Numerical solution of fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions via Chebyshev wavelet method

ABSTRACT

In this paper, the fourth kind Chebyshev wavelets collocation method (FCWM) is applied for solving a class of fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the fourth kind. Moreover, upper bound of error of the fourth kind Chebyshev wavelets expansion is given. Based on the collocation technique, the fourth kind Chebyshev wavelets together with Gaussian integration are used to reduce the problem to the solution of a system of algebraic equations. During the process of establishing the expression of the solution, the boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration. Some examples are provided to confirm the reliability and effectiveness of the proposed method.

KEYWORDS:

• Fourth kind Chebyshev wavelets
• fractional integro-differential equations
• fractional calculus
• collocation method
• mixed boundary conditions
• Gaussian quadrature rule

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 65T60
• 26A33
• 45J05
• 34A08

Acknowledgements

All authors contributed equally to the writing of this paper. All authors read and approved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11601076), the Youth Science Foundation of Jiangxi Province (Grant No. 20151BAB211004, 20151BAB211012), and the Science and Technology Project of Jiangxi Provincial Education Department (Grant No. GJJ170473).