Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions
Mahmoud Hamed Taha,
Mohamed Abdel-Latif Ramadan,
Galal Mahrous Moatimid
Issue:
Volume 6, Issue 2, June 2021
Pages:
19-34
Received:
6 March 2021
Accepted:
22 March 2021
Published:
26 May 2021
Abstract: This paper is concerned with a numerical method based on the improved block-pulse basis functions (IBPFs). It is done mainly to solve linear and nonlinear Volterra and Fredholm integral equations of the second kind. These equations can be simplified into a linear system of algebraic equations by using IBPFs and their operational matrix of integration. After that, the system can be programmed and solved using Mathematica. The changes made to the method obviously improved - as it will be shown in the numerical examples - the time taken by the program to solve the system of algebraic equations. Also, it is reflected in the accuracy of the solution. This modification works perfectly and improved the accuracy over the regular block–pulse basis functions (BPF). A slight change in the intervals of the BPF changes the whole technique to a new easier and more accurate technique. This change has worked well while solving different types of integral equations. The accompanied theorems of the IBPF technique and error estimation are stated and proved. The paper also dealt with the uniqueness and convergence theorems of the solution. Numerical examples are presented to illustrate the efficiency and accuracy of the method. The tables and required graphs are also shown to prove and demonstrate the efficiency.
Abstract: This paper is concerned with a numerical method based on the improved block-pulse basis functions (IBPFs). It is done mainly to solve linear and nonlinear Volterra and Fredholm integral equations of the second kind. These equations can be simplified into a linear system of algebraic equations by using IBPFs and their operational matrix of integrati...
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Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation
Kedir Aliy,
Alemayehu Shiferaw,
Hailu Muleta
Issue:
Volume 6, Issue 2, June 2021
Pages:
35-42
Received:
28 March 2021
Accepted:
11 May 2021
Published:
26 May 2021
Abstract: In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E2), maximum absolute error (E∞) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.
Abstract: In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients t...
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