Numerical solution of integro-differential equations of high-order Fredholm by the simplified reproducing kernel method

ABSTRACT

The key of the reproducing kernel method (RKM) to solve the initial boundary value problem is to construct the reproducing kernel meeting the homogenous initial boundary conditions of the considered problems. The usual method is that the initial boundary conditions must be homogeneous and put them into space. Another common method is to put homogeneous or non-homogeneous conditions directly into the operator. In addition, we give a new numerical method of RKM for dealing with initial boundary value problems, homogeneous conditions are put into space, and for nonhomogeneous conditions, we put them into operators. The focus of this paper is to further verify the reliability and accuracy of the latter two methods. Through solving three numerical examples of integral–differential equations and comparing with other methods, we find that the two methods are useful.

KEYWORDS:

• Initial boundary condition
• integral–differential equations
• numerical methods
• reproducing kernel space
• reproducing kernel methods

2010 AMS SUBJECT CLASSIFICATIONS:

• 34K05
• 34K30
• 47G20
• 45Axx
• 45Bxx

Acknowledgments

The authors thank the reviewers for their careful reading and valuable advice.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper is supported by the Natural Science Foundation of Inner Mongolia [numbers 2017MS0103 and 2015MS0118), the National Natural Science Foundation of China [number 11361037], Numerical analysis of graduate course construction project of Inner Mongolia University of Technology [number KC2014001].