Application of the Hybrid Differential Transform Method to the Nonlinear Equations

Abstract

In this paper, a hybrid method is introduced briefly to predict the behavior of the non-linear partial differential equations. The method is hybrid in the sense that different numerical methods, differential transform and finite differences, are used in different subdomains. Our aim of this approach is to combine the flexibility of differential transform and the efficiency of finite differences. An explicit hybrid method for the transient response of inhomogeneous nonlinear partial differential equations is presented; applying finite difference scheme on the fixed grid size is used to approximate the space discretisation, whereas the differential transform method is used for time operator. Comparison of the efficiency of the different approaches is a very important aspect of this study. In our test cases, the hybrid approach is faster than the corresponding highly optimized finite difference method in two dimensional computations. We compared our hybrid approach’s results with the exact and/or numerical solutions of PDE which obtained from Adomian Decomposition Method. Results show that the hybrid approach may be an important tool to reduce the execution time and memory requirements for large scale computations and get remarkable results in predicting the solutions of nonlinear initial value problems

New Algorithm for the Lid-driven Cavity Flow Problem with Boussinesq-Stokes Suspension

“…The procedure adopted above has facilitated the conversion of the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations ( Jang et al 2000, Cilingir Süngü and Demir 2012a, 2012b. The Gauss-Siedel iterative procedure then used to solve the linear system thus has assured convergence.…”

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A Semi-Discretization Method Based on Finite Difference and Differential Transform Methods to Solve the Time-Fractional Telegraph Equation

“…In 2008, a hybrid method based on the combination of DTM and FDM was presented to solve a nonlinear heat conduction differential equation [34]. Also, in 2012, some nonlinear PDEs were solved with the hybrid method [35]. Arsalan (2020) applied a hybrid scheme to solve the one-dimensional integer-order telegraph equations [36,37].…”

Solving Whitham-Broer-Kaup-Like Equations Numerically by using Hybrid Differential Transform Method and Finite Differences Method

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