Natural Transform Method with Modified ADM for Nonlinear Time Fractional PDEs with Proportional Delay

This paper presents a practical approach for solving nonlinear partial differential equations with both time fractional and proportional delay. These equations appear in many real-world situations, such as viscoelasticity, earthquake, population dynamics, volcanic eruption, and control theory. These problems are challenging, and the fractional operator and the nature of the delay add another layer of difficulty. Because of this, there is a need for efficient numerical methods. The study uses the natural transform along with a Modified Adomian Decomposition Method. The Caputo fractional derivative helps manage the memory effects present in fractional systems. We effectively handle the nonlinear parts using modified Adomian polynomials, and examine our method’s convergence and stability in the Banach sense. To show that our method works well, we test it on carefully chosen benchmark problems involving nonlinear fractional dynamics with proportional delay. These examples demonstrate our method’s capability to manage the challenges of nonlinearity, fractional order, and delay terms. The analysis of absolute, relative errors and and statistical performance measure confirms the accuracy and reliability of the technique, even with few iterations. We also discuss the method’s convergence behavior and how it compares to other numerical methods in terms of efficiency. The results demonstrated that the suggested approach provides accurate results with a limited number of terms and performs better than the other numerical techniques in the literature. The novelty of this work lies in integrating the natural transform with a modified decomposition method designed for fractional-delay systems. We also discuss the limitations and possible extensions of the method, offering insights for future research directions.