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An Introduction to Numerical Methods for the Solutions of Partial Differential Equations 被引量:1
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作者 Manoj Kumar Garima Mishra 《Applied Mathematics》 2011年第11期1327-1338,共12页
Partial differential equations arise in formulations of problems involving functions of several variables such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity, etc. The... Partial differential equations arise in formulations of problems involving functions of several variables such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity, etc. The present paper deals with a general introduction and classification of partial differential equations and the numerical methods available in the literature for the solution of partial differential equations. 展开更多
关键词 partial differential equationS EIGENVALUE FINITE Difference method FINITE Volume method FINITE Element method
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Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems
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作者 Khalid Abd Elrazig Awad Alla Elnour 《Journal of Applied Mathematics and Physics》 2024年第3期877-896,共20页
To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’... To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section. 展开更多
关键词 First-Order differential equations Picard method Taylor Series method numerical Solutions numerical Examples MATLAB Software
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Method of Lines for Third Order Partial Differential Equations 被引量:2
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作者 Mustafa Kudu Ilhame Amirali 《Journal of Applied Mathematics and Physics》 2014年第2期33-36,共4页
The method of lines is applied to the boundary-value problem for third order partial differential equation. Explicit expression and order of convergence for the approximate solution are obtained.
关键词 method of LINES partial differential equation CONVERGENCE Error ESTIMATES
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Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics 被引量:1
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作者 杜明婧 孙宝军 凯歌 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第3期53-57,共5页
This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)metho... This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)method which deals with this problem is very troublesome.This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel(AMPIRK)method for the first time.This method has three obvious advantages which are as follows.Firstly,the piecewise number is reduced.Secondly,the calculation accuracy is improved.Finally,the waste time caused by too many fragments is avoided.Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others.The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics. 展开更多
关键词 time-fractional partial differential equation adaptive multi-step reproducing kernel method method numerical solution
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THE NUMERICAL STABILITY OF THE BLOCK θ-METHODS FOR DELAY DIFFERENTIAL EQUATIONS 被引量:1
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作者 田红炯 匡蛟勋 《Numerical Mathematics A Journal of Chinese Universities(English Series)》 SCIE 2001年第1期1-8,共8页
This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the nume... This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the numerical processes that satisfy an important asymptotic stability condition related to the class of test problems y′(t)=ay(t)+by(t-τ) with a,b∈C, Re(a)<-|b| and τ>0. We prove that the block θ method is GP stable if and only if the method is A stable for ordinary differential equations. Furthermore, it is proved that the P and GP stability are equivalent for the block θ method. 展开更多
关键词 numerical stability block θ methods delay differential equations.
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Mixture of a New Integral Transform and Homotopy Perturbation Method for Solving Nonlinear Partial Differential Equations 被引量:1
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作者 Artion Kashuri Akli Fundo Matilda Kreku 《Advances in Pure Mathematics》 2013年第3期317-323,共7页
In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomi... In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2]. 展开更多
关键词 HOMOTOPY PERTURBATION methods A NEW INTEGRAL Transform Nonlinear partial differential equations He’s POLYNOMIALS
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Crank-Nicolson ADI Galerkin Finite Element Methods for Two Classes of Riesz Space Fractional Partial Differential Equations 被引量:1
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作者 An Chen 《Computer Modeling in Engineering & Sciences》 SCIE EI 2020年第6期917-939,共23页
In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the... In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis. 展开更多
关键词 Fractional partial differential equations Galerkin approximation alternating direction implicit method STABILITY CONVERGENCE
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Application of the Adomian Decomposition Method (ADM) for Solving the Singular Fourth-Order Parabolic Partial Differential Equation 被引量:1
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作者 Béyi Boukary Justin Loufouilou-Mouyedo +1 位作者 Joseph Bonazebi-Yindoula Gabriel Bissanga 《Journal of Applied Mathematics and Physics》 2018年第7期1476-1480,共5页
In this paper, the ADM method is used to construct the solution of the singular fourth-order partial differential equation.
关键词 SBA method SINGULAR FOURTH-ORDER partial differential equation
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The Adomian Decomposition Method for Solving Nonlinear Partial Differential Equation Using Maple 被引量:1
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作者 Dalal Adnan Maturi Honaida Mohammed Malaikah 《Advances in Pure Mathematics》 2021年第6期595-603,共9页
The nonlinear partial differential equation is solved using the Adomian decomposition method (ADM) in this article. A number of examples have been provided to illustrate the numerical results, which is the comparison ... The nonlinear partial differential equation is solved using the Adomian decomposition method (ADM) in this article. A number of examples have been provided to illustrate the numerical results, which is the comparison of the exact and numerical solutions, and it has been discovered through the tables that the amount of error between the exact and numerical solutions is very small and almost non-existent, and the graph also shows how the exact solution of absolutely applies to the numerical solution. This demonstrates the precision of the Adomian decomposition method (ADM) for solving the nonlinear partial differential equation with Maple18. And that in terms of obtaining numerical results, this approach is characterized by ease, speed, and high accuracy. 展开更多
关键词 Nonlinear partial differential equation Adomian Decomposition method Maple18
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Double Elzaki Transform Decomposition Method for Solving Non-Linear Partial Differential Equations 被引量:1
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作者 Moh A. Hassan Tarig M. Elzaki 《Journal of Applied Mathematics and Physics》 2020年第8期1463-1471,共9页
In this paper, we discuss a new method employed to tackle non-linear partial differential equations, namely Double Elzaki Transform Decomposition Method (DETDM). This method is a combination of the Double ELzaki Trans... In this paper, we discuss a new method employed to tackle non-linear partial differential equations, namely Double Elzaki Transform Decomposition Method (DETDM). This method is a combination of the Double ELzaki Transform and Adomian Decomposition Method. This technique is hereafter provided and supported with necessary illustrations, together with some attached examples. The results reveal that the new method is very efficient, simple and can be applied to other non-linear problems. 展开更多
关键词 Double Elzaki Transform Adomian Decomposition method Non-Linear partial differential equations
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On L ∞ Stability and Convergence of Fictitious Domain Method for the Numerical Solution to Parabolic Differential Equation with Derivative Boundary Conditions
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作者 孙志忠 《Journal of Southeast University(English Edition)》 EI CAS 1996年第2期108-111,共4页
This paper investigates some known difference schemes for the numerical solution to parabolic differential equation with derivative boundary conditions by the fictitious domain method.The stability and convergence in... This paper investigates some known difference schemes for the numerical solution to parabolic differential equation with derivative boundary conditions by the fictitious domain method.The stability and convergence in L ∞ are proven. 展开更多
关键词 numerical solution fictitious domain method PARABOLIC differential equation DERIVATIVE boundary condition
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A SOLVING METHOD FOR A SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS WITH AN APPLICATION TO THE BENDING PROBLEM OF A THICK PLATE
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作者 尹益辉 陈刚 陈裕泽 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1999年第11期1259-1265,共7页
A theorem of solving a system of linear non-homogeneous differential equations through integrating and adding its basic solutions is put forward and proved, the mathematical role and physical nature of the theorem is ... A theorem of solving a system of linear non-homogeneous differential equations through integrating and adding its basic solutions is put forward and proved, the mathematical role and physical nature of the theorem is interpreted briefly. As an example, the theorem is applied to solve the problem of thermo-force bending of a thick plate. 展开更多
关键词 partial differential equations integrating method thick plate thermo-force bending
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Numerical Solution for Fractional Partial Differential Equation with Bernstein Polynomials
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作者 Jin-Sheng Wang Li-Qing Liu +1 位作者 Yi-Ming Chen Xiao-Hong Ke 《Journal of Electronic Science and Technology》 CAS 2014年第3期331-338,共8页
A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational ma... A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient. 展开更多
关键词 Absolute error Bernstein polynomials fractional partial differential equation numerical solution operational matrix
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Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions
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作者 徐承龙 郭本瑜 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2008年第3期311-331,共21页
The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are est... The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches. 展开更多
关键词 modified Laguerre orthogonal approximation and interpolation multiple dimensions spectral and pseudospectral methods nonlinear partial differential equations
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Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations
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作者 Ge Hon-Xia Liu Yong-Qing Cheng Rong-Jun 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第1期46-51,共6页
The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin (EFG) method, which is based on the moving least-square approximation. Compared w... The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin (EFG) method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α(0 〈 α≤ 1). The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions. 展开更多
关键词 element-free Galerkin (EFG) method meshless method time fractional partial differential equations
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The Multi-scale Method for Solving Nonlinear Time Space Fractional Partial Differential Equations
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作者 Hossein Aminikhah Mahdieh Tahmasebi Mahmoud Mohammadi Roozbahani 《IEEE/CAA Journal of Automatica Sinica》 EI CSCD 2019年第1期299-306,共8页
In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these fu... In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these functions are reshaped to satisfy on boundary conditions exactly. The Adams fractional method is used to reduce the problem to a system of equations. By multiscale method this system is divided into some smaller systems which have less computations. We get an approximated solution which is more accurate on some subdomains by combining the solutions of these systems. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique, also the stability of the method is discussed. 展开更多
关键词 Adams FRACTIONAL method B-SPLINE WAVELETS MULTI-SCALE method nonlinear FRACTIONAL partial differential equations
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Extended Mapping Transformation Method and Its Applications to Nonlinear Partial Differential Equation(s)
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作者 ZHAO Hong BAI Cheng-Lin 《Communications in Theoretical Physics》 SCIE CAS CSCD 2005年第3X期473-478,共6页
In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary fun... In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained. 展开更多
关键词 nonlinear partial differential equations extended mapping transformation method exact solutions
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Adaptive Single Piecewise Interpolation Reproducing Kernel Method for Solving Fractional Partial Differential Equation
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作者 DU Mingjing 《Journal of Donghua University(English Edition)》 CAS 2022年第5期454-460,共7页
It is well-known that using the traditional reproducing kernel method(TRKM) for solving the fractional partial differential equation(FPDE) is very intractable. In this study, the adaptive single piecewise interpolatio... It is well-known that using the traditional reproducing kernel method(TRKM) for solving the fractional partial differential equation(FPDE) is very intractable. In this study, the adaptive single piecewise interpolation reproducing kernel method(ASPIRKM) is determined to solve the FPDE. This improved method not only improves the calculation accuracy, but also reduces the waste of time. Two numerical examples show that the ASPIRKM is a more time-saving numerical method than the TRKM. 展开更多
关键词 fractional partial differential equation(FPDE) reproducing kernel method(RKM) single piecewise numerical solution
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Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace Decomposition Method and the Pade Approximation
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作者 Magdy Ahmed Mohamed Mohamed Shibl Torky 《American Journal of Computational Mathematics》 2013年第3期175-184,共10页
In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and ... In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions. 展开更多
关键词 NONLINEAR SYSTEM of partial differential equationS The LAPLACE Decomposition method The Pade Approximation The COUPLED SYSTEM of the Approximate equationS for Long WATER Waves The Whitham Broer Kaup Shallow WATER Model The SYSTEM of Hirota-Satsuma COUPLED KdV
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A Uniformly Convergent Numerical Method Using Weak Formulation for Singularly Perturbed Differential Equations
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作者 Weiqun Zhang 《Journal of Mathematics and System Science》 2019年第1期1-6,共6页
A numerical method using weak formulation is proposed to solve singularly perturbed differential equations. The numerical method is applied to both linear and nonlinear perturbation problems. A linear differential equ... A numerical method using weak formulation is proposed to solve singularly perturbed differential equations. The numerical method is applied to both linear and nonlinear perturbation problems. A linear differential equation is solved using its weak formulation with a test space composed of exponential functions matching boundary layers. A nonlinear singular perturbation problem is converted into a system of linear differentiation equations. Then each linear differential equation is solved iteratively. The uniform convergence, which is independent of the singular perturbation parameter, is numerically verified. 展开更多
关键词 SINGULAR PERTURBATION differential equations boundary layers numerical methods WEAK formulation
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