The purpose of this paper is to suggest a numerical technique to solve fractional variational problems (FVPs). These problems are based on Caputo fractional derivatives. Rayleigh–Ritz method is used in this technique. First we approximate the objective function by the trapezoidal rule. Then, the unknown function is expanded in terms of the Bernstein polynomials. By this method, a system of algebraic equations is driven. We provide examples to show the effectiveness of this technique, which is considered in the current study.

In the given research, our main goal is to explore newsoliton solutions in (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation. This was achieved via the extended (G'/G2)-expansion method. For this, we first analyse the given (2 + 1)-dimensional HFSC system of a partial differential equation (PDE) that is obtained by separating the equation into real and imaginary parts. The solution of our equation is determined as rapidly convergent sequences effectively computed by using Mathematica software. Furthermore, a special class of nonlinear wave solutions like periodic, straight, decreasing, dark-singular soliton and dark bright singular soliton were obtained in Heisenberg ferromagnetic dynamics. The amplitude of the soliton is determined by giving different values of parameters. With a suitable choice of parameters, 3D and 2D graphical illustrations are reported. The methodology applied is effective for obtaining exact solutions for various fractional partial differential equations in complex medium. The numerical module of the results obtained is studied, with fascinating figures showing the physical significance of the solutions.