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Academic literature on the topic 'Heat equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 31 August 2024

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Journal articles on the topic "Heat equation"

1

N O, Onuoha. "Transformation of Parabolic Partial Differential Equations into Heat Equation Using Hopf Cole Transform." International Journal of Science and Research (IJSR) 12, no. 6 (June 5, 2023): 1741–43. http://dx.doi.org/10.21275/sr23612082710.

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Zeng, Chulan. "Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations." Communications on Pure & Applied Analysis 21, no. 3 (2022): 749. http://dx.doi.org/10.3934/cpaa.2021197.

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<p style='text-indent:20px;'>In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>. The potentials include all the nonnegative ones. For the first two equations, we prove if <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> satisfies some growth conditions in <inline-formula><tex-math id="M3">\begin{document}$ (x,t)\in \mathrm{M}\times [0,1] $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> is analytic in time <inline-formula><tex-math id="M5">\begin{document}$ (0,1] $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M6">\begin{document}$ \mathrm{M} $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ R^d $\end{document}</tex-math></inline-formula> or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that <inline-formula><tex-math id="M8">\begin{document}$ u(x,t) $\end{document}</tex-math></inline-formula> is analytic in time at <inline-formula><tex-math id="M9">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general.</p><p style='text-indent:20px;'>For the nonlinear heat equation with power nonlinearity of order <inline-formula><tex-math id="M10">\begin{document}$ p $\end{document}</tex-math></inline-formula>, we prove that a solution is analytic in time <inline-formula><tex-math id="M11">\begin{document}$ t\in (0,1] $\end{document}</tex-math></inline-formula> if it is bounded in <inline-formula><tex-math id="M12">\begin{document}$ \mathrm{M}\times[0,1] $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a positive integer. In addition, we investigate the case when <inline-formula><tex-math id="M14">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a rational number with a stronger assumption <inline-formula><tex-math id="M15">\begin{document}$ 0<C_3 \leq |u(x,t)| \leq C_4 $\end{document}</tex-math></inline-formula>. It is also shown that a solution may not be analytic in time if it is allowed to be <inline-formula><tex-math id="M16">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. As a lemma, we obtain an estimate of <inline-formula><tex-math id="M17">\begin{document}$ \partial_t^k \Gamma(x,t;y) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M18">\begin{document}$ \Gamma(x,t;y) $\end{document}</tex-math></inline-formula> is the heat kernel on a manifold, with an explicit estimation of the coefficients.</p><p style='text-indent:20px;'>An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable <inline-formula><tex-math id="M19">\begin{document}$ x $\end{document}</tex-math></inline-formula>, implying that the analyticity of space and time can be independent. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.</p>

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Gupta, Rohit, Rakesh Kumar Verma, and Sanjay Kumar Verma. "Solving Wave Equation and Heat Equation by Rohit Transform (RT)." Journal of Physics: Conference Series 2325, no. 1 (August 1, 2022): 012036. http://dx.doi.org/10.1088/1742-6596/2325/1/012036.

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Abstract The wave equation and the heat equation are widely known differential equations coming to light in engineering, basic and material sciences. The differential equations which represent the wave equation and the heat equation are usually solved by the exact technique or by the approximate technique or by the purely numerical technique. Since the implementation of these techniques is very complex, computationally vigorous, and requires elaborate computations, therefore, for finding the solutions of differential equations depicting the wave equation and the heat equation, there is a need to ask for integral transform techniques. Integral transform techniques render productive means for finding the solutions of problems coming to light in engineering, basic and material sciences. The Rohit transform (RT) is a new integral transformation put forward by the author Rohit Gupta recently in the year 2020 and has been utilized for finding the solutions of problems coming to light in engineering, basic and material sciences like other transform techniques. In this study, the RT is brought in for finding the solutions of the heat equation and the wave equation expressed in terms of differential equations which are generally partial in nature.

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Kochubei, Anatoly N., Yuri Kondratiev, and José Luís da Silva. "On fractional heat equation." Fractional Calculus and Applied Analysis 24, no. 1 (January 29, 2021): 73–87. http://dx.doi.org/10.1515/fca-2021-0004.

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Abstract In this paper, the long-time behavior of the Cesaro mean of the fundamental solution for fractional Heat equation corresponding to random time changes in the Brownian motion is studied. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.

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Kiniry, James R., Sumin Kim, and Henri E. Z. Tonnang. "Back to the Future: Revisiting the Application of an Enzyme Kinetic Equation to Maize Development Nearly Four Decades Later." Agronomy 9, no. 9 (September 19, 2019): 566. http://dx.doi.org/10.3390/agronomy9090566.

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With the recent resurgence in interest in models describing maize (Zea mays L.) development rate responses to temperature, this study uses published data to refit the Poikilotherm equation and compare it to broken stick “heat stress” equations. These data were for the development rate of eight open pollinated maize varieties at diverse sites in Africa. The Poikilotherm equation was applied with the original published parameters and after refitting with the data in this study. The heat stress equation was tested after fitting with just the first variety and after fitting with each variety. The Poikilotherm equation with the original parameter values had large errors in predicting development rates in much of the temperature range. The adjusted Poikilotherm equation did much better with the root-mean-square error (RMSE) decreasing from 0.034 to 0.003 (1/day) for a representative variety. The heat stress equation fit to the first variety did better than the Poikilotherm equation when applied to all the varieties. The heat stress equations fitted separately for each variety did not have an improved fit compared to the one heat stress equation. Thus, separate fitting of such an equation for different varieties may not be necessary. The one heat stress equation, the separate heat stress equation, and the Poikilotherm equation each had a better fit than nonlinear Briere et al. curves. The Poikilotherm equation showed promise, realistically capturing the high, low, and optimum rate values measured. All the equations showed promise to some degree for future applications in simulating the maize development rate. When fitting separate regressions for each variety for the heat stress equations, the base temperatures had a mean of 5.3 °C, similar to a previously published value of 6 °C. The last variety had noticeably different rates than the others. This study demonstrated that a simple approach (the heat stress equation) should be adequate in many cases. It also demonstrated that more detailed equations can be useful when a more mechanistic system is desired. Future research could investigate the reasons for the different development rate response of the last variety and investigate similar varieties.

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Oane, Mihai, Muhammad Arif Mahmood, and Andrei C. Popescu. "A State-of-the-Art Review on Integral Transform Technique in Laser–Material Interaction: Fourier and Non-Fourier Heat Equations." Materials 14, no. 16 (August 22, 2021): 4733. http://dx.doi.org/10.3390/ma14164733.

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Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties. Such wonderful features have enabled heat equations in various fields, including laser and electron beam processing. The integral transform technique (ITT) is a powerful general-purpose semi-analytical/numerical method that transforms partial differential equations into a coupled system of ordinary differential equations. Under this category, Fourier and non-Fourier heat equations can be implemented on both equilibrium and non-equilibrium thermo-dynamical processes, including a wide range of processes such as the Two-Temperature Model, ultra-fast laser irradiation, and biological processes. This review article focuses on heat equation models, including Fourier and non-Fourier heat equations. A comparison between Fourier and non-Fourier heat equations and their generalized solutions have been discussed. Various components of heat equations and their implementation in multiple processes have been illustrated. Besides, literature has been collected based on ITT implementation in various materials. Furthermore, a future outlook has been provided for Fourier and non-Fourier heat equations. It was found that the Fourier heat equation is simple to use but involves infinite speed heat propagation in comparison to the non-Fourier heat equation and can be linked with the Two-Temperature Model in a natural way. On the other hand, the non-Fourier heat equation is complex and involves various unknowns compared to the Fourier heat equation. Fourier and Non-Fourier heat equations have proved their reliability in the case of laser–metallic materials, electron beam–biological and –inorganic materials, laser–semiconducting materials, and laser–graphene material interactions. It has been identified that the material properties, electron–phonon relaxation time, and Eigen Values play an essential role in defining the precise results of Fourier and non-Fourier heat equations. In the case of laser–graphene interaction, a restriction has been identified from ITT. When computations are carried out for attosecond pulse durations, the laser wavelength approaches the nucleus-first electron separation distance, resulting in meaningless results.

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Tahir, M., G. Abbas, Kazuharu Bamba, and M. R. Shahzad. "Dynamics of dissipative self-gravitating source in Rastall gravity." International Journal of Modern Physics A 36, no. 20 (July 13, 2021): 2150153. http://dx.doi.org/10.1142/s0217751x21501530.

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The dynamics of dissipative gravitational collapse of a source is explored in Rastall gravity. The field equations are derived for the geometry and collapsing matter. The dynamical equations are formulated for the heat flux and diffusion approximation. The heat transportation equation is derived by using Müller–Israel–Stewart approach to investigate the effects of heat flux on the collapsing source. Moreover, an equation is found by combining the dynamical and heat transport equation, the consequences of this equation are discussed in detail. Furthermore, the Rastall parameter [Formula: see text] effect is analyzed for the collapse of sphere.

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Owino, Joseph Owuor. "GROUP ANALYSIS OF A NONLINEAR HEAT-LIKE EQUATION." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 11, no. 01 (January 13, 2023): 3113–31. http://dx.doi.org/10.47191/ijmcr/v11i1.03.

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We study a nonlinear heat like equation from a lie symmetry stand point. Heat equation have been employed to study ow of current, information and propagation of heat. The Lie group approach is used on the system to obtain symmetry reductions and the reduced systems studied for exact solutions. Solitary waves have been constructed by use of a linear span of time and space translation symmetries. We also compute conservation laws using multiplier approach and by a conservation theorem due to Ibragimov.

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Hills, Norman L., and John M. Irwin. "Infinite-Order Differential Equations and the Heat Equation." SIAM Journal on Mathematical Analysis 20, no. 2 (March 1989): 430–38. http://dx.doi.org/10.1137/0520029.

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Samdarshi, S. K., and S. C. Mullick. "Analytical Equation for the Top Heat Loss Factor of a Flat-Plate Collector With Double Glazing." Journal of Solar Energy Engineering 113, no. 2 (May 1, 1991): 117–22. http://dx.doi.org/10.1115/1.2929955.

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An analytical equation for the top heat loss factor of a flat-plate collector with double glazing has been developed. The maximum computational errors resulting from the use of this equation are plus or minus three percent compared to numerical solution of the heat balance equations. The equation is considerably more accurate than the currently used semi-empirical equations over the entire range of variables covered. It is found that the computational errors resulting from simplification of the proposed equation by approximation of the individual heat-transfer coefficients are much lower than the errors resulting from the use of semi-empirical equations.

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Dissertations / Theses on the topic "Heat equation"

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Carroll, Andrew. "The stochastic nonlinear heat equation." Thesis, University of Hull, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.310216.

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Jumarhon, Bartur. "The one dimensional heat equation and its associated Volterra integral equations." Thesis, University of Strathclyde, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.342381.

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Wang, Jun. "Integral Equation Methods for the Heat Equation in Moving Geometry." Thesis, New York University, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10618746.

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Many problems in physics and engineering require the solution of the heat equation in moving geometry. Integral representations are particularly appropriate in this setting since they satisfy the governing equation automatically and, in the homogeneous case, require the discretization of the space-time boundary alone. Unlike methods based on direct discretization of the partial differential equation, they are unconditonally stable. Moreover, while a naive implementation of this approach is impractical, several efforts have been made over the past few years to reduce the overall computational cost. Of particular note are Fourier-based methods which achieve optimal complexity so long as the time step Δt is of the same order as Δx, the mesh size in the spatial variables. As the time step goes to zero, however, the cost of the Fourier-based fast algorithms grows without bound. A second difficulty with existing schemes has been the lack of efficient, high-order local-in-time quadratures for layer heat potentials.

In this dissertation, we present a new method for evaluating heat potentials that makes use of a spatially adaptive mesh instead of a Fourier series, a new version of the fast Gauss transform, and a new hybrid asymptotic/numerical method for local-in-time quadrature. The method is robust and efficient for any Δt, with essentially optimal computational complexity. We demonstrate its performance with numerical examples and discuss its implications for subsequent work in diffusion, heat flow, solidification and fluid dynamics.

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Xie, Shuguang School of Mathematics UNSW. "Stochastic heat equations with memory in infinite dimensional spaces." Awarded by:University of New South Wales. School of Mathematics, 2005. http://handle.unsw.edu.au/1959.4/24257.

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This thesis is concerned with stochastic heat equation with memory and nonlinear energy supply. The main motivation to study such systems comes from Thermodynamics, see [85]. The main objective of this work is to study the existence and uniqueness of solutions to such equations and to investigate some fundamental properties of solutions like continuous dependence on initial conditions. In our approach we follow the seminal papers by Da Prato and Clement [10], where the stochastic heat equation with memory is tranformed into an integral equation in a function space and the so-called mild solutions are studied. In the aforementioned papers only linear equations with additive noise were investigated. The main contribution of this work is the extension of this approach to nonlinear equations. Our main tools are the theory of stochastic convolutions as developed in [33] and the theory of resolvent kernels for deterministic linear heat equations with memory, see[10]. Since the solution at time t depends on the whole history of the process up to time t, the resolvent kernel does not define a semigroup of operators in the state space of the process and therefore a ???standard??? theory of stochastic evolution equations as presented in the monograph [33] does not apply. A more delicate analysis of the resolvent kernles and the associated stochastic convolutions is needed. We will describe now content of this thesis in more detail. Introductory Chapters 1 and 2 collect some basic and essentially well known facts about the Wiener process, stochastic integrals, stochastic convolutions and integral kernels. However, some results in Chapter 2 dealing with stochastic convolution with respect to non-homogenous Wiener process are extensions of the existing theory. The main results of this thesis are presented in Chapters 3 and 4. In Chapter 3 we prove the existence and uniqueness of solutions to heat equations with additive noise and either Lipschitz or dissipative nonlinearities. In both cases we prove the continuous dependence of solutions on initial conditions. In Chapter 4 we prove the existence and uniqueness of solutions and continuous dependence on initial conditions for equations with multiplicative noise. The diffusion coefficients defined by unbounded operators are allowed.

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COMI, GIULIA. "Two Fractional Stochastic Problems: Semi-Linear Heat Equation and Singular Volterra Equation." Doctoral thesis, Università degli studi di Pavia, 2019. http://hdl.handle.net/11571/1292026.

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Thompson, Jeremy R. (Jeremy Ray). "Physical Motivation and Methods of Solution of Classical Partial Differential Equations." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc277898/.

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We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.

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Zhang, Junchi. "GPU computing of Heat Equations." Digital WPI, 2015. https://digitalcommons.wpi.edu/etd-theses/515.

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There is an increasing amount of evidence in scientific research and industrial engineering indicating that the graphic processing unit (GPU) has a higher efficiency and a stronger ability over CPUs to process certain computations. The heat equation is one of the most well-known partial differential equations with well-developed theories, and application in engineering. Thus, we chose in this report to use the heat equation to numerically solve for the heat distributions at different time points using both GPU and CPU programs. The heat equation with three different boundary conditions (Dirichlet, Neumann and Periodic) were calculated on the given domain and discretized by finite difference approximations. The programs solving the linear system from the heat equation with different boundary conditions were implemented on GPU and CPU. A convergence analysis and stability analysis for the finite difference method was performed to guarantee the success of the program. Iterative methods and direct methods to solve the linear system are also discussed for the GPU. The results show that the GPU has a huge advantage in terms of time spent compared with CPU in large size problems.

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Zähle, Henryk. "Stochastic heat equation and catalytic super Brownian motion." [S.l.] : [s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=972728163.

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Huntul, Mousa Jaar M. "Determination of unknown coefficients in the heat equation." Thesis, University of Leeds, 2018. http://etheses.whiterose.ac.uk/22491/.

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The purpose of this thesis is to find the numerical solutions of one or multiple unknown coefficient identification problems in the governing heat transfer parabolic equations. These inverse problems are numerically solved subject to various types of overdetermination conditions such as the heat flux, nonlocal observation, mass/energy specification, additional temperature measurement, Cauchy data, general integral type over-determination, Stefan condition and heat momentum of the first, second and third order. The main difficulty associated with solving these inverse problems is that they are ill-posed since small changes in the input data can result in enormous changes in the output solution, therefore traditional techniques fail to provide accurate and stable solutions. Throughout this thesis, the finite-difference method (FDM) with the Crank-Nicolson (C-N) scheme is mainly used as a direct solver except in Chapters 8 and 9 where an alternating direction explicit (ADE) method is employed in order to deal with the two-dimensional heat equation. An explicit forward time central space (FTCS) method is also employed in Chapter 2 for the extension to higher dimensions. The treatment for solving a degenerate parabolic equation, which vanishes at the initial moment of time is discussed in Chapter 6. The inverse problems investigated are discretised using FDM or ADE and recast as nonlinear least-squares minimization problems with lower and upper simple bounds on the unknown coefficients. The resulting optimization problems are numerically solved using the \emph{lsqnonlin} routine from MATLAB optimization toolbox. The stability of the numerical solutions is investigated by introducing random noise into the input data which yields unstable results if no regularization is employed. The regularization method is included (where necessary) in order to reduce the influence of measurement errors on the numerical results. The choice of the regularization parameter(s) is based on the L-curve method, on the discrepancy principle criterion or on trial and error.

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Hayman, Kenneth John. "Finite-difference methods for the diffusion equation." Title page, table of contents and summary only, 1988. http://web4.library.adelaide.edu.au/theses/09PH/09phh422.pdf.

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Books on the topic "Heat equation"

1

Seizō, Itō. Diffusion equations. Providence, R.I: American Mathematical Society, 1992.

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Bejenaru, Ioan. Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions. Providence, Rhode Island: American Mathematical Society, 2013.

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Lawler, Gregory F. Random walk and the heat equation. Providence, R.I: American Mathematical Society, 2010.

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Wazwaz, Abdul-Majid. Partial differential equations: Methods and applications. Lisse: Balkema, 2001.

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Straughan, B. Heat waves. New York: Springer, 2011.

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Seizō, Itō. Diffusion equations: Seizō Itō ; translated by Seizō Itō. Providence, R.I: American Mathematical Society, 1992.

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Berline, Nicole. Heat kernels and Dirac operators. Berlin: Springer-Verlag, 1992.

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Berline, Nicole. Heat kernels and Dirac operators. 2nd ed. Berlin: Springer, 1996.

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Hans, Triebel, ed. Hybrid function spaces, heat and Navier-Stokes equations. Zürich: European Mathematical Society, 2014.

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Sowers, R. B. Short-time geometry of random heat kernels. Providence, R.I: American Mathematical Society, 1998.

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Book chapters on the topic "Heat equation"

1

Bassanini, Piero, and Alan R. Elcrat. "Heat Equation." In Theory and Applications of Partial Differential Equations, 53–101. Boston, MA: Springer US, 1997. http://dx.doi.org/10.1007/978-1-4899-1875-8_3.

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Fursaev, Dmitri, and Dmitri Vassilevich. "Heat Equation." In Theoretical and Mathematical Physics, 67–94. Dordrecht: Springer Netherlands, 2011. http://dx.doi.org/10.1007/978-94-007-0205-9_4.

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Brenig, Wilhelm. "Rate Equations (Master Equation, Stosszahlansatz)." In Statistical Theory of Heat, 158–63. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/978-3-642-74685-7_32.

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Uddin, Naseem. "Heat Conduction Equation." In Heat Transfer, 37–46. Boca Raton: CRC Press, 2023. http://dx.doi.org/10.1201/9781003428404-2.

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Bellman, Richard, and George Adomian. "The Heat Equation." In Partial Differential Equations, 110–19. Dordrecht: Springer Netherlands, 1985. http://dx.doi.org/10.1007/978-94-009-5209-6_10.

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Borthwick, David. "The Heat Equation." In Introduction to Partial Differential Equations, 97–110. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-48936-0_6.

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Kress, Rainer. "The Heat Equation." In Linear Integral Equations, 152–62. New York, NY: Springer New York, 1999. http://dx.doi.org/10.1007/978-1-4612-0559-3_9.

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Arendt, Wolfgang, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander. "The Heat Equation." In Vector-valued Laplace Transforms and Cauchy Problems, 395–410. Basel: Springer Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-5075-9_6.

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DiBenedetto, Emmanuele. "The Heat Equation." In Partial Differential Equations, 225–91. Boston, MA: Birkhäuser Boston, 1995. http://dx.doi.org/10.1007/978-1-4899-2840-5_6.

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DiBenedetto, Emmanuele. "The Heat Equation." In Partial Differential Equations, 135–81. Boston: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4552-6_6.

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Conference papers on the topic "Heat equation"

1

BARHOUMI, Abdessatar, Habib OUERDIANE, and Hafedh RGUIGUI. "GENERALIZED EULER HEAT EQUATION." In Proceedings of the 29th Conference. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814295437_0008.

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CHUNG, SOON-YEONG. "HEAT EQUATION VIA GENERALIZED FUNCTIONS." In Proceedings of Modelling and Control of Mechanical Systems. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776594_0005.

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Sadybekov, Makhmud, and Aidyn Kassymov. "An isoperimetric inequality for heat potential and heat equation." In INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2016). Author(s), 2016. http://dx.doi.org/10.1063/1.4959643.

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Alkmim, Nasser, and Lineu José Pedroso. "NUMERICAL IMPLEMENTATION OF HEAT EQUATION CONSIDERING NONLINEAR HEAT SOURCE." In XXXVIII Iberian-Latin American Congress on Computational Methods in Engineering. Florianopolis, Brazil: ABMEC Brazilian Association of Computational Methods in Engineering, 2017. http://dx.doi.org/10.20906/cps/cilamce2017-0423.

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Kumar, Vivek, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "High Accurate Solver for Heat Equation using Modified Equation Approach." In ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010. AIP, 2010. http://dx.doi.org/10.1063/1.3498538.

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Shnaid, Isaac. "Governing Equations for Heat Conduction With Finite Speed of Heat Propagation." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-33855.

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In heat conduction, two different analytical approaches exist. The classical approach is based on a parabolic type Fourier equation with infinite speed of heat propagation. The second approach employs the hyperbolic type governing equation assuming finite speed of heat propagation. This approach requires fundamental modifications of classical thermodynamics which are developed in the frame of extended thermodynamics. In this work, governing equations for heat conduction with finite speed of heat propagation are derived directly from classical thermodynamics. For a linear flow of heat, the developed governing equation is linear and of parabolic type. In a three dimensional case, the system of nonlinear equations is formulated. Analytical solutions of the equations for linear flow of heat are obtained, and their analysis shows characteristic features of heat propagation with finite speed, being fully consistent with classical thermodynamics.

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Tadashi Takakura, Kotaro Takayama, Hiroshige Nishina, Kazuaki Tamura, and and Shinji Muta. "Evapotranspiration Estimate by Heat Balance Equation." In 2005 Tampa, FL July 17-20, 2005. St. Joseph, MI: American Society of Agricultural and Biological Engineers, 2005. http://dx.doi.org/10.13031/2013.19526.

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HongFang Wang and E. R. Hancock. "Probabilistic Relaxation using the Heat Equation." In 18th International Conference on Pattern Recognition (ICPR'06). IEEE, 2006. http://dx.doi.org/10.1109/icpr.2006.947.

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Asllanaj, F., G. Jeandel, J. R. Roche, and D. Schmitt. "Analysis of radiative transfer equation coupled with nonlinear heat conduction equation." In Proceedings of the 4th European Conference. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812777201_0030.

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Shudo, T., and H. Suzuki. "New Heat Transfer Equation Applicable to Hydrogen-Fuelled Engines." In ASME 2002 Internal Combustion Engine Division Fall Technical Conference. ASMEDC, 2002. http://dx.doi.org/10.1115/icef2002-515.

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Equations to describe heat transfer from burning gas to a combustion chamber have been empirically derived from hydrocarbon combustion engines. Previous research has analyzed the applicability of the equations to hydrogen combustion and showed that they calculate a lower cooling loss than experimental values. By focusing on the gas velocity term in the heat transfer equation and investigating replacement terms to better fit to hydrogen combustion, a new equation including rate of heat release in the gas velocity term is proposed. It is shown that the new equation is more applicable to hydrogen combustion than the widely used Woschni’s equation.

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Reports on the topic "Heat equation"

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Garrett, Charles Kristopher. Numerically Solving the Heat Equation. Office of Scientific and Technical Information (OSTI), June 2017. http://dx.doi.org/10.2172/1364581.

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Petkov, Alexander. An Entropy Formula for the Heat Equation on a Quaternionic Contact Manifold. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, October 2019. http://dx.doi.org/10.7546/crabs.2019.10.01.

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Petkov, Alexander. On Some Applications of Entropy Formula for the Heat Equation on a Quaternionic Contact Manifold. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, February 2020. http://dx.doi.org/10.7546/crabs.2020.02.05.

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Forristall, R. Heat Transfer Analysis and Modeling of a Parabolic Trough Solar Receiver Implemented in Engineering Equation Solver. Office of Scientific and Technical Information (OSTI), October 2003. http://dx.doi.org/10.2172/15004820.

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Simon, Gordon, Yi-Ching Yao, and Xizhi Wu. Sequential Tests for the Drift of a Wiener Process with a Smooth Prior, and the Heat Equation. Fort Belvoir, VA: Defense Technical Information Center, October 1987. http://dx.doi.org/10.21236/ada190322.

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Berry, Ray, M. Kunick, David Andrs, Joshua Hansel, and Richard Martineau. Sockeye Heat Pipe Code Theory Development: Based on the 7-Equation, Two-Phase Flow Model of RELAP-7. Office of Scientific and Technical Information (OSTI), September 2019. http://dx.doi.org/10.2172/1875848.

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Guan, Jiajing, Sophia Bragdon, and Jay Clausen. Predicting soil moisture content using Physics-Informed Neural Networks (PINNs). Engineer Research and Development Center (U.S.), August 2024. http://dx.doi.org/10.21079/11681/48794.

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Environmental conditions such as the near-surface soil moisture content are valuable information in object detection problems. However, such information is generally unobtainable at the necessary scale without active sensing. Richards’ equation is a partial differential equation (PDE) that describes the infiltration process of unsaturated soil. Solving the Richards’ equation yields information about the volumetric soil moisture content, hydraulic conductivity, and capillary pressure head. However, Richards’ equation is difficult to approximate due to its nonlinearity. Numerical solvers such as finite difference method (FDM) and finite element method (FEM) are conventional in approximating solutions to Richards’ equation. But such numerical solvers are time-consuming when used in real-time. Physics-informed neural networks (PINNs) are neural networks relying on physical equations in approximating solutions. Once trained, these networks can output approximations in a speedy manner. Thus, PINNs have attracted massive attention in the numerical PDE community. This project aims to apply PINNs to the Richards’ equation to predict underground soil moisture content under known precipitation data.

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Pradhan, Nawa Raj, Charles Wayne Downer, and Sergey Marchenko. User guidelines on catchment hydrological modeling with soil thermal dynamics in Gridded Surface Subsurface Hydrologic Analysis (GSSHA). Engineer Research and Development Center (U.S.), March 2024. http://dx.doi.org/10.21079/11681/48331.

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Climate warming is expected to degrade permafrost in many regions of the world. Degradation of permafrost has the potential to affect soil thermal, hydrological, and vegetation regimes. Projections of long-term effects of climate warming on high-latitude ecosystems require a coupled representation of soil thermal state and hydrological dynamics. Such a coupled framework was developed to explicitly simulate the soil moisture effects of soil thermal conductivity and heat capacity and its effects on hydrological response. In the coupled framework, the Geophysical Institute Permafrost Laboratory (GIPL) model is coupled with the Gridded Surface Subsurface Hydrologic Analysis (GSSHA) model. The new permafrost heat transfer in GSSHA is computed with the GIPL scheme that simulates soil temperature dynamics and the depth of seasonal freezing and thawing by numerically solving a one-dimensional quasilinear heat equation with phase change. All the GIPL input and output parameters and the state variables are set up to be consistent with the GSSHA input-output format and grid distribution data input requirements. Test-case simulated results showed that freezing temperatures reduced soil storage capacity, thereby producing higher peak and lower base flow. The report details the functions and format of required input variables and cards, as a guideline, in GSSHA hydrothermal analysis of frozen soils in permafrost-active areas.

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Wilson, D., Chris Pettit, Vladimir Ostashev, and Matthew Kamrath. Signal power distributions for simulated outdoor sound propagation in varying refractive conditions. Engineer Research and Development Center (U.S.), July 2024. http://dx.doi.org/10.21079/11681/48774.

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Probability distributions of acoustic signals propagating through the near-ground atmosphere are simulated by the parabolic equation method. The simulations involve propagation at four angles relative to the mean wind, with frequencies of 100, 200, 400, and 800 Hz. The environmental representation includes realistic atmospheric refractive profiles, turbulence, and ground interactions; cases are considered with and without parametric uncertainties in the wind velocity and surface heat flux. The simulated signals are found to span a broad range of scintillation indices, from near zero to exceeding ten. In the absence of uncertainties, the signal power (or intensity) is fit well by a two-parameter gamma distribution, regardless of the frequency and refractive conditions. When the uncertainties are included, three-parameter distributions, namely, the compound gamma or generalized gamma, are needed for a good fit to the simulation data. The compound gamma distribution appears preferable because its parameters have a straight forward interpretation related to the saturation and modulation of the signal by uncertainties.

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Novick-Cohen, Amy. Solidation Front/Viscous Phase Transitions, Forwards-Backward Heat Equations. Fort Belvoir, VA: Defense Technical Information Center, July 1989. http://dx.doi.org/10.21236/ada211068.

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