New iterative method for solving non-linear time-fractional Phi-Four equation
DOI:
https://doi.org/10.64497/jssci.123Keywords:
: Fractional differential equations, q-Homotopy Analysis Transform Method (q-HATM), Nonlinear dynamics, Numerical approximation, Convergence analysis, Phi-four equation, New Iterative Method (NIM)Abstract
In this paper, we employ the New Iterative Method (NIM) to obtain approximate solutions of the nonlinear time-fractional Phi-four equation, a model of significant importance in nonlinear science and fractional calculus. Fractional partial differential equations are recognized for their ability to capture memory effects and anomalous dynamics; however, their nonlinear nature makes them challenging to solve analytically. To address this, the proposed approach applies the NIM framework and computes the approximate solutions using Maple 13. The obtained results are compared with the exact analytical solution as well as the q-Homotopy Analysis Transform Method (q-HATM) to assess both accuracy and efficiency. Numerical experiments, illustrated through a series of two-dimensional and three-dimensional graphical representations, demonstrate that the approximate solutions produced by NIM converge rapidly to the exact solution. In addition, the tabulated error analysis confirms the robustness and stability of the method, showing that NIM achieves comparable accuracy with q-HATM while requiring fewer computational steps and avoiding complex procedures. The findings suggest that NIM is not only effective for solving the time-fractional Phi-four equation but also offers a simple, flexible, and efficient tool for tackling a broad class of nonlinear fractional partial differential equations. Consequently, this method holds strong potential for application in physics, engineering, and other fields where fractional models play a crucial role.
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