On the Numerical Solution of Volterra-Fredholm Integral Equations with Exponential Kernal using Chebyshev and Legendre Collocation Methods (original) (raw)
Related papers
International Journal of Applied Mathematical Research, 2014
A smoothing transformation, Legendre and Chebyshev collocation method are presented to solve numerically the Voltterra-Fredholm Integral Equations with Logarithmic Kernel. We transform the Volterra Fredholm integral equations to a system of Fredholm integral equations of the second kind, using a smoothing transformation to cancel the singularities in the kernel, a system Fredholm integral equation with smooth kernel is obtained and will be solved using Legendre and Chebyshev polynomials. This lead to a system of algebraic equations with Legendre or Chebychev coefficients. Thus, by solving the matrix equation, Legendre and Chebychev coefficients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.
Numerical Solution of Volterra Integral Equations Using the Chebyshev-Collocation Spectral Methods
The main purpose of this paper is to submit a new numerical approach for the Volterra integral equations based on a spectral method. The Chebyshev-collocation spectral method is proposed to solve the Volterra integral equations of the second kind and then convergence analysis of proposed method is discussed. Numerical examples show that the approximate solutions have a good degree of accuracy.
Chebyshev collocation treatment of Volterra–Fredholm integral equation with error analysis
Arabian Journal of Mathematics, 2019
This work reports a collocation algorithm for the numerical solution of a Volterra–Fredholm integral equation (V-FIE), using shifted Chebyshev collocation (SCC) method. Some properties of the shifted Chebyshev polynomials are presented. These properties together with the shifted Gauss–Chebyshev nodes were then used to reduce the Volterra–Fredholm integral equation to the solution of a matrix equation. Nextly, the error analysis of the proposed method is presented. We compared the results of this algorithm with others and showed the accuracy and potential applicability of the given method.
A numerical method for solving Fredholm and Volterra integral equations is presented and analyzed. The method is essentially based on making use of Gauss quadrature formula. The second kind Chebyshev polynomials are used as basis functions. The main idea behind our algorithm depends on reducing the solution of the integral equation into a solution of algebraic system of equations which can be solved by a suitable numerical solver. Some illustrative examples are included to demonstrate the validity and applicability of the suggested algorithm.
Mathematical Sciences, 2012
AbstractWhen we discretize nonlinear Volterra integral equations using some numerical, such as collocation methods, the arising algebraic systems are nonlinear. Applying quasilinear technique to the nonlinear Volterra integral equations gives raise to linear Volterra integral equations. The solutions of these equations yield a functional sequence quadratically convergent to the solution. Then, we use collocation method based on Chebyshev polynomials and a modified Clenshaw-Curtis quadrature and obtain a numerical solution. Error analysis has been performed, and the method has been applied on three numerical examples.
Solving Nonlinear Volterra-Fredholm Integral Equations Using an Accurate Spectral Collocation Method
Tatra Mountains Mathematical Publications, 2021
In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both L∞ and weighted L2 norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods
2016
A Sinc collocation method for numerical solution of Volterra-Fredholm integral equations of the second kind is developed by incorporating a variable transformation of double exponential order into the Sinc function expansion technique. The derived Sinc collocation formula is used to convert a Volterra-Fredholm integral equation defined on a finite interval into a set of algebraic equations. Numerical examples are presented to show the rapid convergence and exceptional accuracy of the method.