In this work, we have considered the Riccati–Bernoulli sub-ODE method for obtaining the exact solutions of nonlinear fractional-order differential equations. The fractional derivatives are described in Jumarie’smodified Riemann–Liouville sense. The space–time fractional modified equal width (mEW) equation and timefractional generalised Hirota–Satsuma coupled Korteweg–de Vries (KdV) equations are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations (ODEs), which were obtained from the nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

In this article, analytical and numerical solutions to the simplified modified Camassa–Holm (MCH) equation by using the Riccati–Bernoulli (RB) sub-ODE method and moving mesh method are obtained. Some new solutions are given. The discrtisation of the presented model is introduced in the form of finite difference operators. Some 3D graphs for the presented solution are plotted with the aid of the Matlab software for suitable values of parameters. We introduce the comparison between numerical and exact solutions.

In this article, we applied the unified solver method to extract stochastic solutions of a new stochastic extension of nonlinear Schrödinger equation. This solver gives the closed formula in explicit form. The acquired stochastic solutions may be applicable for explaining some phenomena in many fields of applied sciences. The presented results illustrate that the proposed solver is efficient and adequate. Moreover, the constraint conditions are utilised to verify the existence of solutions. Chi-square statistical distribution is chosen to represent the dispersion random input. In order to illustrate the dynamical behaviour of random solutions, the expectation value and their variance are depicted graphically using suitable parameters.