This article is concerned with developing a method to find a numerical solution of a one-dimensional variable-order non-linear partial integro-differential equation (PIDE) viz., reaction-advection-diffusion equation with initial and boundary conditions. The proposed numerical scheme shifted Legendre collocation method is based on operational matrices. The operational matrices using one-dimensional wavelets are derived to solve the said variable-order model.First operational matrices have been introduced for integration and variableorder derivatives using one-dimensional Legendre wavelets (LWs). After that, using the shifted Legendre collocation points, the model is reduced to a system of algebraic equations, which are solved using Newton-Cotes method. The error is then calculated by comparing the numerical solution obtained from the system of algebraic equations and the known exact solution of an existing problem to validate the efficiency of the proposed numerical scheme. The main contribution of the article is the graphical exhibitions of the solution profile for different variable order derivatives in presence of different values of the parameters.

In the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.

The present article contains the study of fixed time synchronization of octonion valued neural networks (OVNNs) with time varying delays. Firstly, OVNNs are decomposed into eight real valued systems of equations. Using some lemmas and Lyapunov functional, sufficient criteria have been derived for the fixed time synchronization. Moreover a suitable non-linear controller was created to keep the drive response system in synchronization state. Finally, a numerical example is performed to demonstrate and validate the theoretical findings.