Spectral Galerkin schemes for a class of multi-order fractional pantograph equations. (English) Zbl 1456.65129
Summary: In this paper, we study and present a spectral numerical technique for solving a general class of multi-order fractional pantograph equations with varying coefficients and systems of pantograph equations. In this study, the spectral Galerkin approach in combination with the properties of shifted Legendre polynomials is used to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with fractional pantograph equations via spectral Galerkin approach. The errors and convergence of the adopted approach are rigorously analyzed. The efficiency and accuracy of the technique are tested by considering five different examples, to ensure that the suggested approach is more accurate than the existing other techniques. The obtained results in this paper are comparing favorably with those published by other researchers and with the existing exact solutions, whenever possible.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35C10 | Series solutions to PDEs |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 35R11 | Fractional partial differential equations |
Keywords:
spectral Galerkin method; shifted Legendre polynomials; pantograph equation; fractional calculus; Caputo fractional derivativeReferences:
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