NEW APPLICATIONS OF THE CAPUTO FRACTIONAL DERIVATIVE TO OBTAIN NUMERICAL SOLUTION OF THE COUPLED FRACTIONAL KORTEWEG–DE VRIES MODEL USING QUINTIC B-SPLINE FUNCTIONS

A fractional differential equation accurately represents nature when considering the symmetry properties that represent physical models. In this paper, we examine an approximate solution to the fractional coupled Korteweg–de Vries (CKdV) equation using Crank–Nicolson and quintic B-spline techniques on uniform mesh points. Fractional CKdV equation represents a mathematical model for shallow water wave surfaces. Quintic B-spline techniques are applied to spatial derivatives on uniform mesh points, while the temporal derivative is calculated using Caputo’s formula with the forward finite difference approach. Moreover, the Crank–Nicolson scheme is employed, as it is an unconditionally stable method that ensures reasonable accuracy. The discretization strategy used in this study is unconditionally stable when applied using the von Neumann technique. Additionally, the convergence analysis is examined; it has an order of O(h² + τ²). New numerical test problems demonstrate the method’s feasibility and the approximated solutions are shown to agree well with the exact solutions. Convergence order can be examined through graphical representations, which facilitate numerical analysis. Furthermore, the proposed method requires less memory storage for computing numerical solutions, making it more efficient.