on the specific partial differential equation, as well as the initial conditions and the boundary conditions.
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Yoon, Sungha, Darae Jeong, Chaeyoung Lee, et al. "Fourier-Spectral Method for the Phase-Field Equations." Mathematics 8, no. 8 (2020): 1385. http://dx.doi.org/10.3390/math8081385.
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In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A.
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Kazakov, Alexander. "Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type." Symmetry 13, no. 5 (2021): 871. http://dx.doi.org/10.3390/sym13050871.
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The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.
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Elshegmani, Zieneb Ali, Rokiah Rozita Ahmad, Saiful Hafiza Jaaman, and Roza Hazli Zakaria. "Transforming Arithmetic Asian Option PDE to the Parabolic Equation with Constant Coefficients." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–6. http://dx.doi.org/10.1155/2011/401547.
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Arithmetic Asian options are difficult to price and hedge, since at present, there is no closed-form analytical solution to price them. Transforming the PDE of the arithmetic the Asian option to a heat equation with constant coefficients is found to be difficult or impossible. Also, the numerical solution of the arithmetic Asian option PDE is not very accurate since the Asian option has low volatility level. In this paper, we analyze the value of the arithmetic Asian option with a new approach using means of partial differential equations (PDEs), and we transform the PDE to a parabolic equation with constant coefficients. It has been shown previously that the PDE of the arithmetic Asian option cannot be transformed to a heat equation with constant coefficients. We, however, approach the problem and obtain the analytical solution of the arithmetic Asian option PDE.
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Yermekkyzy, L. "VARIATIVE SOLUTION OF THE COEFFICIENT INVERSE PROBLEM FOR THE HEAT EQUATIONS." BULLETIN Series of Physics & Mathematical Sciences 72, no. 4 (2020): 23–27. http://dx.doi.org/10.51889/2020-4.1728-7901.03.
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One of the main types of inverse problems for partial differential equations are problems in which the coefficients of the equations or the quantities included in them must be determined using some additional information. Such problems are called coefficient inverse problems for partial differential equations. Coefficient inverse problems (identification problems) have become the subject of close study, especially in recent years. Interest in them is caused primarily by their important applied values. They find applications in solving problems of planning the development of oil fields (determining the filtration parameters of fields), in creating new types of measuring equipment, in solving problems of environmental monitoring, etc. The standard formulation of the coefficient inverse problem contains a functional (discrepancy), physics. When formulating the statements of inverse problems, the statements of direct problems are assumed to be known. The solution to the problem is sought from the condition of its minimum. Inverse problems for partial differential equations can be posed in variational form, i.e., as optimal control problems for the corresponding systems. A variational statement of one coefficient inverse problem for a one-dimensional heat equation is considered. By the solution of the boundary value problem for each fixed control coefficient we mean a generalized solution from the Sobolev space. The questions of correctness of the considered coefficient inverse problem in the variational setting are investigated.
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Secer, Aydin. "Numerical Solution and Simulation of Second-Order Parabolic PDEs with Sinc-Galerkin Method Using Maple." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/686483.
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An efficient solution algorithm for sinc-Galerkin method has been presented for obtaining numerical solution of PDEs with Dirichlet-type boundary conditions by using Maple Computer Algebra System. The method is based on Whittaker cardinal function and uses approximating basis functions and their appropriate derivatives. In this work, PDEs have been converted to algebraic equation systems with new accurate explicit approximations of inner products without the need to calculate any numeric integrals. The solution of this system of algebraic equations has been reduced to the solution of a matrix equation system via Maple. The accuracy of the solutions has been compared with the exact solutions of the test problem. Computational results indicate that the technique presented in this study is valid for linear partial differential equations with various types of boundary conditions.
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Cebers, Andrejs, and Harijs Kalis. "NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD." Mathematical Modelling and Analysis 18, no. 1 (2013): 80–96. http://dx.doi.org/10.3846/13926292.2013.756835.
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Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions is obtained using method of lines for solving a large system of ordinary differential equations (ODE).
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Walker, Anthony S., and Kyle E. Niemeyer. "Applying the Swept Rule for Solving Two-Dimensional Partial Differential Equations on Heterogeneous Architectures." Mathematical and Computational Applications 26, no. 3 (2021): 52. http://dx.doi.org/10.3390/mca26030052.
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The partial differential equations describing compressible fluid flows can be notoriously difficult to resolve on a pragmatic scale and often require the use of high-performance computing systems and/or accelerators. However, these systems face scaling issues such as latency, the fixed cost of communicating information between devices in the system. The swept rule is a technique designed to minimize these costs by obtaining a solution to unsteady equations at as many possible spatial locations and times prior to communicating. In this study, we implemented and tested the swept rule for solving two-dimensional problems on heterogeneous computing systems across two distinct systems and three key parameters: problem size, GPU block size, and work distribution. Our solver showed a speedup range of 0.22–2.69 for the heat diffusion equation and 0.52–1.46 for the compressible Euler equations. We can conclude from this study that the swept rule offers both potential for speedups and slowdowns and that care should be taken when designing such a solver to maximize benefits. These results can help make decisions to maximize these benefits and inform designs.
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Okino, T., T. Shimozaki, R. Fukuda, and Hiroki Cho. "Analytical Solutions of the Boltzmann Transformation Equation." Defect and Diffusion Forum 322 (March 2012): 11–31. http://dx.doi.org/10.4028/www.scientific.net/ddf.322.11.
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The so-called continuity equation derived by Fick is one of the most fundamental and extremely important equations in physics and/or in materials science. As is well known, this partial differential equation is also called the diffusion equation or the heat conduction equation and is applicable to physical phenomena of the conservation system. Incorporating the parabolic law relevant to a random movement into it, Boltzmann obtained the ordinary differential equation (B-equation). Matano then applied the B-equation to the analysis of the nonlinear problem for the interdiffusion experiment. The empirical Boltzmann-Matano (B-M) method has been successful in the metallurgical field. However, the nonlinear B-equation was not mathematically solved for a long time. Recently, the analytical solutions of the B-equation were obtained in accordance with the results of the B-M method. In the present study, an applicable limitation of the B-equation to the interdiffusion problems is investigated from a mathematical point of view.
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Bahri, Mawardi, Ryuichi Ashino, and Rémi Vaillancourt. "Continuous quaternion fourier and wavelet transforms." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 04 (2014): 1460003. http://dx.doi.org/10.1142/s0219691314600030.
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A two-dimensional (2D) quaternion Fourier transform (QFT) defined with the kernel [Formula: see text] is proposed. Some fundamental properties, such as convolution, Plancherel and vector differential theorems, are established. The heat equation in quaternion algebra is presented as an example of the application of the QFT to partial differential equations. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.
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Matus, Piter. "Stability of Difference Schemes for Nonlinear Time-dependent Problems." Computational Methods in Applied Mathematics 3, no. 2 (2003): 313–29. http://dx.doi.org/10.2478/cmam-2003-0020.
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AbstractIn the present paper, a priori estimates of the stability in the sense of the initial data of the difference schemes approximating quasilinear parabolic equations and nonlinear transfer equation have been obtained. The basic point is connected with the necessity of estimating all derivatives entering into the nonlinear part of the difference equations. These estimates have been proved without any assumptions about the properties of the differential equations and depend only on the behavior of the initial and boundary conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for the finite instant of time t 6 t0 connected with the fact that the solution of the Riccati equation becomes infinite. is already associated with the behavior of the second derivative of the initial function and coincides with the time of the exact solution destruction (heat localization in the peaking regime). A close relation between the stability and convergence of the difference scheme solution is given. Thus, not only a priori estimates for stability have been established, but it is also shown that the obtained conditions permit exact determination of the time of destruction of the solution of the initial boundary value problem for the original nonlinear differential equation in partial derivatives. In the present paper, concrete examples confirming the theoretical conclusions are given.
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Aziz, Taha. "On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique." Mathematical Problems in Engineering 2021 (June 14, 2021): 1–10. http://dx.doi.org/10.1155/2021/9974073.
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The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.
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Nakshatrala, K. B., H. Nagarajan, and M. Shabouei. "A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations." Communications in Computational Physics 19, no. 1 (2016): 53–93. http://dx.doi.org/10.4208/cicp.180615.280815a.
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AbstractTransient diffusion equations arise in many branches of engineering and applied sciences (e.g., heat transfer and mass transfer), and are parabolic partial differential equations. It is well-known that these equations satisfy important mathematical properties like maximum principles and the non-negative constraint, which have implications in mathematical modeling. However, existing numerical formulations for these types of equations do not, in general, satisfy maximum principles and the non-negative constraint. In this paper, we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion equation. The proposed methodology is based on the method of horizontal lines in which the time is discretized first. This results in solving steady anisotropic diffusion equation with decay equation at every discrete time-level. We also present other plausible temporal discretizations, and illustrate their shortcomings in meeting maximum principles and the non-negative constraint. The proposed methodology can handle general computational grids with no additional restrictions on the time-step. We illustrate the performance and accuracy of the proposed methodology using representative numerical examples. We also perform a numerical convergence analysis of the proposed methodology. For comparison, we also present the results from the standard single-field semi-discrete formulation and the results from a popular software package, which all will violate maximum principles and the non-negative constraint.
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Vlasenko, Larisa A., Anatoly G. Rutkas, and Arkady A. Chikrii. "STOCHASTIC DIFFERENTIAL GAMES IN DISTRIBUTED SYSTEMS WITH DELAY." Journal of Automation and Information sciences 1 (January 1, 2021): 41–54. http://dx.doi.org/10.34229/0572-2691-2021-1-4.
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We study a differential game of approach in a delay stochastic system. The evolution of the system is described by Ito`s linear stochastic differential equation in Hilbert space. The considered Hilbert spaces are assumed to be real and separable. The Wiener process takes values in a Hilbert space and has a nuclear symmetric positive covariance operator. The pursuer and evader controls are non-anticipating random processes, taking on values, generally, in different Hilbert spaces. The operator multiplying the system state is the generator of an analytic semigroup. Solutions of the equation are represented with the help of a formula of variation of constants by the initial data and the control block. The delay effect is taken into account by summing shift type operators. To study the differential game, the method of resolving functions is extended to case of delay stochastic systems in Hilbert spaces. The technique of set-valued mappings and their selectors is used. We consider the application of obtained results in abstract Hilbert spaces to systems described by stochastic partial differential equations with time delay. By taking into account a random external influence and time delay, we study the heat propagation process with controlled distributed heat source and leak.
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Oyanader, Mario A., and Pedro E. Arce. "Role of Aspect Ratio and Joule Heating within the Fluid Region Near a Cylindrical Electrode in Electrokinetic Remediation: A Numerical Solution based on the Boundary Layer Model." International Journal of Chemical Reactor Engineering 11, no. 2 (2013): 687–99. http://dx.doi.org/10.1515/ijcre-2012-0007.
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Abstract This contribution focuses on the analysis of the hydrodynamics taking place near an electrode of cylindrical geometry in electrokinetic applications. Both the temperature development conditions and Joule heating effect are included. A boundary layer approach has been used to model the hydrodynamic in the system. This is based on the heat transfer model, the continuity equation, and the Navier–Stokes equation. The resulting set of two partial differential equations mutually coupled is solved applying the Von Karman integral approximation. A numerical solution of the differential–integral model is used to illustrate the behavior of the systems under a variety of conditions.
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Yang, Cheng-Dong, Jianlong Qiu, and Jun-Wei Wang. "RobustH∞Control for a Class of Nonlinear Distributed Parameter Systems via Proportional-Spatial Derivative Control Approach." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/631071.
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This paper addresses the problem of robustH∞control design via the proportional-spatial derivative (P-sD) control approach for a class of nonlinear distributed parameter systems modeled by semilinear parabolic partial differential equations (PDEs). By using the Lyapunov direct method and the technique of integration by parts, a simple linear matrix inequality (LMI) based design method of the robustH∞P-sD controller is developed such that the closed-loop PDE system is exponentially stable with a given decay rate and a prescribedH∞performance of disturbance attenuation. Moreover, a suboptimalH∞controller is proposed to minimize the attenuation level for a given decay rate. The proposed method is successfully employed to address the control problem of the FitzHugh-Nagumo (FHN) equation, and the achieved simulation results show its effectiveness.
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Kaber, Sidi-Mahmoud, Amine Loumi, and Philippe Parnaudeau. "Parallel Solution of Linear Systems." East Asian Journal on Applied Mathematics 6, no. 3 (2016): 278–89. http://dx.doi.org/10.4208/eajam.210715.250316a.
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AbstractComputational scientists generally seek more accurate results in shorter times, and to achieve this a knowledge of evolving programming paradigms and hardware is important. In particular, optimising solvers for linear systems is a major challenge in scientific computation, and numerical algorithms must be modified or new ones created to fully use the parallel architecture of new computers. Parallel space discretisation solvers for Partial Differential Equations (PDE) such as Domain Decomposition Methods (DDM) are efficient and well documented. At first glance, parallelisation seems to be inconsistent with inherently sequential time evolution, but parallelisation is not limited to space directions. In this article, we present a new and simple method for time parallelisation, based on partial fraction decomposition of the inverse of some special matrices. We discuss its application to the heat equation and some limitations, in associated numerical experiments.
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Sun, Shuai, Mario Miscuglio, Xiaoxuan Ma, et al. "Induced homomorphism: Kirchhoff’s law in photonics." Nanophotonics 10, no. 6 (2021): 1711–21. http://dx.doi.org/10.1515/nanoph-2020-0655.
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Abstract When solving, modeling or reasoning about complex problems, it is usually convenient to use the knowledge of a parallel physical system for representing it. This is the case of lumped-circuit abstraction, which can be used for representing mechanical and acoustic systems, thermal and heat-diffusion problems and in general partial differential equations. Integrated photonic platforms hold the prospective to perform signal processing and analog computing inherently, by mapping into hardware specific operations which relies on the wave-nature of their signals, without trusting on logic gates and digital states like electronics. Here, we argue that in absence of a straightforward parallelism a homomorphism can be induced. We introduce a photonic platform capable of mimicking Kirchhoff’s law in photonics and used as node of a finite difference mesh for solving partial differential equation using monochromatic light in the telecommunication wavelength. Our approach experimentally demonstrates an arbitrary set of boundary conditions, generating a one-shot discrete solution of a Laplace partial differential equation, with an accuracy above 95% with respect to commercial solvers. Our photonic engine can provide a route to achieve chip-scale, fast (10 s of ps), and integrable reprogrammable accelerators for the next generation hybrid high-performance computing. Summary A photonic integrated platform which can mimic Kirchhoff’s law in photonics is used for approximately solve partial differential equations noniteratively using light, with high throughput and low-energy levels.
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Azimi, Mohammadreza, and Rouzbeh Riazi. "Flow and heat transfer of MHD graphene oxide-water nanofluid between two non-parallel walls." Thermal Science 21, no. 5 (2017): 2095–104. http://dx.doi.org/10.2298/tsci150513100a.
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The steady 2-D heat transfer and flow between two non-parallel walls of a graphene oxide nanofluid in presence of uniform magnetic field are investigated in this paper. The analytical solution of the non-linear problem is obtained by Galerkin optimal homotopy asymptotic method. At first a similarity transformation is used to reduce the partial differential equations modeling the flow and heat transfer to ordinary non-linear differential equation systems containing the semi angle between the plate?s parameter, Reynolds number, the magnetic field strength, nanoparticle volume fraction, Eckert and Prandtl numbers. Finally, the obtained analytical results have been compared with results achieved from previous works in some cases.
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Maes, Frederick, and Marián Slodička. "Some Inverse Source Problems of Determining a Space Dependent Source in Fractional-Dual-Phase-Lag Type Equations." Mathematics 8, no. 8 (2020): 1291. http://dx.doi.org/10.3390/math8081291.
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The dual-phase-lag heat transfer models attract a lot of interest of researchers in the last few decades. These are used in problems arising from non-classical thermal models, which are based on a non-Fourier type law. We study uniqueness of solutions to some inverse source problems for fractional partial differential equations of the Dual-Phase-Lag type. The source term is supposed to be of the form h(t)f(x) with a known function h(t). The unknown space dependent source f(x) is determined from the final time observation. New uniqueness results are formulated in Theorem 1 (for a general fractional Jeffrey-type model). Here, the variational approach was used. Theorem 2 derives uniqueness results under weaker assumptions on h(t) (monotonically increasing character of h(t) was removed) in a case of dominant parabolic behavior. The proof technique was based on spectral analysis. Section Modified Model for τq>τT shows that an analogy of Theorem 2 for dominant hyperbolic behavior (fractional Cattaneo–Vernotte equation) is not possible.