112 research outputs found
Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations
This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE
A Legendary Polynomial Approach to Solutions of Volterra Integro-Differential Equations with Delay
This paper presents a numerical computation approach to the solution of Volterra Integro-Differential Equation with delay via legendary Polynomial method. Computational code was developed and implemented on a Mathematical Software (Matlab 2009b) to solve the problem. The accuracy, efficiency and effectiveness of the method of solution were ascertained by comparing the solution obtained with the exact solution in the literature and it showed that the present results were in agreement with both the exact and existing result
Multiple Perturbed Collocation Tau Method for Solving Nonlinear Integro-Differential Equations
The purpose of the study was to investigate the numerical solution of non-linear Fredholm and Volterra integro-differential equations by the proposed method called Multiple Perturbed Collocation Tau Method (MPCTM). We assumed a perturbed approximate solution in terms of Chebyshev polynomial basis function and then determined the derivatives of the perturbed approximate solution which are then substituted into the special classes of the problems considered. Thus, resulting into n-folds integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Newton’s method which are then substituted back into the approximate solution.Illustrative examples are given to demonstrate the efficiency, computational cost and accuracy of the method. The results obtained with some numerical examples are compared favorable with some existing numerical methods in literature and with the exact solutions where they are known in closed form.Keywords: Nonlinear Problems, Tau Method, Integro-Differential, Newton’s method
Spectral collocation method for compact integral operators
We propose and analyze a spectral collocation method for integral
equations with compact kernels, e.g. piecewise smooth kernels and
weakly singular kernels of the form We prove that 1) for integral equations, the convergence
rate depends on the smoothness of true solutions . If
satisfies condition (R): }, we obtain a geometric rate of convergence; if
satisfies condition (M): ,
we obtain supergeometric rate of convergence for both Volterra
equations and Fredholm equations and related integro differential
equations; 2) for eigenvalue problems, the convergence rate depends
on the smoothness of eigenfunctions. The same convergence rate for
the largest modulus eigenvalue approximation can be obtained.
Moreover, the convergence rate doubles for positive compact
operators. Our numerical experiments confirm our theoretical
results
Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions
This paper presents an efficient spectral method for solving the fractional
Fredholm integro-differential equations. The non-smoothness of the solutions to
such problems leads to the performance of spectral methods based on the
classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low
order of convergence. For this reason, the development of classic numerical
methods to solve such problems becomes a challenging issue. Since the
non-smooth solutions have the same asymptotic behavior with polynomials of
fractional powers, therefore, fractional basis functions are the best candidate
to overcome the drawbacks of the accuracy of the spectral methods. On the other
hand, the fractional integration of the fractional polynomials functions is in
the class of fractional polynomials and this is one of the main advantages of
using the fractional basis functions. In this paper, an implicit spectral
collocation method based on the fractional Chelyshkov basis functions is
introduced. The framework of the method is to reduce the problem into a
nonlinear system of equations utilizing the spectral collocation method along
with the fractional operational integration matrix. The obtained algebraic
system is solved using Newton's iterative method. Convergence analysis of the
method is studied. The numerical examples show the efficiency of the method on
the problems with smooth and non-smooth solutions in comparison with other
existing methods
A novel third kind Chebyshev wavelet collocation method for the numerical solution of stochastic fractional Volterra integro-differential equations
In the formulation of natural processes like emissions, population development, financial markets, and the mechanical systems, in which the past affect both the present and the future, Volterra integro-differential equations appear. Moreover, as many phenomena in the real world suffer from disturbances or random noise, it is normal and healthy for them to go from probabilistic models to stochastic models. This article introduces a new approach to solve stochastic fractional Volterra integro-differential equations based on the operational matrix method of Chebyshev wavelets of third kind and stochastic operational matrix of Chebyshev wavelets of third kind. Also, we have given the convergence and error analysis of the proposed method. A variety of numerical experiments are carried out to demonstrate our theoretical findings.Publisher's Versio
Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients
AbstractThe main aim of this paper is to apply the Legendre polynomials for the solution of the linear Fredholm integro-differential-difference equation of high order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results
- …