Research Article  Open Access
Piecewise Approximate Analytical Solutions of HighOrder Singular Perturbation Problems with a Discontinuous Source Term
Abstract
A reliable algorithm is presented to develop piecewise approximate analytical solutions of third and fourthorder 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 zeroorder 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 highorder singular perturbation problems with a discontinuous source term.
1. Introduction
Many mathematical problems that model reallife 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 secondorder singular perturbation boundary value problems (SPBVPs) [1â€“3]. But only few authors have developed numerical methods for higher order SPBVPs (see, e.g., [4â€“10]). 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 piecewiseuniform mesh finite difference method [11â€“14] and fitted mesh finite element method [15, 16] for third and fourthorder 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 highorder 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., [17â€“21]). There is no need for discretization, perturbations, and further large computational work and roundoff 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 fourthorder 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 zeroorder 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 highorder 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 thorder 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.

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.
ThirdOrder SPBVP [13]. Find such that where , , and are sufficiently smooth functions on satisfying the following conditions:where â€‰â€‰isâ€‰â€‰arbitrarilyâ€‰â€‰closeâ€‰â€‰toâ€‰â€‰1, for some .
FourthOrder SPBVP [11]. Find such that where , , and are sufficiently smooth functions on satisfying the following conditions:where sâ€‰â€‰arbitrarilyâ€‰â€‰closeâ€‰â€‰toâ€‰â€‰1,â€‰â€‰forâ€‰â€‰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. ZeroOrder 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 zeroorder 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 thorder 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 thorder 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 thirdorder SPBVP from [13, 16]whereUsing the present method with 5thorder DTM, the piecewise analytical solution is given by
Example 2. Consider the thirdorder SPBVP with variable coefficients from [13, 16]whereUsing the present method with 5thorder DTM, the piecewise analytical solution is given by
Example 3. Consider the fourthorder SPBVP from [13, 16] whereUsing the present method with 5thorder DTM, the piecewise analytical solution is given byThe numerical solution for each example is presented overall the problem domain as shown in Figures 1â€“3.
The corresponding maximum pointwise errors are taken to bewhere is the obtained approximate solution using thorder 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 2â€“7. The numerical results in Tables 2â€“7 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.

