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International Journal of Differential Equations
Volume 2016 (2016), Article ID 1015634, 12 pages
http://dx.doi.org/10.1155/2016/1015634
Research Article

Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term

1Department of Mathematics, Faculty of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
2Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shibin El-Kom 32511, Egypt

Received 29 July 2016; Accepted 5 October 2016

Academic Editor: Patricia J. Y. Wong

Copyright © 2016 Essam R. El-Zahar. 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.

Abstract

A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term.

1. Introduction

Many mathematical problems that model real-life phenomena cannot be solved completely by analytical means. Some of the most important mathematical problems arising in applied mathematics are singular perturbation problems. These problems commonly occur in many branches of applied mathematics such as transition points in quantum mechanics, edge layers in solid mechanics, boundary layers in fluid mechanics, skin layers in electrical applications, and shock layers in fluid and solid mechanics. The numerical treatment of these problems is accompanied by major computational difficulties due to the presence of sharp boundary and/or interior layers in the solution. Therefore, more efficient and simpler computational methods are required to solve these problems.

For the past two decades, many numerical methods have appeared in the literature, which cover mostly second-order singular perturbation boundary value problems (SPBVPs) [13]. But only few authors have developed numerical methods for higher order SPBVPs (see, e.g., [410]). However, most of them have concentrated on problems with smooth data. In fact some authors have developed numerical methods for problems with discontinuous data which gives rise to an interior layer in the exact solution of the problem, in addition to the boundary layer at the outflow boundary point. Most notable among these methods are piecewise-uniform mesh finite difference method [1114] and fitted mesh finite element method [15, 16] for third- and fourth-order SPBVPs with a discontinuous source term. The aim of this paper is to employ a semianalytical method which is Differential Transform Method (DTM) as an alternative to existing methods for solving high-order SPBVPs with a discontinuous source term.

DTM is introduced by Zhou [17] in a study of electric circuits. This method is a formalized modified version of Taylor series method where the derivatives are evaluated through recurrence relations and not symbolically as the traditional Taylor series method. The method has been used effectively to obtain highly accurate solutions for large classes of linear and nonlinear problems (see, e.g., [1721]). There is no need for discretization, perturbations, and further large computational work and round-off errors are avoided. Additionally, DTM does not generate secular terms (noise terms) and does not need analytical integrations as other semianalytical methods like HPM, HAM, ADM, or VIM and so DTM is an attractive tool for solving differential equations.

In this paper, a reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and DTM. First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical method for treating high-order singular perturbation problems with a discontinuous source term.

2. Differential Transform Method for ODE System

Let us describe the DTM for solving the following system of ODEs:subject to the initial conditionsLet be the interval over which we want to find the solution of (1)-(2). In actual applications of the DTM, the th-order approximate solution of (1)-(2) can be expressed by the finite serieswherewhich implies that is negligibly small. Using some fundamental properties of DTM (Table 1), the ODE system (1)-(2) can be transformed into the following recurrence relations:where is the differential transform of the function , for . Solving the recurrence relation (5), the differential transform , , can be easily obtained.

Table 1: Some fundamental operations of DTM.

3. Description of the Method

Motivated by the works of [11, 13, 16], we, in the present paper, suggest an asymptotic semianalytic method which is DTM to develop piecewise approximate analytical solutions for the following class of SPBVPs.

Third-Order SPBVP [13]. Find such that where , , and are sufficiently smooth functions on satisfying the following conditions:where   is  arbitrarily  close  to  1, for some .

Fourth-Order SPBVP [11]. Find such that where , , and are sufficiently smooth functions on satisfying the following conditions:where s  arbitrarily  close  to  1,  for  some. For both the problems defined above, , , , , , and . It is assumed that is sufficiently smooth on and its derivatives have discontinuity at the point and the jump at is given as .

3.1. Zero-Order Asymptotic Expansion Approximations

The SPBVP (6) can be transformed into an equivalent problem of the formwhere and [14].

Similarly the SPBVP (8) can be transformed intowhere and [11].

Remark 1. Hereafter, only the above systems (10) and (11) are considered.

Using some standard perturbation methods [11, 13, 16, 22] one can construct an asymptotic expansion for the solution of (10) and (11) as follows.

Find a continuous function of the terminal value reduced system of (10) such thatThat is, find a smooth function on that satisfies the following equivalent reduced BVP:Then findDefine on aswhere .

Similarly one can construct an asymptotic expansion for the solution of (11). In fact, for this problem is the solution of the terminal value reduced system That is, in particular, find a smooth function on that satisfies the following equivalent reduced BVP:Then find .

Define on aswhere

Theorem 2 (see [11, 13]). The zero-order asymptotic expansion defined above for the solution of (10) and (11) satisfies the inequality

Now, in order to obtain piecewise analytical solutions of (10) and (11), we only need to obtain piecewise analytical solutions of the terminal value reduced systems (12) and (16), that is, the solution of equivalent reduced BVPs (13) and (17).

3.2. Piecewise Approximate Analytical Solutions

The solution of BVP (13) can be represented as a piecewise solution form:Thus the BVP (13) is transformed intowith continuity and smoothness conditions , and , are unknown constants.

Applying th-order DTM on (22) results in the recurrence relationswhere , , , , , and are the differential transform of , , , , , and , respectively, and and values are determined from the transformed continuity and smoothness conditions:The recurrence relations (23) with transformed conditions (24) represent a system of algebraic equations in the coefficients of the power series solution of the reduced BVP (13) and the unknowns and . Solving this algebraic system, the piecewise smooth approximate solution of (13) is obtained and given byAnd thus, the piecewise approximate analytical solution of (10) is obtained and given bywhere .

Similarly the reduced BVP (17) can be transformed intowith continuity and smoothness conditions , , and , , are unknown constants.

Applying th-order DTM on (27) results in the recurrence relationswhere the unknown constants , , and are determined from the transformed continuity and smoothness conditions:And the piecewise approximate analytical solution of (11) is obtained and given bywhere

3.3. Error Estimate

The error estimate of the present method has two sources: one from the asymptotic approximation and the other from the truncated series approximation by DTM.

Theorem 3. Let be the solution of (10). Further let be the approximate solution (26). Then

Proof. Since the DTM is a formalized modified version of the Taylor series method, then we have a bounded error given by From Theorem 2 and the above bounded error, we haveSince the singular perturbation parameter is extremely small, the present method works well for singular perturbation problems.

Remark 4. A similar statement is true for the solution of (11) and the approximate solution (30).

4. Illustrating Examples

In this section we will apply the method described in the previous section to find piecewise approximate analytical solutions for three SPBVPs with a discontinuous source term.

Example 1. Consider the third-order SPBVP from [13, 16]whereUsing the present method with 5th-order DTM, the piecewise analytical solution is given by

Example 2. Consider the third-order SPBVP with variable coefficients from [13, 16]whereUsing the present method with 5th-order DTM, the piecewise analytical solution is given by

Example 3. Consider the fourth-order SPBVP from [13, 16] whereUsing the present method with 5th-order DTM, the piecewise analytical solution is given byThe numerical solution for each example is presented overall the problem domain as shown in Figures 13.
The corresponding maximum pointwise errors are taken to bewhere is the obtained approximate solution using th-order DTM over a uniform mesh , , , and is our numerical reference solution obtained using DTM with order .
The computed maximum pointwise errors and for the above solved BVPs are given in Tables 27. The numerical results in Tables 27 agree with the theoretical ones present in this paper where the obtained solutions and their derivatives converge rapidly to the reference solutions with increasing the order of the DTM.

Table 2: Maximum pointwise errors and for the solution of Example 1.
Table 3: Maximum pointwise errors and for the first derivative solution of Example 1.
Table 4: Maximum pointwise errors and for the solution of Example 2.
Table 5: Maximum pointwise errors and for the first derivative solution of Example 2.
Table 6: Maximum pointwise errors and for the solution of Example 3.
Table 7: Maximum pointwise errors and for the second derivative solution of Example 3.
Figure 1: Graphs of the approximate solution and its first derivative for Example 1 at and .
Figure 2: Graphs of the approximate solution and its first derivative for Example 2 at and .
Figure 3: Graphs of the approximate solution and the second derivative for Example 3 at and .

5. Conclusion

We have presented a new reliable algorithm to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion SPBVPs with a discontinuous source term. The algorithm is based on constructing a zero-order asymptotic expansion of the solution and the DTM which provides the solutions in terms of convergent series with easily computable components. The original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion for the solution of the transformed system is constructed. For simplicity, the result terminal value reduced system is replaced by its equivalent reduced BVP with suitable continuity and smoothness conditions. Then a piecewise smooth solution of the reduced BVP is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented and shows that the method results in high-order convergence for small values of the singular perturbation parameter. We have applied the method on three SPBVPs and the piecewise analytical solution is presented for each one overall the problem domain. The numerical results confirm that the obtained solutions and their derivatives converge rapidly to the reference solutions with increasing the order of the DTM. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order SPBVPs with a discontinuous source term. The method is based on a straightforward procedure, suitable for engineers.

Competing Interests

The author declares that he has no competing interests.

References

  1. M. K. Kadalbajoo and K. C. Patidar, “A survey of numerical techniques for solving singularly perturbed ordinary differential equations,” Applied Mathematics and Computation, vol. 130, no. 2-3, pp. 457–510, 2002. View at Publisher · View at Google Scholar · View at Scopus
  2. M. Kumar, P. Singh, and H. K. Mishra, “A recent survey on computational techniques for solving singularly perturbed boundary value problems,” International Journal of Computer Mathematics, vol. 84, no. 10, pp. 1439–1463, 2007. View at Publisher · View at Google Scholar · View at Scopus
  3. K. K. Sharma, P. Rai, and K. C. Patidar, “A review on singularly perturbed differential equations with turning points and interior layers,” Applied Mathematics and Computation, vol. 219, no. 22, pp. 10575–10609, 2013. View at Publisher · View at Google Scholar · View at Scopus
  4. S. Valarmathi and N. Ramanujam, “An asymptotic numerical method for singularly perturbed third-order ordinary differential equations of convection-diffusion type,” Computers and Mathematics with Applications, vol. 44, no. 5-6, pp. 693–710, 2002. View at Publisher · View at Google Scholar · View at Scopus
  5. J. C. Roja and A. Tamilselvan, “Numerical method for singularly perturbed third order ordinary differential equations of convection-diffusion type,” Numerical Mathematics: Theory, Methods and Applications, vol. 7, no. 3, pp. 265–287, 2014. View at Publisher · View at Google Scholar · View at Scopus
  6. V. Shanthi and N. Ramanujam, “A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations,” Computers and Mathematics with Applications, vol. 47, no. 10-11, pp. 1673–1688, 2004. View at Publisher · View at Google Scholar · View at Scopus
  7. M. Cui and F. Geng, “A computational method for solving third-order singularly perturbed boundary-value problems,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 896–903, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. M. Kumar and S. Tiwari, “An initial-value technique to solve third-order reaction-diffusion singularly perturbed boundary-value problems,” International Journal of Computer Mathematics, vol. 89, no. 17, pp. 2345–2352, 2012. View at Publisher · View at Google Scholar · View at Scopus
  9. M. I. Syam and B. S. Attili, “Numerical solution of singularly perturbed fifth order two point boundary value problem,” Applied Mathematics and Computation, vol. 170, no. 2, pp. 1085–1094, 2005. View at Publisher · View at Google Scholar · View at Scopus
  10. E. R. El-Zahar, “Approximate analytical solutions of singularly perturbed fourth order boundary value problems using differential transform method,” Journal of King Saud University—Science, vol. 25, no. 3, pp. 257–265, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. V. Shanthi and N. Ramanujam, “Asymptotic numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 252–266, 2006. View at Publisher · View at Google Scholar · View at Scopus
  12. V. Shanthi and N. Ramanujam, “An asymptotic numerical method for fourth order singular perturbation problems with a discontinuous source term,” International Journal of Computer Mathematics, vol. 85, no. 7, pp. 1147–1159, 2008. View at Publisher · View at Google Scholar · View at Scopus
  13. T. Valanarasu and N. Ramanujam, “An asymptotic numerical method for singularly perturbed third-order ordinary differential equations with a weak interior layer,” International Journal of Computer Mathematics, vol. 84, no. 3, pp. 333–346, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. T. Valanarasu and N. Ramanujam, “Asymptotic numerical method for singularly perturbed third order ordinary differential equations with a discontinuous source term,” Novi Sad Journal of Mathematics, vol. 37, no. 2, pp. 41–57, 2007. View at Google Scholar
  15. A. R. Babu and N. Ramanujam, “An asymptotic finite element method for singularly perturbed third and fourth order ordinary differential equations with discontinuous source term,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 372–380, 2007. View at Publisher · View at Google Scholar · View at Scopus
  16. A. R. Babu and N. Ramanujam, “An asymptotic finite element method for singularly perturbed higher order ordinary differential equations of convection-diffusion type with discontinuous source term,” Journal of Applied Mathematics & Informatics, vol. 26, no. 5-6, pp. 1057–1069, 2008. View at Google Scholar
  17. J. K. Zhou, Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
  18. Z. Šmarda, J. Diblík, and Y. Khan, “Extension of the differential transformation method to nonlinear differential and integro-differential equations with proportional delays,” Advances in Difference Equations, vol. 2013, article 69, 2013. View at Publisher · View at Google Scholar · View at Scopus
  19. E. R. El-Zahar, “Applications of adaptive multi step differential transform method to singular perturbation problems arising in science and engineering,” Applied Mathematics & Information Sciences, vol. 9, no. 1, pp. 223–232, 2015. View at Publisher · View at Google Scholar · View at Scopus
  20. H. Fatoorehchi, H. Abolghasemi, and R. Zarghami, “Analytical approximate solutions for a general nonlinear resistor-nonlinear capacitor circuit model,” Applied Mathematical Modelling, vol. 39, no. 19, pp. 6021–6031, 2015. View at Publisher · View at Google Scholar · View at Scopus
  21. Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, and G. H. E. Duchamp, “A multi-step differential transform method and application to non-chaotic or chaotic systems,” Computers and Mathematics with Applications, vol. 59, no. 4, pp. 1462–1472, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. A. H. Nayfeh, Introduction to Perturbation Methods, John Wiley and Sons, New York, NY, USA, 1981.