Matrix method based on the second kind Chebyshev polynomials for solving time fractional diffusion-wave equations

Abstract

In this paper, the second kind Chebyshev polynomials (SKCPs) basis is used to solve time-fractional diffusion-wave equations with damping. We present some notations and definitions of the fractional calculus and introduce some basic properties of the SKCPs. Bivariate shifted SKCPs are defined and the operational matrix of fractional integration and some other needed operational matrices are constructed. Our approach uses the properties of bivariate shifted SKCPs to transform the considered problem to a matrix equation without using any collocation points. The main characteristic of this technique is that only a small number of the basic functions is needed to obtain a satisfactory result. An estimation of the error is given in the sense of Sobolev norms. Numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.

Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type

“…The matrix P in (1.2) is called the operational matrix of integration. The operational matrix of integration P has already been obtained for several types of two-variable basis functions such as bivariate shifted Legendre Nemati et al (2013), bivariate shifted Chebyshev Nemati and Sedaghat (2016), two-dimensional triangular Khajehnasiri (2016), Babolian et al (2010), two-dimensional Bernstein Shekarabi et al (2015), twodimensional block pulse Maleknejad and Mahdiani (2011), Najafalizadeh and Ezzati (2017), hybrid of block-pulse functions and Taylor series Mirzaee and Hoseini (2014).…”

A numerical approach for solving fractional optimal control problems with mittag-leffler kernel

“…With when ω = 1 and μ=−cos h(2)+2sin h(2)/2cos(2+sin h(2)) . This problem is solved with different methods given in Lotfi et al (2011) , Nemati (2016) , Oh and Luus (1989) which include spectral method based on the second-kind Chebyshev polynomials, operational matrix, and orthogonal collocation method, respectively. Hence, in Tables 3 and 4 , the numerical results obtained via the proposed method are compared with the results given in Lotfi et al (2011) , Nemati (2016) , Oh and Luus (1989) at some selected points when ω = 1 .…”

“…This problem is solved with different methods given in Lotfi et al (2011) , Nemati (2016) , Oh and Luus (1989) which include spectral method based on the second-kind Chebyshev polynomials, operational matrix, and orthogonal collocation method, respectively. Hence, in Tables 3 and 4 , the numerical results obtained via the proposed method are compared with the results given in Lotfi et al (2011) , Nemati (2016) , Oh and Luus (1989) at some selected points when ω = 1 . Also, by considering M = 5 , the numerical solutions obtained by different values of ω together with the exact solution are displayed in Figure 4, and comparison of L ∞ -norm and L 2 -norm errors is reported in Figure 5 and Table 5 .…”

“…Orthogonal basis functions have been generally used to achieve approximate solution for many problems in various fields of science. Approximation of the solution using these functions is known as a useful tool in solving many classes of equations numerically, for example, differential equations (Ganji et al, 2020a; Jafari et al, 2019; Sadeghi et al, 2020), partial differential equations (Tajadodi, 2020; Tuan et al, 2020b), integro-differential equations (Ganji et al, 2020b; Jafari et al, 2021; Tuan et al, 2020a), and FOCPs (Heydari et al, 2020; Lotfi et al, 2011; Nemati, 2016) of various orders (fixed, fractional, or variable order).…”

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

“…In [10,11], Sinc-finite difference method and Sinc-Chebyshev method were employed for solving the fractional diffusion-wave equations respectively. Recently methods based on operational matrix of Jacobi and Chebyshev polynomials were proposed to deal with the fractional diffusion-wave equations ( [12][13][14]). In [15], the authors applied fractional order Legendre functions method depending on the choices of two parameters to solve the fractional diffusion-wave equations.…”