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ABSTRACT In the numerical solution of linear Volterra integral equations, two kinds of errors occur. If we use the collocation method, these errors are the collocation and numerical quadrature errors. Each error has its own effect in the... more
ABSTRACT In the numerical solution of linear Volterra integral equations, two kinds of errors occur. If we use the collocation method, these errors are the collocation and numerical quadrature errors. Each error has its own effect in the accuracy of the obtained numerical solution. In this study we obtain an error bound that is sum of these two errors and using this error bound the relation between the smoothness of the kernel in the equation and also the length of the integration interval and each of these two errors are considered. Concluded results also are observed during the solution of some numerical examples.
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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... 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.
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Research Interests: Applied Mathematics, Numerical Analysis, Quadrature, Eigenvalues, Numerical Integration, and 9 moreNumerical Analysis and Computational Mathematics, Eigenvalue Problems, Preconditioning, Toeplitz matrix, Integral Equation, Fredholm Integral Equation Second Kind, Numerical Solution, Singular Integral Equation, and Condition number
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ABSTRACT We consider a stochastic linear heat conduction problem so, it is reduced to a special integral equation of the second kind. Our aim, in this paper is to give stable numerical algorithms for approximating solution of integral... more
ABSTRACT We consider a stochastic linear heat conduction problem so, it is reduced to a special integral equation of the second kind. Our aim, in this paper is to give stable numerical algorithms for approximating solution of integral equation and heat conduction.
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ABSTRACT In this paper the operational matrix of triangular functions for fractional order integration in the Caputo sense is derived. Also, the operational matrix of fractional integration is applied for solving multi-order fractional... more
ABSTRACT In this paper the operational matrix of triangular functions for fractional order integration in the Caputo sense is derived. Also, the operational matrix of fractional integration is applied for solving multi-order fractional differential equations, Abel’s integral equations and nonlinear integro-differential equations. This technique is a successful method because of reducing such problems to solve a system of algebraic equations; so, the problem can be solved directly. The advantage of this method is low cost of setting up the equations . Illustrative examples demonstrate accuracy and efficiency of the method.
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Abstract: The Adomian decomposition method is used and applied to the mathematical model of a biosensor. This model consists of a heat equation with non-linear and non-local boundary conditions. To obtain a canonical form of Adomian, an... more
Abstract: The Adomian decomposition method is used and applied to the mathematical model of a biosensor. This model consists of a heat equation with non-linear and non-local boundary conditions. To obtain a canonical form of Adomian, an equivalent non-linear Volterra integral equation with a weakly singular kernel is set up. In addition, the asymptotic behaviour of the solution as t? 0 and t?• by asymptotic decomposition is obtained. Finally, numerical results are given which support the theoretical results.
ABSTRACT In this paper we apply the fixed point method to solve some nonlinear functional Volterra integral equations which appear in many physical, chemical, and biological problems. In each iteration of this method, cubic... more
ABSTRACT In this paper we apply the fixed point method to solve some nonlinear functional Volterra integral equations which appear in many physical, chemical, and biological problems. In each iteration of this method, cubic semi-orthogonal compactly supported B-spline wavelets are used as basis functions to approximate the solution. Also, the convergence of this numerical method is investigated and some examples are presented to show the accuracy and convergence of the method.
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In this article, we use improved operational matrix of block pulse functions on interval [0, 1) to solve Volterra integral and integro-differential equations of convolution type without solving any system and projection method. We first... more
In this article, we use improved operational matrix of block pulse functions on interval [0, 1) to solve Volterra integral and integro-differential equations of convolution type without solving any system and projection method. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by improved operational matrix of integration. Numerical examples show that the
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International Journal of Computer Mathematics Vol. 85, No. 1, January 2008, 143154 ... Numerical solution of Urysohn integral equations using the iterated collocation method ... KHOSROW MALEKNEJAD*, HESAMODDIN DERILI§ and SAEED SOHRABI¶
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ABSTRACT In this paper, we obtain stochastic operational matrix of block pulse functions on interval [0,1)[0,1) to solve stochastic Volterra–Fredholm integral equations. By using block pulse functions and their stochastic operational... more
ABSTRACT In this paper, we obtain stochastic operational matrix of block pulse functions on interval [0,1)[0,1) to solve stochastic Volterra–Fredholm integral equations. By using block pulse functions and their stochastic operational matrix of integration, the stochastic Volterra–Fredholm integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution. We prove that the rate of convergence is O(h)O(h). Furthermore, the results show that the approximate solutions have a good degree of accuracy.
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ABSTRACT We consider a paper of Banaś and Sadarangani (2008) [11] which deals with monotonicity properties of the superposition operator and their applications. An application of the monotonicity properties is to study the solvability of... more
ABSTRACT We consider a paper of Banaś and Sadarangani (2008) [11] which deals with monotonicity properties of the superposition operator and their applications. An application of the monotonicity properties is to study the solvability of a quadratic Volterra integral equation. In this paper, we prepare an efficient numerical technique based on the fixed point method and quadrature rules to approximate a solution for quadratic Volterra integral equation. Then convergence of numerical scheme is proved by some theorems and some numerical examples are given to show applicability and accuracy of the numerical method and guarantee the theoretical results.
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ABSTRACT In this study, we present an existence of solutions for some nonlinear functional- integral equations which include many key integral and functional equations that appear in nonlinear analysis and its applications. By using the... more
ABSTRACT In this study, we present an existence of solutions for some nonlinear functional- integral equations which include many key integral and functional equations that appear in nonlinear analysis and its applications. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo’s theorem to obtain the mentioned aims in Banach algebra.
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ABSTRACT In this paper, we consider solving matrix systems arising from the linear integral equations and PDE by two recent parallel techniques. The systems are indefinite due to linear constraints imposed on the fluid velocity. The first... more
ABSTRACT In this paper, we consider solving matrix systems arising from the linear integral equations and PDE by two recent parallel techniques. The systems are indefinite due to linear constraints imposed on the fluid velocity. The first approach, known as the multilevel algorithm, employs a hierarchical technique to compute the constrained linear space for the unknowns, followed by the iterative solution of a positive definite reduced problem. The second approach exploits the banded structure of sparse matrices to obtain a different reduced system which is determined by the unknowns common to adjacent block rows. Although the reduced system in this approach may still be indefinite, the algorithm converges to the solution at an accelerated rate.These methods have two desirable characteristics, namely, robust numerical convergence and efficient parallelizability.
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ABSTRACT Volterra type integral equations are appeared in many engineering fields, so that, we select Volterra integral equation of the first kind and wavelets as basis functions to estimate a solution for this kind of equations. In this... more
ABSTRACT Volterra type integral equations are appeared in many engineering fields, so that, we select Volterra integral equation of the first kind and wavelets as basis functions to estimate a solution for this kind of equations. In this procedure, we use collocation method as a projection method to convert integral equation to the system of linear equations. Finally, some numerical examples indicate the accuracy of this method.