Numerical solution of nonlinear partial quadratic integro-differential equations of fractional order via hybrid of block-pulse and parabolic functions. (English) Zbl 1418.65152
Summary: In this paper, an effective numerical approach based on a new two-dimensional hybrid of parabolic and block-pulse functions (2D-PBPFs) is presented for solving nonlinear partial quadratic integro-differential equations of fractional order. Our approach is based on 2D-PBPFs operational matrix method together with the fractional integral operator, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton’s iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro-differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is \({O}({h}^3)\). The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35R09 | Integro-partial differential equations |
| 35R11 | Fractional partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65H10 | Numerical computation of solutions to systems of equations |
| 65R20 | Numerical methods for integral equations |
Keywords:
block-pulse functions; fractional calculus; nonlinear quadratic integral equation; operational matrix; parabolic functions; partial differential equationReferences:
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