Sains Malaysiana 49(4)(2020): 953-962

http://dx.doi.org/10.17576/jsm-2020-4904-24

Using Hybrid of Block-Pulse Functions and Bernoulli Polynomials to Solve Fractional Fredholm-Volterra Integro-Differential Equations

(Menggunakan Fungsi Blok-Denyut Hibrid dan Polinomial Bernoulli untuk Menyelesaikan Persamaan Pembezaan-Integro Fredholm-Volterra Pecahan)

ABBAS SAADATMANDI* & SAMIYE AKHLAGHI

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran

Received: 31 May 2019/Accepted: 5 January 2020

ABSTRACT

Fractional integro-differential equations have been the subject of significant interest in science and engineering problems. This paper deals with the numerical solution of classes of fractional Fredholm-Volterra integro-differential equations. The fractional derivative is described in the Caputo sense. We consider a hybrid of block-pulse functions and Bernoulli polynomials to approximate functions. The fractional integral operator for these hybrid functions together with the Legendre-Gauss quadrature is used to reduce the computation of the solution of the problem to a system of algebraic equations. Several examples are given to show the validity and applicability of the proposed computational procedure.

Keywords: Bernoulli polynomials; Block-pulse functions; fractional integro-differential equations; hybrid functions; Caputo derivative

ABSTRAK

Persamaan pembezaan integro pecahan telah menjadi subjek penting dalam masalah sains dan kejuruteraan. Makalah ini berkaitan dengan penyelesaian berangka kelas persamaan pembezaan integro Fredholm-Volterra pecahan. Terbitan pecahan diterangkan dalam maksud Caputo. Fungsi hibrid blok-denyutan dan polinomial Bernoulli dipertimbangkan untuk penghampiran fungsi. Pengoperasi kamiran pecahan untuk fungsi hibrid bersama-sama dengan kuadratur Legendre-Gauss digunakan untuk mengurangkan pengiraan penyelesaian masalah kepada sistem persamaan algebra. Beberapa contoh diberikan untuk menunjukkan kesahihan dan kebolehgunaan prosedur pengiraan yang dicadangkan.

Kata kunci: Fungsi blok-denyutan; fungsi hibrid; persamaan pembezaan integro pecahan; polinomial Bernoulli; terbitan Caputo

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*Corresponding author; email: saadatmandi@kashanu.ac.ir