A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions. (English) Zbl 1453.65198
Summary: Our aim in this paper is to develop a Legendre-Jacobi collocation approach for a nonlinear system of two-point boundary value problems with derivative orders at most two on the interval \((0,T)\). The scheme is constructed based on the reduction of the system considered to its equivalent system of Volterra-Fredholm integral equations. The spectral rate of convergence for the proposed method is established in both \(L^2\)- and \(L^{\infty}\)- norms. The resulting spectral method is capable of achieving spectral accuracy for problems with smooth solutions and a reasonable order of convergence for non-smooth solutions. Moreover, the scheme is easy to implement numerically. The applicability of the method is demonstrated on a variety of problems of varying complexity. To the best of our knowledge, the spectral solution of such a nonlinear system of fractional differential equations and its associated nonlinear system of Volterra-Fredholm integral equations has not yet been studied in literature in detail. This gap in the literature is filled by the present paper.
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 45D05 | Volterra integral equations |
| 45B05 | Fredholm integral equations |
| 34A08 | Fractional ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65R20 | Numerical methods for integral equations |
Keywords:
system of fractional differential equations; Fredholm integral equations; boundary value problems; convergence analysisReferences:
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