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Journal articles on the topic 'Differential equations, Partial Numerical solutions'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 August 2024

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1

ALdarawi, Iman, Banan Maayah, Eman Aldabbas, and Eman Abuteen. "Numerical Solutions of Some Classes of Partial Differential Equations of Fractional Order." European Journal of Pure and Applied Mathematics 16, no. 4 (October 30, 2023): 2132–44. http://dx.doi.org/10.29020/nybg.ejpam.v16i4.4928.

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This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

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2

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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3

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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4

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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5

Kurbonov, Elyorjon, Nodir Rakhimov, Shokhabbos Juraev, and Feruza Islamova. "Derive the finite difference scheme for the numerical solution of the first-order diffusion equation IBVP using the Crank-Nicolson method." E3S Web of Conferences 402 (2023): 03029. http://dx.doi.org/10.1051/e3sconf/202340203029.

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In the article, a differential scheme is created for the the first-order diffusion equation using the Crank-Nicolson method. The stability of the differential scheme was checked using the Neumann method. To solve the problem numerically, stability intervals were found using the Neman method. This work presents an analysis of the stability of the Crank-Nicolson scheme for the two-dimensional diffusion equation using Von Neumann stability analysis. The Crank-Nicolson scheme is a widely used numerical method for solving partial differential equations that combines the explicit and implicit schemes. The stability analysis is an important factor to consider when choosing a numerical method for solving partial differential equations, as numerical instability can cause inaccurate solutions. We show that the Crank-Nicolson scheme is unconditionally stable, meaning that it can be used for a wide range of parameters without being affected by numerical instability. Overall, the analysis and implementation presented in this work provide a framework for designing and analyzing numerical methods for solving partial differential equations using the Crank-Nicolson scheme. The stability analysis is crucial for ensuring the accuracy and reliability of numerical solutions of partial differential equations.

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Balamuralitharan, S., and . "MATLAB Programming of Nonlinear Equations of Ordinary Differential Equations and Partial Differential Equations." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 773. http://dx.doi.org/10.14419/ijet.v7i4.10.26114.

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My idea of this paper is to discuss the MATLAB program for various mathematical modeling in ordinary differential equations (ODEs) and partial differential equations (PDEs). Idea of this paper is very useful to research scholars, faculty members and all other fields like engineering and biology. Also we get easily to find the numerical solutions from this program.

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7

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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8

Zou, Guang-an. "Numerical solutions to time-fractional stochastic partial differential equations." Numerical Algorithms 82, no. 2 (November 5, 2018): 553–71. http://dx.doi.org/10.1007/s11075-018-0613-0.

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9

Secer, Aydin. "Sinc-Galerkin method for solving hyperbolic partial differential equations." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 8, no. 2 (July 24, 2018): 250–58. http://dx.doi.org/10.11121/ijocta.01.2018.00608.

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In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

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10

Wang, Zhigang, Xiaoting Liu, Lijun Su, and Baoyan Fang. "Numerical Solutions of Convective Diffusion Equations using Wavelet Collocation Method." Advances in Engineering Technology Research 1, no. 1 (May 17, 2022): 192. http://dx.doi.org/10.56028/aetr.1.1.192.

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Some partial differential equations appear in many application fields. Therefore, the discussion of numerical solutions of those partial differential equations using numerical methods becomes a valuable and important issue in numerical simulation. In numerical methods, the wavelet-collocation method has been frequently developed for solving PDEs, and the algorithm has yielded substantial results. However, theoretical research of the numerical solution has been rarely discussed yet. In this paper, the numerical solution of convective diffusion equations using the wavelet-collocation method is established, and its existence and uniqueness are derived.

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11

Shoyeb Ali Sayyed, Gosavi Ganesh Vishnu,. "Basis Function Approaches for Numerical Solutions of Nonlinear Partial Differential Equations." Mathematical Statistician and Engineering Applications 72, no. 2 (October 29, 2023): 106–21. http://dx.doi.org/10.17762/msea.v72i2.2809.

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PDEs (Partial Differential Dynamics) are essential within a number of fields of physics and mathematics because they give a mathematical model of numerous natural events. PDEs are the fundamental domains of application research. At the moment, substantial emphasis is placed upon creating accurate along with analytical answers for regressive PDEs. Numerous methods have been used recently to determine the accurate answers of complicated incomplete differential equations. We use these approaches in order to provide accurate answers for two regressive equations with partial differentials. The primary goal as well as motive for doing the suggested research is to illustrate the significance as well as usefulness of the relative quantification approach to Basic Unit Techniques of various nonlinear systems.

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12

Yazıcı, Muhammet, and Harun Selvitopi. "Numerical methods for the multiplicative partial differential equations." Open Mathematics 15, no. 1 (November 22, 2017): 1344–50. http://dx.doi.org/10.1515/math-2017-0113.

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Abstract We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.

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13

BENALI, ABDELKADER. "NUMERICAL RESOLUTION OF NON-LINEAR EQUATIONS." Journal of Science and Arts 23, no. 3 (September 30, 2023): 721–28. http://dx.doi.org/10.46939/j.sci.arts-23.3-a14.

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In this study, we have employed the highly significant hyperbolic tangent (tanh) method to conduct an in-depth analysis of nonlinear coupled KdV systems of partial differential equations. In comparison to existing sophisticated approaches, this proposed method yields more comprehensive exact solutions for traveling waves without requiring excessive additional effort. We have successfully applied this method to two examples drawn from the literature of nonlinear partial differential equation systems.

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14

Al-Smadi, Mohammed, Asad Freihat, Hammad Khalil, Shaher Momani, and Rahmat Ali Khan. "Numerical Multistep Approach for Solving Fractional Partial Differential Equations." International Journal of Computational Methods 14, no. 03 (April 13, 2017): 1750029. http://dx.doi.org/10.1142/s0219876217500293.

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In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep reduced differential transformation method (MRDTM), is a simple amendment of the reduced differential transformation method, in which it is treated as an algorithm in a sequence of small intervals, in order to hold out accurate approximate solutions over a longer time frame compared to the traditional RDTM. The fractional derivatives are described in the Caputo sense, while the behavior of solutions for different values of fractional order [Formula: see text] compared with exact solutions is shown graphically. The analysis is accompanied by four test examples to demonstrate that the proposed approach is reliable, fully compatible with the complexity of these equations, and can be strongly employed for many other nonlinear problems in fractional calculus.

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15

Ismail, Muhammad, Mujeeb ur Rehman, and Umer Saeed. "Green-Haar method for fractional partial differential equations." Engineering Computations 37, no. 4 (December 12, 2019): 1473–90. http://dx.doi.org/10.1108/ec-05-2019-0234.

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Purpose The purpose of this study is to obtain the numerical scheme of finding the numerical solutions of arbitrary order partial differential equations subject to the initial and boundary conditions. Design/methodology/approach The authors present a novel Green-Haar approach for the family of fractional partial differential equations. The method comprises a combination of Haar wavelet method with the Green function. To handle the nonlinear fractional partial differential equations the authors use Picard technique along with Green-Haar method. Findings The results for some numerical examples are documented in tabular and graphical form to elaborate on the efficiency and precision of the suggested method. The obtained results by proposed method are compared with the Haar wavelet method. The method is better than the conventional Haar wavelet method, for the tested problems, in terms of accuracy. Moreover, for the convergence of the proposed technique, inequality is derived in the context of error analysis. Practical implications The authors present numerical solutions for nonlinear Burger’s partial differential equations and two-term partial differential equations. Originality/value Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

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16

Günerhan, Hatıra, and Ercan Çelik. "Analytical and approximate solutions of Fractional Partial Differential-Algebraic Equations." Applied Mathematics and Nonlinear Sciences 5, no. 1 (March 30, 2020): 109–20. http://dx.doi.org/10.2478/amns.2020.1.00011.

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AbstractIn this paper, we have extended the Fractional Differential Transform method for the numerical solution of the system of fractional partial differential-algebraic equations. The system of partial differential-algebraic equations of fractional order is solved by the Fractional Differential Transform method. The results exhibit that the proposed method is very effective.

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17

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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18

Touchent, Kamal Ait, Zakia Hammouch, and Toufik Mekkaoui. "A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives." Applied Mathematics and Nonlinear Sciences 5, no. 2 (July 1, 2020): 35–48. http://dx.doi.org/10.2478/amns.2020.2.00012.

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AbstractIn this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.

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19

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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20

Mohammed, Mohammed, Mohammed Abed Daim .., and Ahmed Hadi Hussain. "LRPS Method for Solving Linear Partial Differential Equations and Neutrosophic Differential Equations of Fractional Order with Numerical Solutions." International Journal of Neutrosophic Science 24, no. 4 (2024): 257–67. http://dx.doi.org/10.54216/ijns.240419.

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In this work, fractional partial equations' and neutrosophic fractional partial equations analytical series solutions are presented, we consider the fractional derivative in the meaning of Caputo in these formulas. We offer a novel objective method the LRPS which is a strong instrument for precise analytically and numerical solutions to these problems by setting an excellent example, we stress precision, effectiveness, and application style, also we can find exact answers when there is a pattern between the series' parts; alternatively, we can only offer approximations. The Mathematica application is used to assess the numerical and graphical findings to make sure the solutions generated are accurate and that the approach can be modified to solve this kind of this problem. The findings obtained demonstrated that our current procedure is appropriate and efficient for resolving PDEs.

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21

Khatri Ghimire, B., H. Y. Tian, and A. R. Lamichhane. "Numerical solutions of elliptic partial differential equations using Chebyshev polynomials." Computers & Mathematics with Applications 72, no. 4 (August 2016): 1042–54. http://dx.doi.org/10.1016/j.camwa.2016.06.012.

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22

Liu, Nanshan, and En-Bing Lin. "Legendre wavelet method for numerical solutions of partial differential equations." Numerical Methods for Partial Differential Equations 26, no. 1 (January 2010): 81–94. http://dx.doi.org/10.1002/num.20417.

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23

Zhang, Wenbo, and Wei Gu. "Parameter Estimation for Several Types of Linear Partial Differential Equations Based on Gaussian Processes." Fractal and Fractional 6, no. 8 (August 8, 2022): 433. http://dx.doi.org/10.3390/fractalfract6080433.

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This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to solve the inverse problem, where we encode the distribution information of the data into the kernels and construct an efficient data learning machine. We then estimate the unknown parameters of the partial differential Equations (PDEs), which include high-order partial differential equations, partial integro-differential equations, fractional partial differential equations and a system of partial differential equations. Finally, several numerical tests are provided. The results of the numerical experiments prove that the data-driven methods based on Gaussian processes not only estimate the parameters of the considered PDEs with high accuracy but also approximate the latent solutions and the inhomogeneous terms of the PDEs simultaneously.

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24

Ghafoor, Abdul. "Numerical Solutions of Time Dependent Partial Differential Equations via HAAR Wavelets." Natural and Applied Sciences International Journal (NASIJ) 1, no. 1 (December 31, 2020): 39–52. http://dx.doi.org/10.47264/idea.nasij/1.1.4.

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An effective wavelet based scheme coupled with finite difference is used for the solution of two nonlinear time dependent problems namely: Burgers' and Boussinesq equations. These equations have wide-spread application in many fields such as viscous medium, turbulence , uid dynamics, infiltration phenomena etc. The proposed scheme convert the partial differential equations (PDE) to system of algebraic equations. The obtained system can be solved easily. In this paper convergence of the scheme is also discussed to show validity of the technique. Effectiveness of the scheme is shown with the help of test problems. Numerical results verify that the suggested scheme is more accurate, convenient, fast and require low computational cost.

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25

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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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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27

CHEN, C. S., SUNGWOOK LEE, and C. S. HUANG. "DERIVATION OF PARTICULAR SOLUTIONS USING CHEBYSHEV POLYNOMIAL BASED FUNCTIONS." International Journal of Computational Methods 04, no. 01 (March 2007): 15–32. http://dx.doi.org/10.1142/s0219876207001096.

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In this paper, we propose a simple and direct numerical procedure to obtain particular solutions for various types of differential equations. This procedure employs the power series expansion of a differential operator. Chebyshev polynomials are selected as basis functions for the approximation of the inhomogeneous terms of the given partial differential equation. This numerical scheme provides a highly efficient and accurate approximation for the evaluation of a particular solution for a variety of classes of partial differential equations. To demonstrate the effectiveness of the proposed scheme, we couple the method of fundamental solutions to solve a modified Helmholtz equation with irregular boundary configuration. The solutions were observed to have high accuracy.

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Wang, Jiao. "Polynomials for numerical solutions of space-time fractional differential equations (of the Fokker–Planck type)." Engineering Computations 36, no. 9 (November 11, 2019): 2996–3015. http://dx.doi.org/10.1108/ec-02-2019-0061.

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Purpose Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type. Design/methodology/approach The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations. Findings Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique. Originality/value Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

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Kadhim, Murtadha A., Allah Bakhsh Yazdani Cherati, and Mohammed Sahib Mechee. "The numerical solutions for a class of fifth-order partial differential equations via generalized RKM methods." Journal of Interdisciplinary Mathematics 27, no. 4 (2024): 729–35. http://dx.doi.org/10.47974/jim-1762.

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This paper’s RKM method is explicitly constructed for solving particular fifth-order ordinary differential equations (ODEs) [1]. The proposed method has been used to solve the system of fifth-order (ODEs) numerically by converting the fifth-order partial differential equations. The algorithm of the proposed method is introduced and then used to study the numerical problems, which showed that the approximated solutions using the proposed approach agree well with the exact solutions.

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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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Amaratunga, Kevin, John R. Williams, Sam Qian, and John Weiss. "Wavelet-Galerkin solutions for one-dimensional partial differential equations." International Journal for Numerical Methods in Engineering 37, no. 16 (August 30, 1994): 2703–16. http://dx.doi.org/10.1002/nme.1620371602.

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Iqbal, Sajad, Mohammed K. A. Kaabar, and Francisco Martínez. "A Novel Homotopy Perturbation Algorithm Using Laplace Transform for Conformable Partial Differential Equations." Mathematical Problems in Engineering 2021 (December 14, 2021): 1–13. http://dx.doi.org/10.1155/2021/2573067.

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In this article, the approximate analytical solutions of four different types of conformable partial differential equations are investigated. First, the conformable Laplace transform homotopy perturbation method is reformulated. Then, the approximate analytical solution of four types of conformable partial differential equations is presented via the proposed technique. To check the accuracy of the proposed technique, the numerical and exact solutions are compared with each other. From this comparison, we conclude that the proposed technique is very efficient and easy to apply to various types of conformable partial differential equations.

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Nur Aisyah Azeman, Siti, and . "Dual Solutions in the Boundary Layer Flow and Heat Transfer in the Presence of Thermal Radiation with Suction Effect." International Journal of Engineering & Technology 7, no. 4.33 (December 9, 2018): 17. http://dx.doi.org/10.14419/ijet.v7i4.33.23475.

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The dual solutions in the boundary layer flow and heat transfer in the presence of thermal radiation is quantitatively studied. The governing partial differential equations are derived into a system of ordinary differential equations using a similarity transformation, and afterward numerical solution obtained by a shooting technique. Dual solutions execute within a certain range of opposing and assisting flow which related to these numerical solutions. The similarity equations have two branches, upper or lower branch solutions, within a certain range of the mixed convection parameters. Further numerical results exist in our observations which enable to discuss the features of the respective solutions.

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Pyanylo, Yaroslav, and Galyna Pyanylo. "Analysis of approaches to mass-transfer modeling n non-stationary mode." Physico-mathematical modelling and informational technologies, no. 28, 29 (December 27, 2019): 55–64. http://dx.doi.org/10.15407/fmmit2020.28.055.

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A significant number of natural and physical processes are described by differential equations in partial derivatives or systems of differential equations in partial derivatives. Numerical methods have been found to find their solutions. Partial derivatives systems are solved mainly by reducing the order of the system of equations or reducing it to one differential equation. This procedure leads to an increase in the order of the differential equation. There are various restrictions and errors that can lead to additional solutions, boundary conditions for intermediate derivatives, and so on. The work is devoted to the analysis of such situations and ways of exit.

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Soomro, Inayatullah, Rabnawaz Mallah, Muhammad Javed Iqbal, Waqar Ahmed, and Abdul Ghafoor. "Discretization of Laplacian Operator on 19 Points Stencil Using Cylindrical Mesh System with the Help of Explicit Finite Difference Scheme." Pakistan Journal of Engineering, Technology & Science 11, no. 2 (December 15, 2023): 1–10. http://dx.doi.org/10.22555/pjets.v11i2.1019.

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Abstract. In research and technology, discretization is essential for describing and numerically assessing mathematical models. To provide promising numerical formulations with computationally efficient, mathematical models, which are frequently constructed using partial differential equations need to be discretized on curved meshes. Achieving isotropic discrete solutions is crucial for numerical evaluations of partial differential equations in many mathematical models, since it guarantees stability, correctness, and efficiency. This work revolves around the Laplacian operator, a fundamental mathematical operation in models such as Cell Dynamic simulations, Self-Consistent Field theory, and Image Processing. In this work, we utilize the explicit finite difference approach due of its correctness and computational ease. Using a cylindrical mesh structure, the Laplacian is discretized on a 19-point stencil with mixed derivatives. On cylindrical mesh systems, the numerical formulations developed here can be used to approximate solutions of partial differential equations of the first and second order. Significant isotropy, stability, and precision in computational findings for the Laplacian operator are guaranteed by the computational molecule proposed in this work. Potential applications of this formulation include the numerical analysis of different mathematical models employing the Laplacian operator.

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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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Patrício, Fernanda Simões, Miguel Patrício, and Higinio Ramos. "Extrapolating for attaining high precision solutions for fractional partial differential equations." Fractional Calculus and Applied Analysis 21, no. 6 (December 19, 2018): 1506–23. http://dx.doi.org/10.1515/fca-2018-0079.

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Abstract This paper aims at obtaining a high precision numerical approximation for fractional partial differential equations. This is achieved through appropriate discretizations: firstly we consider the application of shifted Legendre or Chebyshev polynomials to get a spatial discretization, followed by a temporal discretization where we use the Implicit Euler method (although any other temporal integrator could be used). Finally, the use of an extrapolation technique is considered for improving the numerical results. In this way a very accurate solution is obtained. An algorithm is presented, and numerical results are shown to demonstrate the validity of the present technique.

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Motsa, S. S., F. G. Awad, Z. G. Makukula, and P. Sibanda. "The Spectral Homotopy Analysis Method Extended to Systems of Partial Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/241594.

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The spectral homotopy analysis method is extended to solutions of systems of nonlinear partial differential equations. The SHAM has previously been successfully used to find solutions of nonlinear ordinary differential equations. We solve the nonlinear system of partial differential equations that model the unsteady nonlinear convective flow caused by an impulsively stretching sheet. The numerical results generated using the spectral homotopy analysis method were compared with those found using the spectral quasilinearisation method (SQLM) and the two results were in good agreement.

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Jeong, Darae, Seungsuk Seo, Hyeongseok Hwang, Dongsun Lee, Yongho Choi, and Junseok Kim. "Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations." Discrete Dynamics in Nature and Society 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/359028.

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We briefly review and investigate the performance of various boundary conditions such as Dirichlet, Neumann, linear, and partial differential equation boundary conditions for the numerical solutions of the Black-Scholes partial differential equation. We use a finite difference method to numerically solve the equation. To show the efficiency of the given boundary condition, several numerical examples are presented. In numerical test, we investigate the effect of the domain sizes and compare the effect of various boundary conditions with pointwise error and root mean square error. Numerical results show that linear boundary condition is accurate and efficient among the other boundary conditions.

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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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Guo, Yanan, Xiaoqun Cao, Bainian Liu, and Mei Gao. "Solving Partial Differential Equations Using Deep Learning and Physical Constraints." Applied Sciences 10, no. 17 (August 26, 2020): 5917. http://dx.doi.org/10.3390/app10175917.

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The various studies of partial differential equations (PDEs) are hot topics of mathematical research. Among them, solving PDEs is a very important and difficult task. Since many partial differential equations do not have analytical solutions, numerical methods are widely used to solve PDEs. Although numerical methods have been widely used with good performance, researchers are still searching for new methods for solving partial differential equations. In recent years, deep learning has achieved great success in many fields, such as image classification and natural language processing. Studies have shown that deep neural networks have powerful function-fitting capabilities and have great potential in the study of partial differential equations. In this paper, we introduce an improved Physics Informed Neural Network (PINN) for solving partial differential equations. PINN takes the physical information that is contained in partial differential equations as a regularization term, which improves the performance of neural networks. In this study, we use the method to study the wave equation, the KdV–Burgers equation, and the KdV equation. The experimental results show that PINN is effective in solving partial differential equations and deserves further research.

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Awati, Vishwanath Basavaraj. "Approximate analytical solutions of MHD viscous flow." Journal of Naval Architecture and Marine Engineering 13, no. 1 (June 15, 2016): 79–87. http://dx.doi.org/10.3329/jname.v13i1.24387.

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The paper presents the semi-numerical solution for the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet caused by boundary layer of an incompressible viscous flow. The governing three partial differential equations of momentum equations are reduced into ordinary differential equation (ODE) by using a classical similarity transformation along with appropriate boundary conditions. Both nonlinearity and infinite interval demand novel mathematical tools for their analysis. We use fast converging Dirichlet series and Method of stretching of variables for the solution of these nonlinear differential equations. These methods have the advantages over pure numerical methods for obtaining the derived quantities accurately for various values of the parameters involved at a stretch and also they are valid in much larger parameter domain as compared with HAM, HPM, ADM and the classical numerical schemes.

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Alqurashi, Turkia Dhawi. "Construction Solutions of Ordinary and Partial Differential Equations using the Analytical and Numerical Methods." Academic Journal of Research and Scientific Publishing 3, no. 33 (January 5, 2022): 103–20. http://dx.doi.org/10.52132/ajrsp.e.2022.33.5.

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In this paper we studied the solution of partial differential equations using numerical methods. The paper includes study of the solving partial differential equations of the type of parabolic, elliptic and hyperbolic, and the method of the net was used for the numerical nods, which represents a case of finite differences. We have two types of solution which are the internal solution and boundary solution. The internal solution is based on the internal nodes of the net. The boundary solution depends on the boundary nodes of the net, in addition to finding the analytical solution of the equations to compare the results. We also discussed solving the problem of Laplace, Poisson, for the importance of these equations in the applied side; Mat lab was used to find the values of tables for the values of border differences. We have derived a new formula for the solution of partial differential equations containing three independent variables.

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Ellerby, F. B., and C. Johnson. "Numerical Solutions of Partial Differential Equations by the Finite Element Method." Mathematical Gazette 73, no. 463 (March 1989): 59. http://dx.doi.org/10.2307/3618226.

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Taşbozan, Orkun, and Gizem Bayaslı. "Numerical Solutions of Conformable Partial Differential Equations By Homotopy Analysis Method." Afyon Kocatepe University Journal of Sciences and Engineering 18, no. 3 (December 1, 2018): 842–51. http://dx.doi.org/10.5578/fmbd.67600.

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Liu, Tongjing, Pengxiang Diwu, Rui Liu, Liwu Jiang, and Baoyi Jiang. "Fast Algorithm of Numerical Solutions for Strong Nonlinear Partial Differential Equations." Advances in Mechanical Engineering 6 (January 1, 2014): 936490. http://dx.doi.org/10.1155/2014/936490.

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Because of a high mobility ratio in the chemical and gas flooding for oil reservoirs, the problems of numerical dispersion and low calculation efficiency also exist in the common methods, such as IMPES and adaptive implicit methods. Therefore, the original calculation process, “one-step calculation for pressure and multistep calculation for saturation,” was improved by introducing a velocity item and using the fractional flow in a direction to calculate the saturation. Based on these developments, a new algorithm of numerical solution for “one-step calculation for pressure, one-step calculation for velocity, and multi-step calculation for fractional flow and saturation” was obtained, and the convergence condition for the calculation of saturation was also proposed. The simulation result of a typical theoretical model shows that the nonconvergence occurred for about 6,000 times in the conventional algorithm of IMPES, and a high fluctuation was observed in the calculation steps. However, the calculation step of the fast algorithm was stabilized for 0.5 d, indicating that the fast algorithm can avoid the nonconvergence caused by the saturation that was directly calculated by pressure. This has an important reference value in the numerical simulations of chemical and gas flooding for oil reservoirs.

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Nakao, Mitsuhiro T. "NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS." Numerical Functional Analysis and Optimization 22, no. 3-4 (June 30, 2001): 321–56. http://dx.doi.org/10.1081/nfa-100105107.

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Li, Yongjin, and Kamal Shah. "Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations." Advances in Mathematical Physics 2017 (2017): 1–14. http://dx.doi.org/10.1155/2017/1535826.

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We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

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Liu, Xinzhi, Yau Shu Wong, and Ji Xingzhi. "Monotone iterations for numerical solutions of nonlinear elliptic partial differential equations." Applied Mathematics and Computation 50, no. 1 (July 1992): 59–91. http://dx.doi.org/10.1016/0096-3003(92)90012-p.

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Zhu, Hongqing, Huazhong Shu, and Meiyu Ding. "Numerical solutions of partial differential equations by discrete homotopy analysis method." Applied Mathematics and Computation 216, no. 12 (August 2010): 3592–605. http://dx.doi.org/10.1016/j.amc.2010.05.005.

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