In this paper fuzzy piecewise CUBIC interpolation is constructed for fuzzy data based on B-SPLINE basis functions. We add two new additional conditions which guarantee uniqueness of fuzzy B-SPLINE interpolation.Other conditions are imposed on the interpolation data to guarantee that the interpolation function to be a well-defined fuzzy function. Finally some examples are given to illustrate the proposed method.

In this paper, we consider a numerical method based on extended CUBIC B-SPLINE basis functions for the determination of an unknown boundary condition in the inverse second-order one-dimensional hyperbolic telegraph equation. Extended CUBIC B-SPLINE (ExCuBs) is an extension of CUBIC B-SPLINE consisting of a parameter, we combined it with the Tikhonov regularization method to obtain a numerically stable solution. The convergence and stability of the technique are proved and shown that it is established under suitable assumptions and accurate order $O(k+h^2)$. The numerical results have been compared with those obtained by the CUBIC B-SPLINE method to verify the accurate nature of our method.

The collocation method based on the exponential CUBIC B-SPLINEs (ECB-SPLINEs) together with the Crank-Nicolson formula is used to solve nonlinear coupled Burgers' equation (CBE). This method is tested by studying three di, erent problems. The proposed scheme is compared with some existing methods. It produced accurate results with the suitable selection of the free parameter of the ECB-SPLINE function. It produces accurate results. Stability of the fully discretized CBE is investigated by the Von Neumann analysis.

In this paper, a numerical method is proposed to approximate the numeric solutions of nonlinear Fisher's reaction–diffusion equation with modified CUBIC B-SPLINE collocation method. The method is based on collocation of modified CUBIC B-SPLINEs over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply modified CUBIC B-SPLINEs for spatial variable and derivatives, which produce a system of first-order ordinary differential equations. We solve this system by using SSP-RK54 scheme. The proposed method needs less storage space that causes less accumulation of numerical errors. The numerical approximate solution to the nonlinear Fisher's reaction–diffusion equation has been computed without using any transformation and linearization process. Illustrative three test examples are included to establish the effectiveness and pertinence of the technique. Easy and economical implementation is the strength of this method.

In this study, an e, ective collocation method based on CUBIC B-SPLINE has been im-plemented to get the numerical solutions for the non-linear Fisher's equation. After separating this scheme with this method, the stability of the method was proven. To check the e, ciency and accuracy of the proposed method, some numerical problems have been considered. The numerical results are found in good agreement with the exact solutions.

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.