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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.

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) [

DTM is introduced by Zhou [

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.

Let us describe the DTM for solving the following system of ODEs:

Some fundamental operations of DTM.

Original function | Transformed function |
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Motivated by the works of [

The SPBVP (

Similarly the SPBVP (

Hereafter, only the above systems (

Using some standard perturbation methods [

Find a continuous function

Similarly one can construct an asymptotic expansion for the solution of (

Define

The zero-order asymptotic expansion

Now, in order to obtain piecewise analytical solutions of (

The solution

Applying

Similarly the reduced BVP (

Applying

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.

Let

Since the DTM is a formalized modified version of the Taylor series method, then we have a bounded error given by

A similar statement is true for the solution of (

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.

Consider the third-order SPBVP from [

Consider the third-order SPBVP with variable coefficients from [

Consider the fourth-order SPBVP from [

The corresponding maximum pointwise errors are taken to be

The computed maximum pointwise errors

Maximum pointwise errors

| Approximation order of DTM, | |||
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4 | 6 | 8 | 10 | |

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Maximum pointwise errors

| Approximation order of DTM, | |||
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4 | 6 | 8 | 10 | |

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Maximum pointwise errors

| Approximation order of DTM, | |||
---|---|---|---|---|

4 | 6 | 8 | 10 | |

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Maximum pointwise errors

| Approximation order of DTM, | |||
---|---|---|---|---|

4 | 6 | 8 | 10 | |

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| | | | |

| | | | |

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Maximum pointwise errors

| Approximation order of DTM, | |||
---|---|---|---|---|

4 | 6 | 8 | 10 | |

| | | | |

| | | | |

| | | | |

| | | | |

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Maximum pointwise errors

| Approximation order of DTM, | |||
---|---|---|---|---|

4 | 6 | 8 | 10 | |

| | | | |

| | | | |

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Graphs of the approximate solution

Graphs of the approximate solution

Graphs of the approximate solution

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.

The author declares that he has no competing interests.