Solving partial differential equations by two-dimensional differential transform method
Applied Mathematics and Computation
On solving the initial-value problems using the differential transformation method
Applied Mathematics and Computation
Two-dimensional differential transform for partial differential equations
Applied Mathematics and Computation
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The main difficulty in solving nonlinear differential equations by the differential transformation method (DTM) is how to treat complex nonlinear terms. This method can be easily applied to simple nonlinearities, e.g, polynomials, however obstacles exist for treating complex nonlinearities. In the latter case, a technique has been recently proposed to overcome this difficulty, which is based on obtaining a differential equation satisfied by this nonlinear term and then applying the differential transformation method to this obtained differential equation. Accordingly, if a differential equation has n-nonlinear terms, then this technique must be separately repeated for each nonlinear term, i.e., n-times, consequently a system of n-recursive relations is required. This significantly increases the computational budget. We instead propose a general symbolic formula to treat any analytic nonlinearity. The new formula can be easily applied when compared with the only other available technique. We also show that this formula has the same mathematical structure as the Adomian polynomials but with constants instead of variable components. Several nonlinear ordinary differential equations are solved to demonstrate the reliability and efficiency of the improved DTM method, which increases its applicability.