Sains Malaysiana 48(12)(2019): 2807–2815

http://dx.doi.org/10.17576/jsm-2019-4812-22

Study on Numerical Solution of a Variable Order Fractional Differential Equation based on Symmetric Algorithm

(Kajian Penyelesaian Berangka Peringkat Berubah Persamaan Pembezaan Pecahan berdasarkan Algoritma Simetri)

Jingrui Liu & Dongyang Pan*

The School of Mathematics and Computer Science, Xinyang Vocational and Technical College, Xinyang, 464000, China

Diserahkan: 21 Februari 2019/ Diterima: 23 Disember 2019

ABSTRAK

As the class of fractional differential equations with changing order has attracted more attention and attention in the fields of research and engineering, it is important to study its numerical solutions. Numerical solution algorithm for a class of fractional differential equations with transformed arrays based on the proposed symmetry algorithm. The symmetry classification is used for the class of values of the boundary problem of the fractional differential equation with the order of change. A fully symmetric classification of the boundary value problem for a class of fractional differential equations with variable sequences is determined by using a fully symmetric differential sequence sorting algorithm. The problem of the boundary value of the fractional differential equation with the transformed order is reduced to the initial value of the ordinary differential equation. The Legendre polynomial method is used to solve the numerical solution of the starting value of the differential equation. The common differential equation is transformed into a matrix series product by a different operator matrix. The matrix products are converted to algebraic equations by discrete variables. By solving the equations, the numerical solution of the starting value of the common differential equation is obtained.

Keywords: Boundary value problem; differential equation; numerical solution; operator matrix; symmetric algorithm; variable fractional order

ABSTRAK

Oleh kerana kelas persamaan pembezaan pecahan dengan susunan berubah telah menarik banyak perhatian dan perhatian dalam bidang penyelidikan dan kejuruteraan, ia amat penting untuk mengkaji penyelesaian berangkanya. Algoritma penyelesaian berangka untuk kelas persamaan pembezaan pecahan dengan transformasi tatasusunan berdasarkan algoritma simetri yang dicadangkan. Pengelasan simetri digunakan untuk nilai kelas masalah sempadan persamaan pembezaan pecahan dengan susunan berubah. Pengelasan simetrik sepenuhnya masalah nilai sempadan untuk kelas persamaan pembezaan pecahan dengan jujukan pemboleh ubah ditentukan dengan menggunakan algoritma pengisihan jujukan pembezaan simetrik sepenuhnya. Masalah nilai sempadan persamaan pembezaan pecahan dengan peringkat berubah dikurangkan kepada masalah nilai awal persamaan pembezaan biasa. Kaedah polinomial Legendre digunakan untuk menyelesaikan penyelesaian berangka masalah nilai permulaan persamaan pembezaan. Persamaan pembezaan biasa diubah menjadi produk siri matriks oleh pengendali matriks lain. Produk matriks ditukar kepada persamaan algebra oleh variat diskret. Dengan menyelesaikan persamaan, penyelesaian berangka nilai permulaan persamaan pembezaan biasa diperoleh.

Kata kunci: Algoritma simetri; masalah nilai sempadan; matriks pengendali; penyelesaian berangka; peringkat pecahan berubah; persamaan pembezaan

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*Pengarang untuk surat-menyurat; email: paneastsun@163.com