Differential Transform Technique for Higher Order Boundary Value Problems

This paper presents the approximate solution of higher order boundary value problems by differential transform method. Two examples are considered to illustrate the efficiency of this method. The results converge rapidly to the exact solution and are shown in tables and graphs.


Introduction
Recently, studies showed that higher order boundary value problems arise in the areas of fluid dynamics, hydrodynamics and hydromagnetic stability and other applied sciences.Specifically, fifth-order boundary value problems arise in viscoelastic fluid.The problem was considered by Wazwaz (2001) using the decomposition method.Caglar et al (2006) solved it via B-spline interpolation and compared the results with finite element and finite volume methods and Triphathi (2012).Sixth order boundary value problems occur in astrophysics and it has attracted the attention of researchers like Wazwaz (2001) who investigated it using modified decomposition method, He (2003) used variational approach method and Erturk (2007) approached it via differential transformation method.However, seventh order boundary value problems that arise in modeling induction motors with two rotor circuits was considered by Siddiqi et. al (2012) while eight-order boundary value problem which occur in hydrodynamic and hydromagnetic stability was also studied by Siddiqi et al (1996) and Mohammad-Jawad (2010).Other authors who have also studied higher order boundary value problems include Wazwaz (2000), Othman et al. (2010) and Mohyud-Din (2010).
The differential transform method is applied in this work to solve boundary value problems of ninth and twelfth orders.This method was proposed by Zhou (1986).Some authors who have also adopted this method include, Opanuga et al (2014) on systems of ordinary differential equations, also Opanuga et al (2015) applied it in numerical solution of two-point boundary value problems, Edeki et al (2014) analyzed linear and nonlinear differential equations and finally Edeki et al (2015), in transformed Cauchy-Euler equidimensional equations of homogenous type.
The exact solution for the bvp is The differential transformation of equation ( 4) is given as and differential transformation of the boundary conditions yield Using the boundary conditions (8) in the transformed equation ( 7) at 0 x = , we obtain the solution of ( ) u t , for 9 t ≥ We finally obtain the following series of equation using the inverse transformation equation ( 3  The exact solution for the boundary -value problem is written as The differential transform of equation ( 12) is given as and the differential transformation of the boundary conditions yield , ( 4) , ( 6) , ( 8) (1), Using the transformed boundary conditions ( 16) in equation ( 15) at 0 x = , we obtain the series solution ( ) u t , for 12 t ≥ .The constants A, B, C and D are evaluated by using the boundary conditions ( 16) at 1 x = to give the system of equations below.(17) Solving the system of equations yield the following: 0 .9 9 9 9 9 8 3 6 1 4 , 1 .00 0 0 1 6 1 7 5, 0 .9 9 9 8 4 0 7 3 2 2 , 1 .00 1 5 5 8 9 9 , 0 .98 5 1 0 1 1 4 0 3, 1 .1 3 2 1 1 2 4 7 3 We then obtain the following series solution using the inverse transformation equation (3) up to

Concluding Remarks
This paper has applied differential transform method to solve ninth-order and twelfth-order boundary value problems.The method is easy to apply, accurate and efficient.This is evident from table 1 and the graphical representations of the solution which show strong agreement with the exact solution.

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The constants A, B, C and D can be determined by using the boundary conditions (8

Table 2 .
Numerical solution for example