A wavelet approach for the multi-term time fractional diffusion-wave equation

Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations

“…The absolute errors in solutions are tabulated in Table 7; we compare the absolute errors of present method with the wavelet method in Soltani Sarvestani et al (2019). We see that results of our method are more accurate than the results of obtained by wavelet approach.…”

“…The absolute errors in the solution are tabulated in 2 .6136 × 10 −7 1 320 7.0080 × 10 −4 Table 3 Comparing of present scheme with implicit numerical scheme (Kanth and 1.0172 × 10 −4 Table 5 Comparing of present scheme with two high-order numerical algorithms (Dehghan et al 2015) at different values of N , β = 1.7, β 1 = 1.2 and t = 1 for Example 1 4 .3888 × 10 −7 Fig. 1 The error function at t = 1 and N = 9 and β 1 = 1.2 for Example 1 Tables 5 and 6; comparison between obtained results of present scheme with wavelet approach (Soltani Sarvestani et al 2019) shows that the results of our method is more accurate than proposed method in Soltani Sarvestani et al (2019).…”

“…Most fractional differential equations do not have exact analytical solutions, so for tackling such approximation and numerical schemes must be applied. There are many researchers which have been interested to develop novel numerical methods for fractional partial differential equations (Ren and Wang 2017;Xing and Yan 2018;Zhang and Yang 2018;Sakar et al 2018;Mirzaee and Samadyar 2018) such as explicit finite difference (Shen et al 2011;Sousa 2012;Zhang and Yang 2018;Costa and Pereira 2018), implicit finite difference (Burrage et al 2012;Karatay et al 2011;Sunarto et al 2014), compact finite difference (Cui 2012;Wang and Ren 2019;Wang 2015), finite element (Ford et al 2011;Jiang and Ma 2011;Li and Yang 2017), spline (Arshed 2017;Siddiqi and Arshed 2015;Qiao and Xu 2018), Fourier analysis (Li et al 2018), radial basis functions (Golbabai et al 2019;Ahmadi et al 2017;Dehghan et al 2016;Hosseini et al 2016;Ghehsareh et al 2018), wavelets (Heydari et al 2015;Kargar and Saeedi 2017;Soltani Sarvestani et al 2019), sinc radial basis function (Permoon et al 2016), local radial basis function-generated finite difference (Nikan et al 2020b;Nikan et al 2020a) and spectral methods (Rashidinia and Mohmedi 2018;Yang et al 2018;Zaky 2018a, b;Aghdam et al 2020)…”

“…In addition, a high-order spectral method for the multi-term time fractional diffusion equations has been investigated in works by Zheng et al (2016). In some complete different overview, a wavelet approach for the multi-term time fractional diffusion-wave equation is given by Soltani Sarvestani et al (2019). In reports of Kanth and Garg (2019), an implicit numerical scheme for a class of multi-term time fractional diffusion equation is presented.…”

The Galerkin spectral element method for the solution of two‐dimensional multiterm time fractional diffusion‐wave equation

“…For this equation, Jiang et al 9 used the method of separating variables in a finite domain and derived the analytical solutions with three kinds of nonhomogeneous boundary conditions. Some computationally effective implicit numerical methods are proposed in Liu et al 10 Two numerical methods based on a fourth-order compact finite difference procedure and Galerkin spectral method are given in Dehghan et al 11 A Galerkin method based on the second kind Chebyshev wavelets is proposed in Sarvestani et al 12 Note that the methods of in the previous studies [9][10][11][12] deal with only the one-dimensional case of Equation (1). A meshless collocation method, which uses the moving least squares reproducing kernel particle approximation to construct the shape functions for spatial approximation, is given in Salehi.…”