scholarly journals Solving Operator Equation Based on Expansion Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
A. Aminataei ◽  
S. Ahmadi-Asl ◽  
M. Pakbaz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Pseudospectral Method ◽  
Polynomial Basis ◽  
Operator Equations ◽  
Partial Differential ◽  
General Operator ◽  
Spectral And Pseudospectral Methods ◽  
Boundary Operators ◽  
General Basis

To date, researchers usually use spectral and pseudospectral methods for only numerical approximation of ordinary and partial differential equations and also based on polynomial basis. But the principal importance of this paper is to develop the expansion approach based on general basis functions (in particular case polynomial basis) for solving general operator equations, wherein the particular cases of our development are integral equations, ordinary differential equations, difference equations, partial differential equations, and fractional differential equations. In other words, this paper presents the expansion approach for solving general operator equations in the form Lu+Nu=g(x),x∈Γ, with respect to boundary condition Bu=λ, where L, N and B are linear, nonlinear, and boundary operators, respectively, related to a suitable Hilbert space, Γ is the domain of approximation, λ is an arbitrary constant, and g(x)∈L2(Γ) is an arbitrary function. Also the other importance of this paper is to introduce the general version of pseudospectral method based on general interpolation problem. Finally some experiments show the accuracy of our development and the error analysis is presented in L2(Γ) norm.

scholarly journals A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 270
Author(s):  
Cheng-Yu Ku ◽  
Jing-En Xiao ◽  
Chih-Yu Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Radial Basis Function ◽  
Collocation Method ◽  
Basis Function ◽  
Numerical Solutions ◽  
Polynomial Basis ◽  
Partial Differential ◽  
Minimum Order ◽  
Radial Basis

In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

scholarly journals Numerical simulation method of lines together with a pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mushtaq Ali ◽  
Mohammed Almoaeet ◽  
Basim Karim Albuohimad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Ordinary Differential Equations ◽  
Method Of Lines ◽  
Pseudospectral Method ◽  
Partial Differential ◽  
Content Type ◽  
Jacobi Functions ◽  
Left And Right

PurposeThis study aims to use new formula derived based on the shifted Jacobi functions have been defined and some theorems of the left- and right-sided fractional derivative for them have been presented.Design/methodology/approachIn this article, the authors apply the method of lines (MOL) together with the pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative (SFPDEs). Then, using the collocation nodes to reduce the SFPDEs to the system of ordinary differential equations, which can be solved by the ode45 MATLAB toolbox.FindingsApplying the MOL method together with the pseudospectral discretization method converts the space-dependent on fractional partial differential equations to the system of ordinary differential equations.Originality/valueThis paper contributes to gain choosing the shifted Jacobi functions basis with special parameters a, b and give the authors this opportunity to obtain the left- and right-sided fractional differentiation matrices for this basis exactly. The results of the examples are presented in this article. The authors found that the method is efficient and provides accurate results, and the authors found significant implications for success in the science, technology, engineering and mathematics domain.

A Multi-Domain Bivariate Pseudospectral Method for Evolution Equations

2017 ◽  
Vol 14 (04) ◽  
pp. 1750041 ◽  
Author(s):  
V. M. Magagula ◽  
S. S. Motsa ◽  
P. Sibanda
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Pseudospectral Method ◽  
Analytic Solutions ◽  
Partial Differential ◽  
Various Boundary Conditions

In this paper, we present a new general approach for solving nonlinear evolution partial differential equations. The novelty of the approach is in the combination of spectral collocation and Lagrange interpolation polynomials with Legendre–Gauss–Lobatto grid points to descritize and solve equations in piece-wise defined intervals. The method is used to solve several nonlinear evolution partial differential equations, namely, the modified KdV–Burgers equation, modified KdV equation, Fisher’s equation, Burgers–Fisher equation, Burgers–Huxley equation and the Fitzhugh–Nagumo equation. The results are compared with known analytic solutions to confirm accuracy, convergence and to get a general understanding of the performance of the method. In all the numerical experiments, we report a high degree of accuracy of the numerical solutions. Strategies for implementing various boundary conditions are discussed.

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