Lerch matrix collocation method for 2D and 3D Volterra type integral and second order partial integro differential equations together with an alternative error analysis and convergence criterion based on residual functions

A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of collocation approximations on arbitrary and quasi‐uniform grids is studied theoretically and numerically.

Download Full-text

A Novel Collocation Method Based on Residual Error Analysis for Solving Integro-Differential Equations Using Hybrid Dickson and Taylor Polynomials

Sains Malaysiana ◽

10.17576/jsm-2017-4602-19 ◽

2017 ◽

Vol 46 (02) ◽

pp. 335-347 ◽

Cited By ~ 10

Author(s):

ÖMÜR KIVANÇ KÜRKÇÜ

Keyword(s):

Differential Equations ◽

Error Analysis ◽

Collocation Method ◽

Residual Error ◽

Taylor Polynomials

Download Full-text

Solving Nonlinear Fractional Integro-Differential Equations of Volterra Type Using Novel Mathematical Matrices

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4029281 ◽

2015 ◽

Vol 10 (6) ◽

Cited By ~ 3

Author(s):

Farshid Mirzaee ◽

Saeed Bimesl ◽

Emran Tohidi

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Collocation Method ◽

Operational Matrix ◽

Volterra Type

In this paper, the operational matrix of Euler functions for fractional derivative of order β in the Caputo sense is derived. Via this matrix, we develop an efficient collocation method for solving nonlinear fractional Volterra integro-differential equations. Illustrative examples are given to demonstrate the validity and applicability of the proposed method, and the comparisons are made with the existing results.

Download Full-text

Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.2166147 ◽

2005 ◽

Vol 1 (2) ◽

pp. 178-185 ◽

Cited By ~ 13

Author(s):

Pankaj Kumar ◽

Om P. Agrawal

Keyword(s):

Integral Equation ◽

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Numerical Scheme ◽

Numerical Solutions ◽

Volterra Type ◽

Entire Domain ◽

Type Integral ◽

Fractional Differential

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1<α<2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra-type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra-type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve two problems, linear and nonlinear, using two different types of polynomials, cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results for problem 1 and the numerical results obtained using another scheme for problem 2. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

Download Full-text

Lyapunov Type Integral Inequalities for Certain Differential Equations

Georgian Mathematical Journal ◽

10.1515/gmj.1997.139 ◽

1997 ◽

Vol 4 (2) ◽

pp. 139-148

Author(s):

B. G. Pachpatte

Keyword(s):

Asymptotic Behavior ◽

Differential Equations ◽

Second Order ◽

Integral Inequalities ◽

Asymptotic Behavior Of Solutions ◽

Second Order Differential Equations ◽

Elementary Analysis ◽

Type Integral ◽

Zeros Of Solutions

Abstract In the present paper we establish Lyapunov type integral inequalities related to the zeros of solutions of certain second-order differential equations by using elementary analysis. We also present some immediate applications of our results to study the asymptotic behavior of solutions of the corresponding differential equations.

Download Full-text