Abstract
An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient.
Highlights
Fractional differential equation (FDE) is an extension of integer-order model
Compared with the classical integerorder differential equation, FDE provides an excellent instrument for the description of memory and hereditary properties of various materials and processes
Researchers have pointed out that the fractional calculus plays an important role in modeling and many systems in interdisciplinary fields can be elegantly described with the help of fractional derivative, such as dynamics of earthquakes [1], viscoelastic systems [2], biological systems [3, 4], diffusion model [5], chaos [6], wave propagation [7], and partial bed-load transport [8]
Summary
Fractional differential equation (FDE) is an extension of integer-order model. Compared with the classical integerorder differential equation, FDE provides an excellent instrument for the description of memory and hereditary properties of various materials and processes. Fu et al [10] presented a Laplace transformed boundary particle method for numerical modeling of time fractional diffusion equations. The development of numerical algorithms to solve variable order FDE is necessary. Shen et al [15] have given an approximate scheme for the variable order time fractional diffusion equation. A numerical method based on Legendre polynomials was presented for a class of variable order FDEs in [20]. The second kind of Chebyshev polynomials have been paid less attention for solving variable order FDE. We will solve a kind of variable order fractional differential-integral equations (FDIEs) defined on the interval [0, R] (R > 0) based on the second kind of Chebyshev polynomials.
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