Numerical solutions of linear and nonlinear higher-order boundary value problems by differential transformation method and Adomian decomposition method

In this research, we proposed the generalization of differential transformation method for solving higher-order boundary value problems of higher-order linear and non-linear differential equations. In particular, we extended the existing method to solve th-order boundary value problems of th-order linear and th-order non-linear differential equations. To verify the proposed theorems, we provided proof for each theorem. In addition, one of the main objectives of this research is to measure the accuracy level of the proposed method. To achieve this objective, we provided some numerical examples for each proposed theorem. We solved each problem in the numerical examples by using the proposed method and modified Adomian decomposition method. Then, we compared the results with the exact solution of each problem. The method with smaller error with the exact solution will be considered as an accurate method. From the numerical examples also, we want to observe which method that is simpler and easier to implement. We also consider time consumption to calculate DTM and ADM. To achieve all these objectives, we study rigorously the differential transformation method and the modified Adomian decomposition method for solving higher-order boundary value problems. We focus on differential transformation method for solving linear and nonlinear higherorder boundary value problems. Different types of higher-orders are chosen such as the fourth, fifth, sixth and seventh-order boundary value problems. We observed that,solving each problem by using the modified Adomian decomposition method is very hard and needs more time in calculation. In addition, the calculation of Adomian's polynomial is tedious. On the contrary, the proposed method is simpler and easier to implement. It involves smaller number of computation steps compared to the modified decomposition method. The linearization, discretization or perturbation also is not required during the calculation processes. From the numerical results, we observed that the proposed method gives closer approximation to the exact solution since the error is smaller than the modified decomposition method. This implies that, the proposed method is more accurate than the modified Adomian decomposition method. Besides that, we also solved system of differential equations by using differential transformation method for linear and nonlinear equation. These systems of differential equations can’t be solved by using modified decomposition method since the method is very hard to implement. From the results that we obtained, it can reinforce conclusion made by many researchers that differential transformation method is more efficient and accurate than modified Adomian decomposition method. Therefore, we proved the differential transformation method very successful and powerful tool in numerical solution for the bounded domains.

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