The ultimate goal of this performance study is to provide a  proposed scheme for solving the time-fractional stochastic advection-diffusion equation (TFSADE) of order $\alpha (0\le \alpha <1)$. In this proposed scheme, we utilize an approach based on cubic TRIGONOMETRIC B-SPLINE collocation methods (CTBSCM).  In this study, we replace the existing fractional derivative with the fractional Caputo derivative for time discretization and then replace the first and second derivatives of the equation using cubic TRIGONOMETRIC B-SPLINE functions for spatial discretization. Applying this proposed scheme to TFSADE causes the equation to reduce to the linear system. In the end, the examples show that the order of convergence of the proposed method is $O(\tau ^{2-\alpha}+h^2)$ where $h$ and $\tau$  are the spatial and time step lengths, respectively.

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In the present study, the Modified Equal Width (MEW) wave equation is going to be solved numerically by presenting a new technique based on the collocation finite element method in which TRIGONOMETRIC cubic B-SPLINEs are used as approximate functions. In order to support the present study, three test problems,namely, the motion of a single solitary wave, the interaction of two solitary waves, and the birth of solitons are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the numerical conserved laws as well as the error norms L2 and L∞, .

In this paper, a singularly perturbed one-dimensional initial boundary value problem of a quasilinear Sobolev-type equation is presented. The nonlinear term of the problem is linearized by Newton’s linearization method. Time derivatives are discretized by implicit Euler’s method on nonuniform step size. A uniform TRIGONOMETRIC B-SPLINE collocation method is used to treat the spatial variable. The convergence analysis of the scheme is proved, and the accuracy of the method is of order two in space and order one in time direction, respectively. To test the efficiency of the method, a model example is demonstrated. Results of the scheme are presented in tabular, and the figure indicates the scheme is uniformly convergent and has an initial layer at t = 0.