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Artykuły w czasopismach na temat „Heat equation Numerical solutions”

Autor: Grafiati

Data publikacji: 4 czerwca 2021

Data aktualizacji: 13 lutego 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

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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 stati

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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 efficien

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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

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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 au

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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 equat

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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 exa

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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 equ

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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

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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

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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 me

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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

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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

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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

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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 directi

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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.

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24

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

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26

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 derivativ

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27

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 obta

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28

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

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29

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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30

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 spe

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31

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 arbit

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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 succe

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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 w

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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 complexit

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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

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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 add

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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

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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

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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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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 numerica

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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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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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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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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 lo

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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 an

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