Research Article  Open Access
On the Solutions of Nonlinear HigherOrder Boundary Value Problems by Using Differential Transformation Method and Adomian Decomposition Method
Abstract
We study higherorder boundary value problems (HOBVP) for higherorder nonlinear differential equation. We make comparison among differential transformation method (DTM), Adomian decomposition method (ADM), and exact solutions. We provide several examples in order to compare our results. We extend and prove a theorem for nonlinear differential equations by using the DTM. The numerical examples show that the DTM is a good method compared to the ADM since it is effective, uses less time in computation, easy to implement and achieve high accuracy. In addition, DTM has many advantages compared to ADM since the calculation of Adomian polynomial is tedious. From the numerical results, DTM is suitable to apply for nonlinear problems.
1. Introduction
Recently, many researchers use ADM to approximate numerical solutions. In [1], Wazwaz proposed a modification of ADM method in series solution to accelerate its rapid convergence, and, in [2], Wazwaz also presented several numerical examples of higherorder boundary value problems for firstorder linear equation and secondorder nonlinear equation by applying modified decomposition method.
In addition, Wazwaz [3, 4] provided firstorder linear and secondorder nonlinear problems to solve fifthorder and sixthorder boundary value problems by the modified decomposition method. Later, MeΕ‘troviΔ [5] solved eightorder boundary value problems for firstorder linear and secondorder nonlinear boundary value problems. Similarly, in [6], Hosseini and Jafari used Adomian decomposition method to solve highorder and system of nonlinear differential equations.
The Adomian decomposition method is widely used in applied science to compute the series solution accurately because it provides rapid convergent series to solve the problem. The big application of ADM in research area is stochastic and deterministic problems involving differential, integrodifferential, integral, differential delay, and systems of such equations; for example, see [3, 7]. For example, we use the same method for solving several different problems, such as, in calculus of variations, see [8], for eikonal partial differential equation, see [9], for the FitzhughNagumo equation which models the transmission of nerve impulses, see [10], for linear and nonlinear systems of Volterra functional equations using AdomianPade technique, see [11], for coupled Burgers equations by using AdomianPade technique see [12], for solution of a nonlinear timedelay model in biology by using semianalytical approaches, see [13], for solving the pantograph equation of order , see [14], and for nonclassic problem for onedimensional hyperbolic equation by using the decomposition procedure, see [15].
Further, Ray and Bera in [16] used ADM to solve analytical solution of a fractional diffusion equation. They performed the explicit solution of the equation in the closed form by using initial value problem.
In [17], Ayaz investigated initial value problem of partial differential equation (PDE) to solve twodimensional differential transformation method, and we compare the results with Adomian decomposition method. The results show that the solutions of the present method are exactly the same as the decomposition method, but the calculation of DTM is simple and reduces the difficulty of calculations.
Ayaz in [18] performed two and threedimensional differential transformation methods to find exact solutions of linear and nonlinear partial differential equations. Results are compared to decomposition method, and DTM has less computational effort. After that, ErtΓΌrk and Momani in [19] presented numerical solution by comparing the differential transformation method (DTM) and Adomian decomposition method (ADM) for solving linear and nonlinear fourthorder boundary value problems and proved that DTM is very accurate and efficient in numerical solution.
Recently, Arikoglu and Ozkol solved fractional differential equations by using differential transform method. They applied fractional differential equations to various types of problems such as the BagleyTorvik, Ricatti, and composite fractional oscillation equations; see [20].
In this study, we make comparison among differential transformation method, Adomian decomposition method and exact solutions. We prove that DTM is more powerful technique than ADM and can be applied to nonlinear problems easily.
2. Differential Transformation Method
Suppose that the function is continuously differentiable in the interval for , then we have the following definition.
Definition 2.1. The differential transform of the function for the th derivative is defined as follows: where is the original function and is the transformed function. The inverse differential transform of is defined as Note that, the substitution of (2.1) into (2.2) yields the following equation: which is the Taylor's series for at .
The following theorems are operations of differential transforms.
Theorem 2.2. If , then .
Theorem 2.3. If , then .
Theorem 2.4. If , then .
Theorem 2.5. If , then .
Theorem 2.6. If , then .
Theorem 2.7. If , then .
Theorem 2.8. If , then , where
Theorem 2.9. If , then .
Theorem 2.10. If , then .
Theorem 2.11. If , then .
Theorem 2.12. If , then .
See the details in [21].
3. General Differential Transformation
Now we prove the following generalized theorem.
Theorem 3.1. The general differential transformation for nonlinear thorder BVPs, , is given by
Proof. We prove this theorem by induction method. First of all, we prove the generalization of the differential equation . Let , then
For , we have
By definition, we have
From (3.1), for , we have
This implies that the theorem holds for . Now assume that for , the DT is as follows:
Note that we have
For , we have
Thus, for , we have
For , we have
By definition, we have
From (3.1), for , we have
This implies that the theorem holds for . Now, we prove the generalization of the BVPs,
For that purpose, we fixed .
For and , we have
By definition, we have
From (3.1), for , we have
Now that is the required result; thus, the theorem holds for .
Now assume that for , the DT is as follows:
For , we have
Thus, for , we have
For , we have
From (3.1), for , we have
Note that the theorem holds for .
Example 3.2. By using DTM, we solve the following nonlinear equation of order three for fifthorder BVP:
subject to the boundary conditions
By using Theorem 3.1, (3.22) is transformed to the following equation:
By using (2.1) at , we obtain the following transformed boundary conditions:
where and .
By using the transformed equation (3.24) together with the transformed boundary conditions (3.25), we can easily solve , for . The constants and can be evaluated by using boundary conditions (3.23) at :
These equations give and .
Finally, the following series solution can be formed by applying the inverse transformation in (2.2) up to :
Example 3.3. Next, we solve the following fourthorder nonlinear for fifthorder BVP by using DTM:
subject to the boundary conditions
By using Theorem 3.1, (3.28) is transformed to the following equation:
By using (2.1) at , we obtain the following transformed boundary conditions:
where and .
By using the transformed equation (3.30) together with the transformed boundary conditions (3.31), we can easily solve , for . The constants and can be evaluated by using boundary conditions (3.29) at :
These equations give and .
Finally, the following series solution can be formed by applying the inverse transformation in (2.2) up to :
Example 3.4. We perform the following third order of nonlinear equation for sixthorder BVP by using DTM:
subject to the boundary conditions
By using Theorem 3.1, (3.34) is transformed to the following equation:
By using (2.1) at , we obtain the following transformed boundary conditions:
where , , and .
By using the transformed equation (3.36) together with the transformed boundary conditions (3.37), we can easily solve , for . The constants and can be evaluated by using boundary conditions (3.35) at :
These equations give , , and .
Consequently, the following series solution can be formed by applying the inverse transformation equation in (2.2) up to :
4. Adomian Decomposition Method
Consider the differential equation where is invertible and is the highestorder derivative, is a linear differential operator, where the order of must be greater than , is a nonlinear operator, and is the source term. By using the given conditions and applying the inverse operator to both sides of (4.1), we get the following equation: where the function is arising from integrating the source term and from applying the given conditions which are prescribed.
The series solution of by the standard Adomian method is given as follows: On the other hand, the nonlinear function by an infinite series of polynomial is given as follows: The components are determined by using the following relation: for .
From the above equations, we observe that the component is identified by the function . We can apply modification by assuming that the function is divisible into two parts, and . Let the function be as follows: Under this assumption, we have a slight variation for components and , where assigned to and is combined with the other terms in (4.5) to assign . The modified recursive algorithm is as follows: for . There are several rules that are needed to follow for Adomian polynomials of nonlinear operator : and so on; see [3].
For comparison purpose, we solve the boundary value problems in Examples 3.2, 3.3, and 3.4 by using the Adomian decomposition method.
Example 4.1. By using ADM, we solve the following nonlinear equation of order three for fifthorder BVP: subject to the boundary conditions Equation (4.10) can be rewritten in operator form as follows: Operating with fivefold integral operator on (4.12) and using the boundary conditions at , we obtain the following equation: Then, determine the constants , . Substituting the decomposition series (4.3) for and the series of polynomials (4.4) into (4.13) yields Then, we split the terms into two parts which are assigned to and that are not included under in (4.14). We can obtain the following recursive relation: To determine the constants and , we use the boundary conditions in (4.11) at on the fourterm approximant , where Then, solving the above equation yields Then we get the series solution as follows:
Example 4.2. Now, we solve the following fourthorder nonlinear for fifthorder boundary value problems by using ADM: subject to the boundary conditions Equation (4.19) can be rewritten in operator form as follows: Operating with fivefold integral operator on (4.21) and using the boundary conditions at , we obtain the following equation: Then, determine the constants , . Substituting the decomposition series (4.3) for and the series of polynomials (4.4) into (4.22) yields Then, we split the terms into two parts which are assigned to and that are not included under in (4.23). We can obtain the following recursive relation: To determine the constants and , we use the boundary conditions in (4.20) at on the fourterm approximant , where Then, solving the above equation yields Finally, we get the series solution as follows:
Example 4.3. Finally, we perform third order of nonlinear function for sixthorder BVP for ADM as follows: subject to the boundary conditions Equation (4.28) can be rewritten in operator form as follows: Operating with sixfold integral operator on (4.30) and using the boundary conditions at , we obtain the following equation: Then, determine the constants , , and . Substituting the decomposition series (4.3) for and the series of polynomials (4.4) into (4.31) yields Then similarly, we split the terms into two parts which are assigned to and that are not included under in (4.32). We can obtain the following recursive relation: To determine the constants , , and , we use the boundary conditions in (4.29) at on the fourterm approximant , where Then, solving the above equation yields Finally, we get the series solution as follows:
5. Results
We provide the results of the given examples in Tables 1, 2, and 3.



6. Conclusion
From the results, the proposed method, as well as the Differential Transformation Method, is more accurate than the Adomian Decomposition Method. The errors between the solutions of Differential Transformation Method and the exact solutions are smaller compared to the errors between the solutions of Adomian Decomposition Method and the exact solutions. In addition, Differential Transformation Method also shows less computational effort because it needs less time in calculation. Besides that, it is hard to calculate Adomian polynomials. From the results we obtained, it can reinforce conclusion made by many researchers that Differential Transformation Method is more efficient and accurate than Adomian Decomposition Method. Therefore, we can conclude that Differential Transformation Method is applicable for such problems in the bounded domains. The computations in all examples were performed by using Maple 13.
Acknowledgments
The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant scheme 0501090720RU and Fundamental Research Grant scheme 011109723FR. The authors also thank the referee(s) for very constructive comments and suggestions that improved the paper.
References
 A.M. Wazwaz, βA reliable modification of Adomian decomposition method,β Applied Mathematics and Computation, vol. 102, no. 1, pp. 77β86, 1999. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A.M. Wazwaz, βApproximate solutions to boundary value problems of higher order by the modified decomposition method,β Computers & Mathematics with Applications, vol. 40, no. 67, pp. 679β691, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A.M. Wazwaz, βThe numerical solution of fifthorder boundary value problems by the decomposition method,β Journal of Computational and Applied Mathematics, vol. 136, no. 12, pp. 259β270, 2001. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A.M. Wazwaz, βThe numerical solution of sixthorder boundary value problems by the modified decomposition method,β Applied Mathematics and Computation, vol. 118, no. 23, pp. 311β325, 2001. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. MeΕ‘troviΔ, βThe modified decomposition method for eighthorder boundary value problems,β Applied Mathematics and Computation, vol. 188, no. 2, pp. 1437β1444, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. M. Hosseini and M. Jafari, βA note on the use of Adomian decomposition method for highorder and system of nonlinear differential equations,β Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 1952β1957, 2009. View at: Google Scholar
 M. Dehghan and F. Shakeri, βThe numerical solution of the second PainlevΓ© equation,β Numerical Methods for Partial Differential Equations, vol. 25, no. 5, pp. 1238β1259, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Dehghan and M. Tatari, βThe use of Adomian decomposition method for solving problems in calculus of variations,β Mathematical Problems in Engineering, vol. 2006, Article ID 65379, 9 pages, 2006. View at: Google Scholar  Zentralblatt MATH
 M. Dehghan and R. Salehi, βA seminumeric approach for solution of the eikonal partial differential equation and its applications,β Numerical Methods for Partial Differential Equations, vol. 26, no. 3, pp. 702β722, 2010. View at: Google Scholar  Zentralblatt MATH
 M. Dehghan, J. M. Heris, and A. Saadatmandi, βApplication of semianalytic methods for the FitzhughNagumo equation, which models the transmission of nerve impulses,β Mathematical Methods in the Applied Sciences, vol. 33, no. 11, pp. 1384β1398, 2010. View at: Google Scholar  Zentralblatt MATH
 M. Dehghan, M. Shakourifar, and A. Hamidi, βThe solution of linear and nonlinear systems of Volterra functional equations using AdomianPade technique,β Chaos, Solitons and Fractals, vol. 39, no. 5, pp. 2509β2521, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Dehghan, A. Hamidi, and M. Shakourifar, βThe solution of coupled Burgers' equations using AdomianPade technique,β Applied Mathematics and Computation, vol. 189, no. 2, pp. 1034β1047, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Dehghan and R. Salehi, βSolution of a nonlinear timedelay model in biology via semianalytical approaches,β Computer Physics Communications, vol. 181, no. 7, pp. 1255β1265, 2010. View at: Publisher Site  Google Scholar
 F. Shakeri and M. Dehghan, βApplication of the decomposition method of adomian for solving the pantograph equation of order m,β Zeitschrift fur Naturforschung, vol. 65, no. 5, pp. 453β460, 2010. View at: Google Scholar
 M. Dehghan, βThe solution of a nonclassic problem for onedimensional hyperbolic equation using the decomposition procedure,β International Journal of Computer Mathematics, vol. 81, no. 8, pp. 979β989, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 S. S. Ray and R. K. Bera, βAnalytical solution of a fractional diffusion equation by Adomian decomposition method,β Applied Mathematics and Computation, vol. 174, no. 1, pp. 329β336, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 F. Ayaz, βOn the twodimensional differential transform method,β Applied Mathematics and Computation, vol. 143, no. 23, pp. 361β374, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 F. Ayaz, βSolutions of the system of differential equations by differential transform method,β Applied Mathematics and Computation, vol. 147, no. 2, pp. 547β567, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 V. S. ErtΓΌrk and S. Momani, βComparing numerical methods for solving fourthorder boundary value problems,β Applied Mathematics and Computation, vol. 188, no. 2, pp. 1963β1968, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. Arikoglu and I. Ozkol, βSolution of fractional differential equations by using differential transform method,β Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1473β1481, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 I. H. AbdelHalim Hassan and V. S. ErtΓΌrk, βSolutions of different types of the linear and nonlinear higherorder boundary value problems by differential transformation method,β European Journal of Pure and Applied Mathematics, vol. 2, no. 3, pp. 426β447, 2009. View at: Google Scholar
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Copyright Β© 2011 Che Haziqah Che Hussin and Adem KiliΓ§man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.