Approximate and Exact Solutions of Some Nonlinear Differential Equations Using the Novel Coupling Approach in the Sense of Conformable Fractional Derivative (Contemporary Mathematics, 2024)
Several scientific fields utilize fractional nonlinear partial differential equations to model various phenomena. However, most of these equations lack exact solutions. Consequently, techniques for obtaining approximate solutions, which sometimes yield exact solutions, are essential. In this research, we develop a new approach by combining the homotopy perturbation method (HPM) and the conformable natural transform to solve the gas-dynamic equation (GDE), the Fokker-Planck equation (FPE), and the Swift-Hohenberg equation (SHE) in the context of conformable derivatives. The proposed approach is called the conformable natural homotopy perturbation method (CNHPM). This approach has the advantage of not requiring assumptions about significant or minor physical factors. Consequently, it eliminates some of the constraints associated with conventional perturbation methods and can solve both weak and highly nonlinear problems. We consider the absolute, relative, and residual errors numerically and graphically to assess the correctness of our approach. The results show that our approach serves as a suitable alternative to the approximate methods in the literature for solving fractional differential equations.
Keywords: fokker-planck equation, gas-dynamic equation, swift-hohenberg equation, approximate solutions, exact solutions