Abstract
In this study, the numerical technique based on two-dimensional block pulse functions (2D-BPFs) has been developed to approximate the solution of fractional Poisson type equations with Dirichlet and Neumann boundary conditions. These functions are orthonormal and have compact support on [ 0,1 ]. The proposed method reduces the original problems to a system of linear algebra equations that can be solved easily by any usual numerical method. The obtained numerical results have been compared with those obtained by the Legendre and CAS wavelet methods. In addition an error analysis of the method is discussed. Illustrative examples are included to demonstrate the validity and robustness of the technique.
Highlights
Fractional calculus is an important theoretical branch of mathematical theories [ ], which has been widely applied in various fields such as the complex physical, mechanical, biological, and engineering fields
The exact solutions of many fractional partial differential equations cannot be obtained, so scholars are committed to obtaining their numerical solutions to reflect the exact solutions
Based on the above model, we propose the following fractional Poisson equations model:
Summary
Fractional calculus is an important theoretical branch of mathematical theories [ ], which has been widely applied in various fields such as the complex physical, mechanical, biological, and engineering fields. In [ ], Maleknejad and Mahdiani proposed to solve nonlinear mixed Volterra-Fredholm integral equations with two-dimensional block pulse functions using a direct method. Reference [ ] gave the numerical methods for solving two-dimensional nonlinear integral equations of fractional order by using a twodimensional block pulse operational matrix. We applied two-dimensional block pulse functions to obtain the numerical solutions of fractional Poisson type equations with Dirichlet and Neumann boundary conditions. Section introduces the definitions and properties of two-dimensional block pulse functions. Section introduces the method for solving fractional Poisson type equations. Since each of the two-dimensional block pulse functions takes only one value in its subregion, the D-BPFs can be expanded by the two D-BPFs: φi ,i (x, t) = φi (x)φi (t), where φi (x) and φi (t) are the D-BPFs related to the variables x and t, respectively. Let Dα is the block pulse operational matrix for the fractional differentiation.
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