Trigonometric tension B-spline collocation approximations for time fractional Burgers’ equation

A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers’ equation

“…The three examples have been considered to validate the accuracy and efficiency of the proposed method. It has been observed that the present method provides better results than the methods in [19,35,36]. The graphical results are also presented that confirm the accuracy of the proposed algorithm.…”

“…Next, Example 2 is solved with free parameter p = 0.015 and ν = 1 and for various other parameters. The L 2 and L ∞ error norms depicted in Tables 8 and 9 show that the proposed method results are better than those presented in [19,35,36], and the proposed method is second-order accurate in space variable. It is also observed that both error norms L 2 and L ∞ are decreasing on increasing the space as well as time mesh sizes.…”

“…The stability analysis of the discretized system of the TFNB equation based on the von Neumann method [22,36,39] is established in this section. According to Duhamels' principle [40], it is assumed that the stability analysis of an inhomogeneous problem is an instantaneous consequence of the stability analysis for the subsequent homogeneous problem.…”

“…The authors of [18] used the Hermite cubic spline collocation technique for the computational approximation of Helmholtz and Burgers' equations. The authors of [19,21,35,36] used the quadratic B-spline Galerkin (QBSG) method, balanced space-time Chebyshev spectral collocation method, finite element method based on the cubic B-spline collocation method, and trigonometric tension B-spline collocation method, respectively, for TFNB equations. A practical and accurate technique based on the shifted Gegenbauer polynomials has been presented in [24] to simulate the multidimensional space-fractional coupled Burgers' equations.…”

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

“…The time-fractional derivative and its extension to variant order have many physical applications [18]. Besides the time-fractional diffusion equation, lots of scholars have developed many schemes to cope with the time fractional Burgers' equation, like the operator splitting approach and artificial boundary method [19], the nonuniform Alikhanov formula of the Caputo time fractional derivative and Fourier spectral approximation in space [20], the L1 scheme and the local discontinuous Galerkin method [21], the Lucas polynomials coupled with finite difference method [22], the fourth-order compact difference scheme [23], the L1 implicit difference scheme based on non-uniform meshes [24], a secondorder energy stable and nonuniform time-stepping scheme [25], a collocation approach with trigonometric tension B-splines [26], the cubic B-spline functions and θ-weighted scheme [27], the local projection stabilization virtual element method [28], a compact difference scheme [29], the Caputo-Katugampola fractional derivative by extending the Laplace transform [30], and the tailored finite point method based on exponential basis [31]. For the time-fractional Schrödinger equations, lots of researchers have proposed many algorithms, like the conformable natural transform and the homotopy perturbation method [32], the conformable fractional derivatives modified Khater technique and the Adomian decomposition method [33], the Laplace Adomian decomposition method and the modified generalized Mittag-Leffler function method [34], a Caputo residual power series scheme [35], and the extended Kudryashov method [36].…”