The Fractional Series Solutions for the Conformable Time-Fractional Swift-Hohenberg Equation through the Conformable Shehu Daftardar-Jafari Approach with Comparative Analysis

The major objective of this study is to derive fractional series solutions of the time-fractional Swift-Hohenberg equations (TFSHEs) in the sense of conformable derivative using the conformable Shehu transform (CST) and the Daftardar-Jafari approach (DJA). We call it the conformable Shehu Daftardar-Jafari approach (CSDJA). One of the universal equations used in the description of pattern formation in spatially extended dissipative systems is the Swift-Hohenberg equation. To assess the effectiveness and consistency of the suggested approach, the numerical results are compared with those obtained by the Elzaki decomposition method (EDM) in the sense of relative and absolute error functions, proving that the CSDJA is an effective substitute for techniques that use He’s or Adomian polynomials to solve TFSHEs. The transition from the imprecise solution to the precise solution at various values of fractional-order derivatives is shown using the recurrence error function. Furthermore, the exact and approximative solutions are compared using 2D and 3D graphics and also numerically in the form of relative and absolute error functions. The results show that the procedure is quick, precise, and easy to implement, and it yields outstanding results. The recommended approach’s strength, which gives it an advantage over the Adomian decomposition and homotopy perturbation methods, is its algorithm for dealing with nonlinear problems without the use of Adomian polynomials or He’s polynomials. The advantage of this method is that it does not make any assumptions about physical parameters. As a result, it can be used to solve both weakly and strongly nonlinear problems and circumvent some of the drawbacks of perturbation techniques.