Since various problems in science and engineering fields can be modeled by nonlinear Volterra-Fre... more Since various problems in science and engineering fields can be modeled by nonlinear Volterra-Fredholm integral equations, the main focus of this study is to present an eective numerical method for solving them. This method is based on the hybrid functions of Legen- dre polynomials and block-pulse functions. By using this approach, a nonlinear Volterra-Fredholm integral equation reduces to a nonlinear system of mere algebraic equations. The convergence analysis and as- sociated theorems are also considered. Test problems are provided to illustrate its accuracy and computational eciency.
Computers & Mathematics with Applications, 2010
Two-dimensional orthogonal triangular functions (2D-TFs) are presented as a new set of basis func... more Two-dimensional orthogonal triangular functions (2D-TFs) are presented as a new set of basis functions for expanding 2D functions. Their properties are determined and an operational matrix for integration obtained. Furthermore, 2D-TFs are used to approximate solutions of nonlinear two-dimensional integral equations by a direct method. Since this approach does not need integration, all calculations can be easily implemented, and several advantages in reducing computational burdens arise. Finally, the efficiency of this method will be shown by comparison with some numerical results.
In the present paper, delta functions (DFs) are proposed as a new set of basis functions. Their p... more In the present paper, delta functions (DFs) are proposed as a new set of basis functions. Their properties and relations to well-known triangular functions (TFs) are described. The simplicity and useful properties of newly proposed sets led us to use them with more accuracy and less computational burden. Furthermore, DFs are applied to propose an efficient method for approximating the solution of integral equations systems. Convergence analysis and the rate of convergence have been considered as well. Some numerical examples are provided to illustrate the computational efficiency and accuracy of the method.
Communications in Nonlinear Science and Numerical Simulation, 2010
A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed... more A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of nonlinear Volterra-Fredholm integral equations. The orthogonal triangular functions are utilized as a basis in collocation method to reduce the solution of nonlinear Volterra-Fredholm integral equations to the solution of algebraic equations. Also a theorem is proved for convergence analysis. Some numerical examples illustrate the proposed method.
In this paper, we present a Taylor-series expansion method for a class of Volterra integral equat... more In this paper, we present a Taylor-series expansion method for a class of Volterra integral equations of second kind with smooth or weakly singular kernels. This method use Taylor-series approximation method for integral equation and transform the integral equation to an nth order, linear differential equation. Boundary conditions for differential equation produce in easy way. This method gives an approximate
In this paper, a numerical approach based on an m-set of general, orthogonal triangular functions... more In this paper, a numerical approach based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of Fredholm integral equations of the first kind. By using the orthogonal triangular functions as a basis in Galerkin method, the solution of linear integral equations reduces to a system of algebric equations. If the recent system become ill-conditioned then we will use the preconditioned technique to convert above problem to well-conditioned. The convergence of the proposed method is established. Some numerical examples illustrate the proposed approach.
Beni-Suef University Journal of Basic and Applied Sciences, 2015
A combination of bivariate Chebyshev polynomials and two-dimensional block-pulse functions are in... more A combination of bivariate Chebyshev polynomials and two-dimensional block-pulse functions are introduced and applied for approximating the numerical solution of twodimensional Fredholm integral equations. All calculations in this approach would be easily implemented. The method has the advantage of reducing computational burden. The convergence analysis is given. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the proposed method.
Since various problems in science and engineering fields can be modeled by nonlinear Volterra-Fre... more Since various problems in science and engineering fields can be modeled by nonlinear Volterra-Fredholm integral equations, the main focus of this study is to present an eective numerical method for solving them. This method is based on the hybrid functions of Legen- dre polynomials and block-pulse functions. By using this approach, a nonlinear Volterra-Fredholm integral equation reduces to a nonlinear system of mere algebraic equations. The convergence analysis and as- sociated theorems are also considered. Test problems are provided to illustrate its accuracy and computational eciency.
Computers & Mathematics with Applications, 2010
Two-dimensional orthogonal triangular functions (2D-TFs) are presented as a new set of basis func... more Two-dimensional orthogonal triangular functions (2D-TFs) are presented as a new set of basis functions for expanding 2D functions. Their properties are determined and an operational matrix for integration obtained. Furthermore, 2D-TFs are used to approximate solutions of nonlinear two-dimensional integral equations by a direct method. Since this approach does not need integration, all calculations can be easily implemented, and several advantages in reducing computational burdens arise. Finally, the efficiency of this method will be shown by comparison with some numerical results.
In the present paper, delta functions (DFs) are proposed as a new set of basis functions. Their p... more In the present paper, delta functions (DFs) are proposed as a new set of basis functions. Their properties and relations to well-known triangular functions (TFs) are described. The simplicity and useful properties of newly proposed sets led us to use them with more accuracy and less computational burden. Furthermore, DFs are applied to propose an efficient method for approximating the solution of integral equations systems. Convergence analysis and the rate of convergence have been considered as well. Some numerical examples are provided to illustrate the computational efficiency and accuracy of the method.
Communications in Nonlinear Science and Numerical Simulation, 2010
A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed... more A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of nonlinear Volterra-Fredholm integral equations. The orthogonal triangular functions are utilized as a basis in collocation method to reduce the solution of nonlinear Volterra-Fredholm integral equations to the solution of algebraic equations. Also a theorem is proved for convergence analysis. Some numerical examples illustrate the proposed method.
In this paper, we present a Taylor-series expansion method for a class of Volterra integral equat... more In this paper, we present a Taylor-series expansion method for a class of Volterra integral equations of second kind with smooth or weakly singular kernels. This method use Taylor-series approximation method for integral equation and transform the integral equation to an nth order, linear differential equation. Boundary conditions for differential equation produce in easy way. This method gives an approximate
In this paper, a numerical approach based on an m-set of general, orthogonal triangular functions... more In this paper, a numerical approach based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of Fredholm integral equations of the first kind. By using the orthogonal triangular functions as a basis in Galerkin method, the solution of linear integral equations reduces to a system of algebric equations. If the recent system become ill-conditioned then we will use the preconditioned technique to convert above problem to well-conditioned. The convergence of the proposed method is established. Some numerical examples illustrate the proposed approach.
Beni-Suef University Journal of Basic and Applied Sciences, 2015
A combination of bivariate Chebyshev polynomials and two-dimensional block-pulse functions are in... more A combination of bivariate Chebyshev polynomials and two-dimensional block-pulse functions are introduced and applied for approximating the numerical solution of twodimensional Fredholm integral equations. All calculations in this approach would be easily implemented. The method has the advantage of reducing computational burden. The convergence analysis is given. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the proposed method.
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Papers by Masood Roodaki