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Journal articles on the topic 'Numerical solution of partial differential equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

M.A.Mohamed, M. A. Mohamed. "Numerical Solution of Nonlinear Partial Differential Equation by Legendre Multiwavelet Method." International Journal of Scientific Research 3, no. 2 (June 1, 2012): 1–8. http://dx.doi.org/10.15373/22778179/feb2014/188.

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2

Maset, Stefano. "Numerical solution of retarded functional differential equations as partial differential equations." IFAC Proceedings Volumes 33, no. 23 (September 2000): 133–35. http://dx.doi.org/10.1016/s1474-6670(17)36930-6.

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3

Zhang, Zhao. "Numerical Analysis and Comparison of Gridless Partial Differential Equations." International Journal of Circuits, Systems and Signal Processing 15 (August 31, 2021): 1223–31. http://dx.doi.org/10.46300/9106.2021.15.133.

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In the field of science and engineering, partial differential equations play an important role in the process of transforming physical phenomena into mathematical models. Therefore, it is very important to get a numerical solution with high accuracy. In solving linear partial differential equations, meshless solution is a very important method. Based on this, we propose the numerical solution analysis and comparison of meshless partial differential equations (PDEs). It is found that the interaction between the numerical solutions of gridless PDEs is better, and the absolute error and relative error are lower, which proves the superiority of the numerical solutions of gridless PDEs

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4

Abhyankar, N. S., E. K. Hall, and S. V. Hanagud. "Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations." Journal of Applied Mechanics 60, no. 1 (March 1, 1993): 167–74. http://dx.doi.org/10.1115/1.2900741.

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The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

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5

FALCONE, M. "NUMERICAL METHODS FOR DIFFERENTIAL GAMES BASED ON PARTIAL DIFFERENTIAL EQUATIONS." International Game Theory Review 08, no. 02 (June 2006): 231–72. http://dx.doi.org/10.1142/s0219198906000886.

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In this paper we present some numerical methods for the solution of two-persons zero-sum deterministic differential games. The methods are based on the dynamic programming approach. We first solve the Isaacs equation associated to the game to get an approximate value function and then we use it to reconstruct approximate optimal feedback controls and optimal trajectories. The approximation schemes also have an interesting control interpretation since the time-discrete scheme stems from a dynamic programming principle for the associated discrete time dynamical system. The general framework for convergence results to the value function is the theory of viscosity solutions. Numerical experiments are presented solving some classical pursuit-evasion games.

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6

Jokar, Sadegh, Volker Mehrmann, Marc E. Pfetsch, and Harry Yserentant. "Sparse approximate solution of partial differential equations." Applied Numerical Mathematics 60, no. 4 (April 2010): 452–72. http://dx.doi.org/10.1016/j.apnum.2009.10.003.

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7

T., V., and G. D. Smith. "Numerical Solution of Partial Differential Equations, Finite Difference Methods." Mathematics of Computation 48, no. 178 (April 1987): 834. http://dx.doi.org/10.2307/2007849.

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8

Iserles, A., and G. D. Smith. "Numerical Solution of Partial Differential Equations: Finite Difference Methods." Mathematical Gazette 70, no. 454 (December 1986): 330. http://dx.doi.org/10.2307/3616228.

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9

Vanani, Solat, and Azim Aminataei. "On the Numerical Solution of Fractional Partial Differential Equations." Mathematical and Computational Applications 17, no. 2 (August 1, 2012): 140–51. http://dx.doi.org/10.3390/mca17020140.

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10

Soliman, A. F., A. M. A. EL-ASYED, and M. S. El-Azab. "On The Numerical Solution of Partial integro-differential equations." Mathematical Sciences Letters 1, no. 1 (May 1, 2012): 71–80. http://dx.doi.org/10.12785/msl/010109.

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11

Sewell, Granville, and Harvey Gould. "The Numerical Solution of Ordinary and Partial Differential Equations." Computers in Physics 3, no. 6 (1989): 98. http://dx.doi.org/10.1063/1.4822884.

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12

Qian, Sam, and John Weiss. "Wavelets and the Numerical Solution of Partial Differential Equations." Journal of Computational Physics 106, no. 1 (May 1993): 155–75. http://dx.doi.org/10.1006/jcph.1993.1100.

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13

Hromadka, T. V., and R. J. Whitley. "Error bounds for numerical solution of partial differential equations." Numerical Methods for Partial Differential Equations 7, no. 4 (1991): 339–46. http://dx.doi.org/10.1002/num.1690070404.

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14

Moore, Peter K., Can Ozturan, and Joseph E. Flaherty. "Towards the automatic numerical solution of partial differential equations." Mathematics and Computers in Simulation 31, no. 4-5 (October 1989): 325–32. http://dx.doi.org/10.1016/0378-4754(89)90127-4.

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15

Harker, Matthew, and Paul O'Leary. "Sylvester Equations and the numerical solution of partial fractional differential equations." Journal of Computational Physics 293 (July 2015): 370–84. http://dx.doi.org/10.1016/j.jcp.2014.12.033.

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16

Gaidomak, S. B. "Numerical solution of linear differential-algebraic systems of partial differential equations." Computational Mathematics and Mathematical Physics 55, no. 9 (September 2015): 1501–14. http://dx.doi.org/10.1134/s0965542515060044.

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17

Choudhury, A. H. "Wavelet Method for Numerical Solution of Parabolic Equations." Journal of Computational Engineering 2014 (February 27, 2014): 1–12. http://dx.doi.org/10.1155/2014/346731.

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We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions.

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18

Bar-Sinai, Yohai, Stephan Hoyer, Jason Hickey, and Michael P. Brenner. "Learning data-driven discretizations for partial differential equations." Proceedings of the National Academy of Sciences 116, no. 31 (July 16, 2019): 15344–49. http://dx.doi.org/10.1073/pnas.1814058116.

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The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.

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19

AL-Jawary, Majeed, and Sayl Abd- AL- Razaq. "Analytic and numerical solution for duffing equations." International Journal of Basic and Applied Sciences 5, no. 2 (March 18, 2016): 115. http://dx.doi.org/10.14419/ijbas.v5i2.5838.

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<p>Daftardar Gejji and Hossein Jafari have proposed a new iterative method for solving many of the linear and nonlinear equations namely (DJM). This method proved already the effectiveness in solved many of the ordinary differential equations, partial differential equations and integral equations. The main aim from this paper is to propose the Daftardar-Jafari method (DJM) to solve the Duffing equations and to find the exact solution and numerical solutions. The proposed (DJM) is very effective and reliable, and the solution is obtained in the series form with easily computed components. The software used for the calculations in this study was MATHEMATICA<sup>®</sup> 9.0.</p>

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20

Blanco, Pablo, Paola Gervasio, and Alfio Quarteroni. "Extended Variational Formulation for Heterogeneous Partial Differential Equations." Computational Methods in Applied Mathematics 11, no. 2 (2011): 141–72. http://dx.doi.org/10.2478/cmam-2011-0008.

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AbstractWe address the coupling of an advection equation with a diffusion-advection equation, for solutions featuring boundary layers. We consider non-overlapping domain decompositions and we face up the heterogeneous problem using an extended variational formulation. We will prove the equivalence between the latter formulation and a treatment based on a singular perturbation theory. An exhaustive comparison in terms of solution and computational efficiency between these formulations is carried out.

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21

Mudge, Michael R., Leon Lapidus, and George F. Pinder. "Numerical Solution of Partial Differential Equations in Science and Engineering." Mathematical Gazette 84, no. 499 (March 2000): 187. http://dx.doi.org/10.2307/3621561.

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22

S., L. R., Hans G. Kaper, and Marc Garbey. "Asymptotic Analysis and the Numerical Solution of Partial Differential Equations." Mathematics of Computation 59, no. 199 (July 1992): 303. http://dx.doi.org/10.2307/2153003.

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23

Siddiqi, Shahid S., and Saima Arshed. "Numerical solution of time-fractional fourth-order partial differential equations." International Journal of Computer Mathematics 92, no. 7 (August 18, 2014): 1496–518. http://dx.doi.org/10.1080/00207160.2014.948430.

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24

Gorial, I. I. "NUMERICAL SOLUTION FOR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS OF TWO-SIDED." Journal of Al-Nahrain University Science 12, no. 2 (June 1, 2009): 128–31. http://dx.doi.org/10.22401/jnus.12.2.15.

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25

Ashyralyev, Allaberen, Fadime Dal, and Zehra Pinar. "On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations." Mathematical Problems in Engineering 2009 (2009): 1–11. http://dx.doi.org/10.1155/2009/730465.

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The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this difference scheme and for the first and second orders difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.

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26

孙, 雯雯. "On Numerical Solution of the Memory Dependent Partial Differential Equations." Advances in Applied Mathematics 06, no. 04 (2017): 637–43. http://dx.doi.org/10.12677/aam.2017.64074.

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27

Ramos, J. I. "Asymptotic Analysis and the Numerical Solution of Partial Differential Equations." Applied Mathematical Modelling 16, no. 12 (December 1992): 666. http://dx.doi.org/10.1016/0307-904x(92)90100-h.

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28

Sharma, Deepika, Kavita Goyal, and Rohit Kumar Singla. "A curvelet method for numerical solution of partial differential equations." Applied Numerical Mathematics 148 (February 2020): 28–44. http://dx.doi.org/10.1016/j.apnum.2019.08.029.

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29

Katsikadelis, John T. "The BEM for numerical solution of partial fractional differential equations." Computers & Mathematics with Applications 62, no. 3 (August 2011): 891–901. http://dx.doi.org/10.1016/j.camwa.2011.04.001.

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30

Speleers, Hendrik, Paul Dierckx, and Stefan Vandewalle. "Numerical solution of partial differential equations with Powell–Sabin splines." Journal of Computational and Applied Mathematics 189, no. 1-2 (May 2006): 643–59. http://dx.doi.org/10.1016/j.cam.2005.03.001.

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31

Modani, Manish, Maithili Sharan, and S. Chandra Sekhara Rao. "Numerical solution of elliptic partial differential equations on parallel systems." Applied Mathematics and Computation 195, no. 1 (January 2008): 162–82. http://dx.doi.org/10.1016/j.amc.2007.04.077.

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32

Kolokoltsov, Vassili, Feng Lin, and Aleksandar Mijatović. "Monte carlo estimation of the solution of fractional partial differential equations." Fractional Calculus and Applied Analysis 24, no. 1 (January 29, 2021): 278–306. http://dx.doi.org/10.1515/fca-2021-0012.

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Abstract The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem (CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.

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33

Syazana Saharizan, Nur, and Nurnadiah Zamri. "Numerical solution for a new fuzzy transform of hyperbolic goursat partial differential equation." Indonesian Journal of Electrical Engineering and Computer Science 16, no. 1 (October 1, 2019): 292. http://dx.doi.org/10.11591/ijeecs.v16.i1.pp292-298.

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<p>The main objective of this paper is to present a new numerical method with utilization of fuzzy transform in order to solve various engineering problems that represented by hyperbolic Goursat partial differentical equation (PDE). The application of differential equations are widely used for modelling physical phenomena. There are many complicated and dynamic physical problems involved in developing a differential equation with high accuracy. Some problems requires a complex and time consuming algorithms. Therefore, the application of fuzzy mathematics seems to be appropriate for solving differential equations due to the transformation of differential equations to the algebraic equation which is solvable. Furthermore, development of a numerical method for solving hyperbolic Goursat PDE is presented in this paper. The method are supported by numerical experiment and computation using MATLAB. This will provide a clear picture to the researcher to understand the utilization of fuzzy transform to the hyperbolic Goursat PDE.</p>

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34

Aziz, Imran, and Imran Khan. "Numerical Solution of Diffusion and Reaction–Diffusion Partial Integro-Differential Equations." International Journal of Computational Methods 15, no. 06 (September 2018): 1850047. http://dx.doi.org/10.1142/s0219876218500470.

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In this paper, a collocation method based on Haar wavelet is developed for numerical solution of diffusion and reaction–diffusion partial integro-differential equations. The equations are parabolic partial integro-differential equations and we consider both one-dimensional and two-dimensional cases. Such equations have applications in several practical problems including population dynamics. An important advantage of the proposed method is that it can be applied to both linear as well as nonlinear problems with slide modification. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency and robustness of the proposed method.

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35

ARLUKOWICZ, P., and W. CZERNOUS. "A numerical method of bicharacteristics For quasi-linear partial functional Differential equations." Computational Methods in Applied Mathematics 8, no. 1 (2008): 21–38. http://dx.doi.org/10.2478/cmam-2008-0002.

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Abstract Classical solutions of mixed problems for first order partial functional differential equations in several independent variables are approximated by solutions of an Euler-type difference problem. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that the given functions satisfy the nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from the general model by specializing the given operators.

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36

Ahmad, Hijaz, Ali Akgül, Tufail A. Khan, Predrag S. Stanimirović, and Yu-Ming Chu. "New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations." Complexity 2020 (October 6, 2020): 1–10. http://dx.doi.org/10.1155/2020/8829017.

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The role of integer and noninteger order partial differential equations (PDE) is essential in applied sciences and engineering. Exact solutions of these equations are sometimes difficult to find. Therefore, it takes time to develop some numerical techniques to find accurate numerical solutions of these types of differential equations. This work aims to present a novel approach termed as fractional iteration algorithm-I for finding the numerical solution of nonlinear noninteger order partial differential equations. The proposed approach is developed and tested on nonlinear fractional-order Fornberg–Whitham equation and employed without using any transformation, Adomian polynomials, small perturbation, discretization, or linearization. The fractional derivatives are taken in the Caputo sense. To assess the efficiency and precision of the suggested method, the tabulated numerical results are compared with the standard variational iteration method and the exact solution as well. In addition, numerical results for different cases of the fractional-order α are presented graphically, which show the effectiveness of the proposed procedure and revealed that the proposed scheme is very effective, suitable for fractional PDEs, and may be viewed as a generalization of the existing methods for solving integer and noninteger order differential equations.

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37

Chaadaev, A. B. "ABOUT SOME SUPPLEMENTARY POSSIBILITY FOR NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS." Fine Chemical Technologies 11, no. 1 (February 28, 2016): 75–78. http://dx.doi.org/10.32362/2410-6593-2016-11-1-75-78.

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A substitution of an non-homogeneous term and of a differential operator by the difference of Laplace operators in the direct co-ordinate system and in the turned one in the partial differential equations of first, second and third order is proposed. The numerical solution obtained by solving the substituting equation corresponds to the exact solution of the initial equations.

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38

Baleanu, Dumitru, and Hassan Kamil Jassim. "Exact Solution of Two-Dimensional Fractional Partial Differential Equations." Fractal and Fractional 4, no. 2 (May 12, 2020): 21. http://dx.doi.org/10.3390/fractalfract4020021.

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In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations.

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39

Higdon, Robert L. "Numerical modelling of ocean circulation." Acta Numerica 15 (May 2006): 385–470. http://dx.doi.org/10.1017/s0962492906250013.

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Computational simulations of ocean circulation rely on the numerical solution of partial differential equations of fluid dynamics, as applied to a relatively thin layer of stratified fluid on a rotating globe. This paper describes some of the physical and mathematical properties of the solutions being sought, some of the issues that are encountered when the governing equations are solved numerically, and some of the numerical methods that are being used in this area.

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40

Wu, G., Eric Wai Ming Lee, and Gao Li. "Numerical solutions of the reaction-diffusion equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 2 (March 2, 2015): 265–71. http://dx.doi.org/10.1108/hff-04-2014-0113.

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Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

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41

Gaidomak, S. V. "Erratum to: “Numerical solution of linear differential-algebraic systems of partial differential equations”." Computational Mathematics and Mathematical Physics 55, no. 12 (December 2015): 2100. http://dx.doi.org/10.1134/s0965542515150014.

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42

Yang, Xiaofan, Yaoxin Liu, and Sen Bai. "A numerical solution of second-order linear partial differential equations by differential transform." Applied Mathematics and Computation 173, no. 2 (February 2006): 792–802. http://dx.doi.org/10.1016/j.amc.2005.04.015.

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43

BARUCCI, E., S. POLIDORO, and V. VESPRI. "SOME RESULTS ON PARTIAL DIFFERENTIAL EQUATIONS AND ASIAN OPTIONS." Mathematical Models and Methods in Applied Sciences 11, no. 03 (April 2001): 475–97. http://dx.doi.org/10.1142/s0218202501000945.

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We analyze partial differential equations arising in the evaluation of Asian options. The equations are strongly degenerate partial differential equations in three dimensions. We show that the solution of the no-arbitrage partial differential equation is sufficiently regular and standard numerical methods can be employed to approximate it.

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44

NAKAO, Mitsuhiro. "Numerical Verification of Solutions for Partial Differential Equations." IEICE ESS FUNDAMENTALS REVIEW 2, no. 3 (2009): 19–28. http://dx.doi.org/10.1587/essfr.2.3_19.

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45

Nakao, Mitsuhiro T. "Numerical verification for solutions to partial differential equations." Sugaku Expositions 30, no. 1 (March 17, 2017): 89–109. http://dx.doi.org/10.1090/suga/419.

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46

Seth, G. S., S. Sarkar, and R. Sharma. "Effects of Hall current on unsteady hydromagnetic free convection flow past an impulsively moving vertical plate with Newtonian heating." International Journal of Applied Mechanics and Engineering 21, no. 1 (February 1, 2016): 187–203. http://dx.doi.org/10.1515/ijame-2016-0012.

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Abstract An investigation of unsteady hydromagnetic free convection flow of a viscous, incompressible and electrically conducting fluid past an impulsively moving vertical plate with Newtonian surface heating embedded in a porous medium taking into account the effects of Hall current is carried out. The governing partial differential equations are first subjected to the Laplace transformation and then inverted numerically using INVLAP routine of Matlab. The governing partial differential equations are also solved numerically by the Crank-Nicolson implicit finite difference scheme and a comparison has been provided between the two solutions. The numerical solutions for velocity and temperature are plotted graphically whereas the numerical results of skin friction and the Nusselt number are presented in tabular form for various parameters of interest. The present solution in special case is compared with a previously obtained solution and is found to be in excellent agreement.

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47

VALENCIANO, J., and M. A. J. CHAPLAIN. "AN EXPLICIT SUBPARAMETRIC SPECTRAL ELEMENT METHOD OF LINES APPLIED TO A TUMOUR ANGIOGENESIS SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 14, no. 02 (February 2004): 165–87. http://dx.doi.org/10.1142/s0218202504003155.

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In this paper we consider a numerical solution to Anderson and Chaplain's tumour angiogenesis model1 over two-dimensional complex geometry. The numerical solution of the governing system of non-linear evolutionary partial differential equations is obtained using the method of lines: after a spatial semi-discretisation based on the subparametric Legendre spectral element method is performed, the original system of partial differential equations is replaced by an augmented system of stiff ordinary differential equations in autonomous form, which is then advanced forward in time using an explicit time integrator based on the fourth-order Chebyshev polynomial. Numerical simulations show the convergence of the steady state numerical solution towards the linearly stable steady state analytical solution.

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48

Liu, Yong Qing, Rong Jun Cheng, and Hong Xia Ge. "Element-Free Galerkin (EFG) Method for Time Fractional Partial Differential Equations." Applied Mechanics and Materials 101-102 (September 2011): 343–47. http://dx.doi.org/10.4028/www.scientific.net/amm.101-102.343.

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In this paper, the first order time derivative of time fractional partial differential equations are replaced by the Caputo fractional order derivative. We derive the numerical solution of this equation using the Element-free Galerkin (EFG) method. In order to obtain the discrete equation, a various method is used and the essential boundary conditions are enforced by the penalty method. Numerical examples are presented and the results are in good agreement with exact solutions.

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49

Iqbal, Mazhar, M. T. Mustafa, and Azad A. Siddiqui. "A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/105414.

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Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.

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50

Kamrani, Minoo. "Numerical solution of partial differential equations with stochastic Neumann boundary conditions." Discrete & Continuous Dynamical Systems - B 22, no. 11 (2017): 1–18. http://dx.doi.org/10.3934/dcdsb.2019061.

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