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Journal articles on the topic 'Heat equation Numerical solutions'

Author: Grafiati

Published: 4 June 2021

Last updated: 13 February 2022

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1

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 (December 31, 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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2

Tadeu, A., C. S. Chen, J. António, and Nuno Simões. "A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform." Advances in Applied Mathematics and Mechanics 3, no. 5 (October 2011): 572–85. http://dx.doi.org/10.4208/aamm.10-m1039.

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AbstractFourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

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3

Korpinar, Zeliha. "On numerical solutions for the Caputo-Fabrizio fractional heat-like equation." Thermal Science 22, Suppl. 1 (2018): 87–95. http://dx.doi.org/10.2298/tsci170614274k.

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In this article, Laplace homotopy analysis method in order to solve fractional heat-like equation with variable coefficients, are introduced. Laplace homotopy analysis method, founded on combination of homotopy methods and Laplace transform is used to supply a new analytical approximated solutions of the fractional partial differential equations in case of the Caputo-Fabrizio. The solutions obtained are compared with exact solutions of these equations. Reliability of the method is given with graphical consequens and series solutions. The results show that the method is a powerfull and efficient for solving the fractional heat-like equations with variable coefficients.

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4

Edja, Kouame Beranger, Kidjegbo Augustin Toure, and Brou Jean-Claude Koua. "Numerical Blow-up for A Heat Equation with Nonlinear Boundary Conditions." Journal of Mathematics Research 10, no. 5 (September 6, 2018): 119. http://dx.doi.org/10.5539/jmr.v10n5p119.

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We study numerical approximations of solutions of a heat equation with nonlinear boundary conditions which produce blow-up of the solutions. By a semidiscretization using a finite difference scheme in the space variable we get a system of ordinary differential equations which is an approximation of the original problem. We obtain sufficient conditions which guarantee the blow-up solution of this system in a finite time. We also show that this blow-up time converges to the theoretical one when the mesh size goes to zero. We present some numerical results to illustrate certain point of our work.

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5

Kochneff, Elizabeth, Yoram Sagher, and Kecheng Zhou. "Homogeneous solutions of the heat equation." Journal of Approximation Theory 69, no. 1 (April 1992): 35–47. http://dx.doi.org/10.1016/0021-9045(92)90047-r.

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6

Zhang, K. "On coupling between the Poincaré equation and the heat equation: non-slip boundary condition." Journal of Fluid Mechanics 284 (February 10, 1995): 239–56. http://dx.doi.org/10.1017/s0022112095000346.

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In contrast to the well-known columnar convection mode in rapidly rotating spherical fluid systems, the viscous dissipation of the preferred convection mode at sufficiently small Prandtl numberPrtakes place only in the Ekman boundary layer. It follows that different types of velocity boundary condition lead to totally different forms of the asymptotic relationship between the Rayleigh numberRand the Ekman numberEfor the onset of convection. We extend both perturbation and numerical analyses with the stress-free boundary condition (Zhang 1994) in rapidly rotating spherical systems to those with the non-slip boundary condition. Complete analytical solutions – the critical parameters for the onset of convection and the corresponding flow and temperature structure – are obtained and a new asymptotic relation betweenRandEis derived. While an explicit solution of the Ekman boundary-layer problem can be avoided by constructing a proper surface integral in the case of the stress-free boundary problem, an explicit solution of the spherical Ekman boundary layer is required and then obtained to derive the solvability condition for the present problem. In the corresponding numerical analysis, velocity and temperature are expanded in terms of spherical harmonics and Chebychev functions. Accurate numerical solutions are obtained in the asymptotic regime of smallEandPr, and comparison between the analytical and numerical solutions is then made to demonstrate that a satisfactory quantitative agreement between the analytical and numerical analyses is reached.

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7

Agyeman, Edmund, and Derick Folson. "Algorithm Analysis of Numerical Solutions to the Heat Equation." International Journal of Computer Applications 79, no. 5 (October 18, 2013): 11–19. http://dx.doi.org/10.5120/13736-1535.

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8

Čiegis, Raimondas. "NUMERICAL SOLUTION OF HYPERBOLIC HEAT CONDUCTION EQUATION." Mathematical Modelling and Analysis 14, no. 1 (March 31, 2009): 11–24. http://dx.doi.org/10.3846/1392-6292.2009.14.11-24.

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Hyperbolic heat conduction problem is solved numerically. The explicit and implicit Euler schemes are constructed and investigated. It is shown that the implicit Euler scheme can be used to solve efficiently parabolic and hyperbolic heat conduction problems. This scheme is unconditionally stable for both problems. For many integration methods strong numerical oscillations are present, when the initial and boundary conditions are discontinuous for the hyperbolic problem. In order to regularize the implicit Euler scheme, a simple linear relation between time and space steps is proposed, which automatically introduces sufficient amount of numerical viscosity. Results of numerical experiments are presented.

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9

Mhammad, Aree A., Faraidun K. Hama Salh, and Najmadin W. Abdulrahman. "Numerical Solution for Non-Stationary Heat Equation in Cooling of Computer Radiator System." Journal of Zankoy Sulaimani - Part A 12, no. 1 (November 5, 2008): 97–102. http://dx.doi.org/10.17656/jzs.10199.

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10

Kandel, H. P., J. Kafle, and L. P. Bagale. "Numerical Modelling on the Influence of Source in the Heat Transformation: An Application in the Metal Heating for Blacksmithing." Journal of Nepal Physical Society 7, no. 2 (August 6, 2021): 97–101. http://dx.doi.org/10.3126/jnphyssoc.v7i2.38629.

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Many physical problems, such as heat transfer and wave transfer, are modeled in the real world using partial differential equations (PDEs). When the domain of such modeled problems is irregular in shape, computing analytic solution becomes difficult, if not impossible. In such a case, numerical methods can be used to compute the solution of such PDEs. The Finite difference method (FDM) is one of the numerical methods used to compute the solutions of PDEs by discretizing the domain into a finite number of regions. We used FDMs to compute the numerical solutions of the one dimensional heat equation with different position initial conditions and multiple initial conditions. Blacksmiths fashioned different metals into the desired shape by heating the objects with different temperatures and at different position. The numerical technique applied here can be used to solve heat equations observed in the field of science and engineering.

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11

Davoudi, Mohammad Mahdi, and Andreas Öchsner. "Error Estimates for the Finite Difference Solution of the Heat Conduction Equation: Consideration of Boundary Conditions and Heat Sources." Defect and Diffusion Forum 336 (March 2013): 195–207. http://dx.doi.org/10.4028/www.scientific.net/ddf.336.195.

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This contribution investigates the numerical solution of the steady-state heat conduction equation. The finite difference method is applied to simple formulations of heat sources where still analytical solutions can be derived. Thus, the results of the numerical approach can be related to the exact solutions and conclusions on the accuracy obtained. In addition, the numerical implementation of different forms of boundary conditions, i.e. temperature and flux condition, is compared to the exact solution. It is found that the numerical implementation of coordinate dependent sources gives the exact result while temperature dependent sources are only approximately represented. Furthermore, the implementation of the mentioned boundary conditions gives the same results as the analytical reference solution.

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12

Rosca, Alin V., Natalia C. Rosca, and Ioan Pop. "Numerical simulation of the stagnation point flow past a permeable stretching/shrinking sheet with convective boundary condition and heat generation." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 1 (January 4, 2016): 348–64. http://dx.doi.org/10.1108/hff-12-2014-0361.

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Purpose – The purpose of this paper is the stagnation-point flow driven by a permeable stretching/shrinking surface with convective boundary condition and heat generation. Design/methodology/approach – It is known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, for the present case it is demonstrated that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is constant. Findings – The governing boundary layer equations are transformed to self-similar nonlinear ordinary differential equations using similarity transformations. Numerical results of the resulting equations are obtained using the function bvp4c from Matlab for different values of the governing parameters. In addition an analytical solution has been obtained for the energy equation when heat generation is absent. The streamlines for the upper branch solution show that the pattern is almost similar to the normal stagnation-point flow, but because of the existence of suction and shrinking effect, the flow seems like suck to the permeable wall. Originality/value – Dual solutions are found for negative values of the moving parameter. A stability analysis has been also performed to show that the first upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible. The streamlines for the lower branch solution are also graphically shown.

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13

Gao, Feng, and Xiao-Jun Yang. "Local fractional Euler’s method for the steady heat-conduction problem." Thermal Science 20, suppl. 3 (2016): 735–38. http://dx.doi.org/10.2298/tsci16s3735g.

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In this paper, the local fractional Euler?s method is proposed to consider the steady heat-conduction problem for the first time. The numerical solution for the local fractional heat-relaxation equation is presented. The comparison between numerical and exact solutions is discussed.

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14

Jeong, Darae, Yibao Li, Chaeyoung Lee, Junxiang Yang, Yongho Choi, and Junseok Kim. "Verification of Convergence Rates of Numerical Solutions for Parabolic Equations." Mathematical Problems in Engineering 2019 (June 23, 2019): 1–10. http://dx.doi.org/10.1155/2019/8152136.

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In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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15

Bibi, Khudija, and Tooba Feroze. "Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation." Symmetry 12, no. 3 (March 2, 2020): 359. http://dx.doi.org/10.3390/sym12030359.

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In this article, an invariantized finite difference scheme to find the solution of the heat equation, is developed. The scheme is based on a discrete symmetry transformation. A comparison of the results obtained by the proposed scheme and the Crank Nicolson method is carried out with reference to the exact solutions. It is found that the proposed invariantized scheme for the heat equation improves the efficiency and accuracy of the existing Crank Nicolson method.

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16

Cai, Ruixian, and Na Zhang. "Some Algebraically Explicit Analytical Solutions of Unsteady Nonlinear Heat Conduction." Journal of Heat Transfer 123, no. 6 (March 5, 2001): 1189–91. http://dx.doi.org/10.1115/1.1392990.

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The analytical solutions of nonlinear unsteady heat conduction equation are meaningful in theory. In addition, they are very useful to the computational heat conduction to check the numerical solutions and to develop numerical schemes, grid generation methods and so forth. However, very few explicit analytical solutions have been known for the unsteady nonlinear heat conduction. In order to develop the heat conduction theory, some algebraically explicit analytical solutions of nonlinear heat conduction equation have been derived in this paper, which include one-dimensional and two-dimensional unsteady heat conduction solutions with thermal conductivity, density and specific heat being functions of temperature.

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17

Byun, D. W., and S. Saitoh. "Approximation by the Solutions of the Heat Equation." Journal of Approximation Theory 78, no. 2 (August 1994): 226–38. http://dx.doi.org/10.1006/jath.1994.1074.

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18

Gorskiy, V. V., and A. G. Loktionova. "Simulating Heat Exchange and Friction in a Thin Laminar Boundary Layer of Air over the Lateral Surface of a Blunted Cone Featuring a Low Aspect Ratio." Herald of the Bauman Moscow State Technical University. Series Mechanical Engineering, no. 6 (135) (December 2020): 4–20. http://dx.doi.org/10.18698/0236-3941-2020-6-4-20.

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It is not possible to obtain a high-quality solution to a convective heat transfer problem without numerically integrating the differential equations describing the boundary layer, which involves a whole range of computational issues. Developing relatively simple yet adequately accurate computation methods becomes crucial. Using the effective length method may be considered to be the first step towards solving this problem. This method boasts an accuracy of convective heat transfer calculation that is acceptable in practice, due to which it became prevalent in aircraft design. However, this method is also relatively labour-intensive, although significantly less so than numerical integration of the boundary layer differential equations. The most efficient approach to solving heat transfer and friction problems in engineering practice would be using simple algebraic equations based on fitting the results of rigorous numerical computations or experimental investigations. Regrettably, there is no information published regarding how accurate these equations are for various operation conditions. The paper presents a solution to this problem based on deriving systematic numerical solutions to the boundary layer equations in the most rigorous analytical statement, along with conducting a thorough analysis of the equation accuracy for both the equations derived and previously published

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19

Hindmarsh, Richard C. A., Gwendolyn J. M. C. Leysinger Vieli, and Frédéric Parrenin. "A large-scale numerical model for computing isochrone geometry." Annals of Glaciology 50, no. 51 (2009): 130–40. http://dx.doi.org/10.3189/172756409789097450.

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AbstractA finite-difference model for the calculation of radar layer geometries in large ice masses is presented. Balance velocities are used as coefficients in the age equation and in the heat equation. Solution of the heat equation allows prediction of sliding areas and computation of basal melt rates. Vertical distributions of velocity are parameterized using shape functions. These can be set uniformly, or allowed to vary in space according to the distribution of sliding. The vertical coordinate can either be uniformly distributed within the thickness of the ice, or be uniformly distributed within the flux. The finite-difference scheme results in a large set of linear equations. These are solved using a nested factorization preconditioned conjugate gradient scheme. The convergence properties of some other iteration solution schemes are studied. The output is computations of age and temperature assuming steady state, in large ice masses at high resolution. Age calculations are used to generate isochrones which show the best fit to observed layers. Comparisons with analytical solutions are made, and the influence of the order of the finite-difference approximation and the choice of vertical coordinate on solution accuracy is considered.

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20

Hall, E. J., and R. H. Pletcher. "Application of a Viscous-Inviscid Interaction Procedure to Predict Separated Flows With Heat Transfer." Journal of Heat Transfer 107, no. 3 (August 1, 1985): 557–63. http://dx.doi.org/10.1115/1.3247460.

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A viscous-inviscid interaction procedure is described for predicting heat transfer in separated flows. The separating flow in a rearward-facing step/asymmetric channel expansion is considered. For viscous regions, the boundary layer momentum and continuity equations are solved inversely in a coupled manner by a finite-difference numerical scheme. The streamwise convective term is altered to permit marching the solution through regions of reversed flow. The inviscid flow is computed by numerically solving the Laplace equation for streamfunction in the region bounded by the displacement surfaces used in the inverse boundary layer solution. The viscous and inviscid solutions are repeated iteratively until the edge velocities obtained from both solutions are in agreement. Predictions using this method compare favorably with experimental data and other predictions.

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21

Romão, E. C., M. D. De Campos, J. A. Martins, and L. F. M. De Moura. "APPLICATION OF GALERKIN FINITE ELEMENT METHOD IN THE SOLUTION OF 3D DIFFUSION IN SOLIDS." Revista de Engenharia Térmica 8, no. 2 (December 31, 2009): 79. http://dx.doi.org/10.5380/reterm.v8i2.61919.

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This paper presents the numerical solution by the Galerkin Finite Element Method, on the three-dimensional Laplace and Helmholtz equations, which represent the heat diffusion in solids. For the two applications proposed, the analytical solutions found in the literature review were used in comparison with the numerical solution. The results analysis was made based on the the L2 Norm (average error throughout the domain) and L¥ Norm (maximum error in the entire domain). The two application results, one of the Laplace equation and the Helmholtz equation, are presented and discussed in order to to test the efficiency of the method.

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22

Zhang, Keke. "On coupling between the Poincaré equation and the heat equation." Journal of Fluid Mechanics 268 (June 10, 1994): 211–29. http://dx.doi.org/10.1017/s0022112094001321.

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It has been suggested that in a rapidly rotating fluid sphere, convection would be in the form of slowly drifting columnar rolls with small azimuthal scale (Roberts 1968; Busse 1970). The results in this paper show that there are two alternative convection modes which are preferred at small Prandtl numbers. The two new convection modes are, at leading order, essentially those inertial oscillation modes of the Poincaré equation with the simplest structure along the axis of rotation and equatorial symmetry: one propagates in the eastward direction and the other propagates in the westward direction; both are trapped in the equatorial region. Buoyancy forces appear at next order to drive the oscillation against the weak effects of viscous damping. On the basis of the perturbation of solutions of the Poincaré equation, and taking into account the effects of the Ekman boundary layer, complete analytical convection solutions are obtained for the first time in rotating spherical fluid systems. The condition of an inner sphere exerts an insignificant influence on equatorially trapped convection. Full numerical analysis of the problem demonstrates a quantitative agreement between the analytical and numerical analyses.

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23

Kazakov, A. L., L. F. Spevak, and E. L. Spevak. "On numerical methods for constructing benchmark solutions to a nonlinear heat equation with a singularity." Diagnostics, Resource and Mechanics of materials and structures, no. 5 (October 2020): 26–44. http://dx.doi.org/10.17804/2410-9908.2020.5.026-044.

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The paper deals with the construction of exact solutions to a nonlinear heat equation with degeneration in the case of the zero value of the required function. Generically self-similar solutions and traveling wave solutions are considered, the construction of which reduces to solving Cauchy problems for a nonlinear second-order ordinary differential equation with a singularity before the higher derivative. Two approaches are proposed to solve the Cauchy problems: the analytical solution by the power series method and the numerical solution by the boundary element method on a specified segment. A complex computational experiment is carried out to compare the above two methods with each other and with the finite difference methods, namely the Euler method and the fourth-order Runge-Kutta method. Power series segments are used on the first step of the finite difference solutions in order to resolve the singularity. The comparison of the application domains, the accuracy of the solutions and their dependence on the parameters of a certain problem shows that the boundary element method is the most universal, although not the most accurate for some particular examples. The conclusions drawn allow us to construct benchmark solutions to verify the approximate solutions of the nonlinear heat equation by various methods in a wide range of parameter values.

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Horváth, Róbert. "On the monotonicity conservation in numerical solutions of the heat equation." Applied Numerical Mathematics 42, no. 1-3 (August 2002): 189–99. http://dx.doi.org/10.1016/s0168-9274(01)00150-7.

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25

De Chant, Lawrence J. "An implicit differential equation governing lumped capacitance, radiation dominated, unsteady, heat transfer." International Journal of Numerical Methods for Heat & Fluid Flow 22, no. 7 (September 14, 2012): 896–906. http://dx.doi.org/10.1108/09615531211255770.

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PurposeAlthough most physical problems in fluid mechanics and heat transfer are governed by nonlinear differential equations, it is less common to be confronted with a “so – called” implicit differential equation, i.e. a differential equation where the highest order derivative cannot be isolated. The purpose of this paper is to derive and analyze an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach.Design/methodology/approachHere we discuss an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach. Due to the implicit nature of this problem, standard integration schemes, e.g. Runge‐Kutta, are not conveniently applied to this problem. Moreover, numerical solutions do not provide the insight afforded by an analytical solution.FindingsA predictor predictor‐corrector scheme with secant iteration is presented which readily integrates differential equations where the derivative cannot be explicitly obtained. These solutions are compared to numerical integration of the equations and show good agreement.Originality/valueThe paper emphasizes that although large‐scale, multi‐dimensional time‐dependent heat transfer simulation tools are routinely available, there are instances where unsteady, engineering models such as the one discussed here are both adequate and appropriate.

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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 (August 26, 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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Romeiro, Neyva Maria Lopes Romeiro, Eduardo Oliveira Belinelli, Jesika Magagnin, Paulo Laerte Natti, and Eliandro Rodrigues Cirilo. "Numerical study of different methods applied to the one-dimensional transient heat equation." REMAT: Revista Eletrônica da Matemática 7, no. 1 (April 20, 2021): e3012. http://dx.doi.org/10.35819/remat2021v7i1id4767.

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This article aims to compare the results obtained by applying three numerical methods: Explicit Euler, Crank-Nicolson,and Multi-stage (R11), in the one-dimensional heat diffusion transient equation with different initial and boundary conditions. The discretization process was performed using the finite difference method. In order to guarantee the convergence of the methods used, consistency and stability were verified by Lax theorem. The results are presented in graphs and tables that contain the data of the analytical solution and the numerical solutions. It was observed that the results obtained by R11 method generated solutions with minor errors.

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Saeed, Umer, and Mujeeb ur Rehman. "Assessment of Haar Wavelet-Quasilinearization Technique in Heat Convection-Radiation Equations." Applied Computational Intelligence and Soft Computing 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/454231.

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We showed that solutions by the Haar wavelet-quasilinearization technique for the two problems, namely, (i) temperature distribution equation in lumped system of combined convection-radiation in a slab made of materials with variable thermal conductivity and (ii) cooling of a lumped system by combined convection and radiation are strongly reliable and also more accurate than the other numerical methods and are in good agreement with exact solution. According to the Haar wavelet-quasilinearization technique, we convert the nonlinear heat transfer equation to linear discretized equation with the help of quasilinearization technique and apply the Haar wavelet method at each iteration of quasilinearization technique to get the solution. The main aim of present work is to show the reliability of the Haar wavelet-quasilinearization technique for heat transfer equations.

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Bandrowski, Bartosz, Anna Karczewska, and Piotr Rozmej. "Numerical solutions to integral equations equivalent to differential equations with fractional time." International Journal of Applied Mathematics and Computer Science 20, no. 2 (June 1, 2010): 261–69. http://dx.doi.org/10.2478/v10006-010-0019-1.

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Numerical solutions to integral equations equivalent to differential equations with fractional timeThis paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.

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Ribeiro, S. S., G. Guimarães, A. Fernandes, and G. C. Oliveira. "HEAT CONDUCTION IN MOVING SOLIDS USING GREEN’S FUNCTION." Revista de Engenharia Térmica 14, no. 1 (June 30, 2015): 65. http://dx.doi.org/10.5380/reterm.v14i1.62115.

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Thermal problems involving moving heat sources occur in various engineering applications, such as welding processes, heat treatment furnaces and other treatments. Typically, in these cases, the precise formulation of the numerical solution due to the high complexity of the heat diffusion governing equation, boundary condition, including convection terms. This work proposes a mathematical analysis, analytical solution, verification using other solutions and comparison with numerical solution of a 1D transient thermal model based on Green’n functions, considering a solid moving at a constant speed along a cartesian coordinated.

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Flyer, N., and A. S. Fokas. "A hybrid analytical–numerical method for solving evolution partial differential equations. I. The half-line." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2095 (April 2008): 1823–49. http://dx.doi.org/10.1098/rspa.2008.0041.

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A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.

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Avci, Derya, Eroglu Iskender, and Necati Ozdemir. "Conformable heat equation on a radial symmetric plate." Thermal Science 21, no. 2 (2017): 819–26. http://dx.doi.org/10.2298/tsci160427302a.

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The conformable heat equation is defined in terms of a local and limit-based definition called conformable derivative which provides some basic properties of integer order derivative such that conventional fractional derivatives lose some of them due to their non-local structures. In this paper, we aim to find the fundamental solution of a conformable heat equation acting on a radial symmetric plate. Moreover, we give a comparison between the new conformable and the existing Grunwald-Letnikov solutions of heat equation. The computational results show that conformable formulation is quite successful to show the sub-behaviors of heat process. In addition, conformable solution can be obtained by a analytical method without the need of a numerical scheme and any restrictions on the problem formulation. This is surely a significant advantageous compared to the Grunwald-Letnikov solution.

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Zhang, Yufeng. "Similarity solutions and the computation formulas of a nonlinear fractional-order generalized heat equation." Modern Physics Letters B 33, no. 10 (April 10, 2019): 1950122. http://dx.doi.org/10.1142/s0217984919501227.

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A generalized nonlinear heat equation with the fractional derivative is proposed, whose similarity solutions are derived from a type of special scalar transformation with two parameters. With the help of separated variable method, two special series solutions of the standard heat equation are obtained. Finally, through computation of the left Riemann–Liouville fractional derivative, we obtain two approximated computation formulas of the factional-order ordinary differential equation which could be used to calculate the numerical solutions of the generalized nonlinear heat conduction equation.

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Mikeš, Karel, and Milan Jirásek. "Free Warping Analysis and Numerical Implementation." Applied Mechanics and Materials 825 (February 2016): 141–48. http://dx.doi.org/10.4028/www.scientific.net/amm.825.141.

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This article deals with the mathematical description and numerical implementation of the free warping problem. The solution of the warping problem is given by a warping function obtained by solving the Laplace equation with a corresponding boundary condition. An analytical solution is available only for a limited number of specific cross-sectional shapes such as ellipse or rectangle. For the solution of a general cross section, the Laplace equation must be solved numerically by the finite element method. From a mathematical point of view, the free warping problem can be described in the same way as the heat transfer phenomena, but in the numerical implementation, there are several features specific to warping analysis.The solution algorithm has been implemented in the OOFEM open-source finite element code [1] and verification has been done on several examples with known analytical solutions.

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Лобанов, Игорь, and Igor' Lobanov. "THEORETICAL ANALYTICAL SOLUTION OF PROBLEM ON STATIONARY SUBCRITICAL CURRENT OF GASEOUS HEAT CARRIER IN PIPING BIFURCATIONS OF HEAT-EXCHANGE EQUIPMENT." Bulletin of Bryansk state technical university 2019, no. 9 (October 7, 2019): 25–35. http://dx.doi.org/10.30987/article_5d9317b27868a4.78923465.

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The aim of the paper consists in obtaining analytical solutions of the problem on current parameters in the flow bifurcations of a gaseous heat carrier in tubes of heat-exchange equipment used in air-space, shipbuilding and other engineering. The investigation method consists in the solution of the equation system of momentum, continuity and power.  In the paper there is substantiated a choice of a theoretical model for a current simulation of a gaseous heat carrier in piping bifurcations of heat-exchange equipment with the allowable degree of proximity to an actual current and complexity of essential computations – a thermo-dynamic model of a subcritical, stationary current of compressible gas. There are obtained analytical solutions of the problem on the current of gaseous heat carrier flows in piping bifurcations of heat-exchange equipment used in different fields of engineering, in particular, in aero-space and shipbuilding and so on, whereas earlier took place only numerical solutions of this problem. In this paper the solutions were obtained without application of special functions used at the solution of non-linear and transcendental equations.

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Vazquez-Leal, Hector, Hüseyin Koçak, and Inan Ates. "Rational Approximations for Heat Radiation and Troesch’s Equations." International Journal of Computational Methods 13, no. 03 (May 31, 2016): 1650039. http://dx.doi.org/10.1142/s0219876216500390.

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In this paper, a new tool for the solution of nonlinear differential equations is presented. It is named rational homotopy perturbation method (RHPM). It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. Furthermore, in order to deal with BVP problems, we propose a modification of RHPM method. The obtained results show that RHPM is a powerful tool capable to generate highly accurate rational solutions.

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Talaee, M. R., and V. Sarafrazi. "Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source." Journal of Mechanics 33, no. 1 (June 17, 2016): 65–75. http://dx.doi.org/10.1017/jmech.2016.42.

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AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

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Jiang, Xin, Xiao Gang Wang, Yue Wei Bai, and Chang Tao Pang. "The Method of Fundamental Solutions for the Moving Boundary Problem of the One-Dimension Heat Conduction Equation." Advanced Materials Research 1039 (October 2014): 59–64. http://dx.doi.org/10.4028/www.scientific.net/amr.1039.59.

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The melting of the material is regarded as the moving boundary problem of the heat conduction equation. In this paper, the method of fundamental solution is extended into this kind of problem. The temperature function was expressed as a linear combination of fundamental solutions which satisfied the governing equation and the initial condition. The coefficients were gained by use of boundary condition. When the metal wire was melting, process of the moving boundary was gained through the conversation of energy and the Prediction-Correlation Method. A example was given. The numerical solutions agree well with the exact solutions. In another example, numerical solutions of the temperature distribution of the metal wire were obtained while one end was heated and melting.

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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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Kalyani, P. "Numerical solution of heat equation through double interpolation." IOSR Journal of Mathematics 6, no. 6 (2013): 58–62. http://dx.doi.org/10.9790/5728-0665862.

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Thapa, Narayan, and Michal Gudejko. "Numerical solution of heat equation by spectral method." Applied Mathematical Sciences 8 (2014): 397–404. http://dx.doi.org/10.12988/ams.2014.39502.

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42

Vynnycky, Michael, Je´ro^me Ferrari, and Noam Lior. "Some Analytical and Numerical Solutions to Inverse Problems Applied to Optimizing Phase-Transformation Tracking in Gas Quenching." Journal of Heat Transfer 125, no. 1 (January 29, 2003): 1–10. http://dx.doi.org/10.1115/1.1517271.

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A transient inverse heat conduction problem focused on gas quenching of steel plates and rings is posed and solved, both analytically and numerically. The quenching objective is to calculate the transient convective heat transfer coefficient which would produce an optimized phase transformation cooling curve. The governing nonlinear heat equation is nondimensionalised, and a small parameter, the reciprocal of the Fourier number, is identified. This allows the construction of an analytic solution in the form of an asymptotic series. For higher values of the reciprocal Fourier number, a numerical scheme incorporating the function specification and Keller Box methods is used to generate solutions. Comparison of the results proves favorable, and suggests that for this inverse problem asymptotic methods provide an attractive alternative to solely numerical ones.

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Cesarano, Clemente. "Generalized special functions in the description of fractional diffusive equations." Communications in Applied and Industrial Mathematics 10, no. 1 (January 1, 2019): 31–40. http://dx.doi.org/10.1515/caim-2019-0010.

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Abstract Starting from the heat equation, we discuss some fractional generalizations of various forms. We propose a method useful for analytic or numerical solutions. By using Hermite polynomials of higher and fractional order, we present some operational techniques to find general solutions of extended form to d'Alembert and Fourier equations. We also show that the solutions of the generalized equations discussed here can be expressed in terms of Hermite-based functions.

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Tatari, Mehdi, Mehdi Dehghan, and Mohsen Razzaghi. "Numerical solution of the one-dimensional heat equation on the bounded intervals using fundamental solutions." Numerical Methods for Partial Differential Equations 24, no. 3 (September 11, 2007): 911–23. http://dx.doi.org/10.1002/num.20296.

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45

Gvirtzman, Zohar, and Zvi Garfunkel. "Numerical solutions for the one-dimensional heat-conduction equation using a spreadsheet." Computers & Geosciences 22, no. 10 (December 1996): 1147–58. http://dx.doi.org/10.1016/s0098-3004(96)00052-0.

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46

Burger, J., and C. Machbub. "Comparison of numerical solutions of a one-dimensional non-linear heat equation." Communications in Applied Numerical Methods 7, no. 3 (April 1991): 233–40. http://dx.doi.org/10.1002/cnm.1630070308.

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Chong, Yuxiang, and John B. Walsh. "The Roughness and Smoothness of Numerical Solutions to the Stochastic Heat Equation." Potential Analysis 37, no. 4 (November 3, 2011): 303–32. http://dx.doi.org/10.1007/s11118-011-9257-6.

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48

Nolan, John P. "Stable distributions and green’s functions for fractional diffusions." Fractional Calculus and Applied Analysis 22, no. 1 (February 25, 2019): 128–38. http://dx.doi.org/10.1515/fca-2019-0008.

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Abstract Stable distributions are a class of distributions that have important uses in probability theory. They also have a applications in the theory of fractional diffusions: symmetric stable density functions are the Green’s functions of the fractional heat equation. We describe efficient numerical representations for these Green’s functions, enabling their use in numerical solutions of fractional heat equations. We also describe a new connection between stable laws and the Weyl fractional derivative.

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Aksenov, Boris G., Yuri E. Karyakin, and Svetlana V. Karyakina. "Solution of heat and mass transfer problems with non-linear coefficients." Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy 5, no. 4 (2019): 10–20. http://dx.doi.org/10.21684/2411-7978-2019-5-4-10-20.

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Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.

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Rubio, Diana, Domingo A. Tarzia, and Guillermo F. Umbricht. "Heat Transfer Process with Solid-solid Interface: Analytical and Numerical Solutions." WSEAS TRANSACTIONS ON MATHEMATICS 20 (September 2, 2021): 404–14. http://dx.doi.org/10.37394/23206.2021.20.42.

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This work is aimed at the study and analysis of the heat transport on a metal bar of length L with a solid-solid interface. The process is assumed to be developed along one direction, across two homogeneous and isotropic materials. Analytical and numerical solutions are obtained under continuity conditions at the interface, that is a perfect assembly. The lateral side is assumed to be isolated and a constant thermal source is located at the left-boundary while the right-end stays free allowing the heat to transfer to the surrounding fluid by a convective process. The differences between the analytic solution and temperature measurements at any point on the right would indicate the presence of discontinuities. The greater these differences, the greater the discontinuity in the interface due to thermal resistances, providing a measure of its propagation from the interface and they could be modeled as temperature perturbations. The problem of interest may be described by a parabolic equation with initial, interface and boundary conditions, where the thermal properties, the conductivity and diffusivity coefficients, are piecewise constant functions. The analytic solution is derived by using Fourier methods. Special attention is given to the Sturm-Liouville problem that arises when deriving the solution, since a complicated eigenvalue equation must to be solved. Numerical simulations are conducted by using finite difference schemes where its convergence and stability properties are discussed along with physical interpretations of the results.

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