Uniqueness and approximation of solution for fractional Bagley–Torvik equations with variable coefficients

ABSTRACT

The initial value problem for fractional Bagley–Torvik equations is investigated by considering variable coefficients and the fractional order as $0<\alpha <2$. Making use of the integration method, a Volterra integral equation of the second kind is obtained. Then the contraction operator theorem in Banach spaces is further used to address the uniqueness of the solution for the obtained Volterra integral equation. A novel numerical method is proposed to find the approximate solution of the given Volterra integral equation. Moreover, the convergence and error estimate of the approximate solution are analysed. Finally, some numerical examples are carried out to show the effectiveness of the proposed method by comparing with the existing ones. The developed method will be helpful for finding a good approximation solution of fractional differential equations in practical applications.

KEYWORDS:

• Bagley–Torvik equation
• Volterra integral equation
• uniqueness
• approximate solution
• convergence and error estimate

2000 AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 45D05

Acknowledgments

The authors would like to thank editors and the reviewer for the valuable comments and suggestions of improving the paper. XCZ acknowledges the project of outstanding young teachers' training in higher education institutions of Guangxi, and the project of Guangxi Colleges and Universities Key Laboratory of Mathematics and Its Applications.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [No. 11362002], and the Innovation Project of Guangxi Graduate Education [No. YCSZ2015030].