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Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations. (English) Zbl 06912520
Comput. Appl. Math. 37, No. 2, 2354-2379 (2018); erratum ibid. 37, No. 2, 2380 (2018).
Summary: The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.
Reviewer: Reviewer (Berlin)
65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
45D05 Volterra integral equations
65F10 Iterative numerical methods for linear systems
26A33 Fractional derivatives and integrals
65L03 Numerical methods for functional-differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
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