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We consider a uniform finite difference method for nonlinear singularly perturbed multi-point boundary value problem on Shishkin mesh. The problem is discretized using integral identities, interpolating quadrature rules, exponential basis functions and remainder terms in integral form. We show that this method is the first order convergent in the discrete maximum norm for original problem (independent of the perturbation parameter ε). To illustrate the theoretical results, we solve test problem and we also give the error distributions in the solution in Table 1 and Figures 1-3.

In this paper we shall consider singularly perturbed multi-point nonlinear problem

where,

Singularly perturbed differential equations arise many applications such as, fluid mechanics, chemical-reactor theory, the Navier-Stokes equations of fluid flow at high Reynolds number, control theory, electrical networks, and other physical models. In recent years, singularly perturbed differential equations were studied by many authors in various fields of applied mathematics and engineering. For examples, Cziegis [

In this study we present uniformly convergent difference scheme on an equidistant mesh for the numerical solution of the problem (1)-(3). The difference scheme is constructed by the method integral identities with the use exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form [

In this section, we describe some properties of the solution of (1) with Lemma 2.1. we use

Lemma 2.1.

Let

where

solution of this problem.

So, the solution of the Equation (1) satisfies the inequalities

and

where, C_{0} and C are constants independent of ε.

Proof. We rewrite the Equation (1). Hence, we use intermediate value theorem for

where

Consequently, we obtain the following linear equation,

Now, let

We can write the solution of the Equation (9) as follows

where

First, we prove the estimate

Second, we prove the estimate

According to the Equation (4), λ is a finite number. Then, from the Equations (13), (14) we have the following inequality

we now prove the estimate the Equation (8).

If

and from the Equation (16)

Now, we take derivative of the Equation (9) and if it called

Now, we proceed with the estimation of

We use the following relation for

where

the Equation (21) with the values

In a similar manner, the Equation (21) with the values

We write the solution of the Equations (18), (19) in the form,

where

According to the maximum principle in the Equations (24), (25), we can the following Barrier function,

This Barrier function provides the conditions of the maximum principle and

In a similar manner, according to the maximum principle in the Equation (26), we can write

where

where

after some arragement, we can obtain,

Finally, from

which leads to the Equation (8).

Let us consider the following any non-uniform mesh on

We present some properties of the mesh function

Now, We will construct the difference scheme for the Equation (1). First, we integrate the Equation (1) over

where

If we rearrange the Equation (31) it gives

After doing some calculation

where

and

So, from the Equation (37), the difference scheme is defined by

Now, we define an approximation for the second boundary condition of the Equation (1). We accepted that

where remainder term

By neglecting

We will use the Shishkin mesh to be e -uniform convergent of the difference scheme the Equations (42)-(44). So the Shishkin mesh divides each of the interval

if

where, N is even number,

Let

where

Lemma 4.2. Let

Proof. We evaluate the Equation (38) and the Equation (41), respectively

Consequently,

In the beginning, we consider the case

it then follows from the Equation (50) that

Second, we consider the case _{i} on

and

In the seperate

In the seperate

where for

and so

Analogously for

thus

according to the Equation (54) and the Equation (55), we can rewrite the the Equation (53)

In the seperate

The last estimate is for

We rewrite the the Equation (50) for

We take integrate in the Equation (58) and so

where

we rewrite the Equation (59) with the Equation (60) and the Equation (61), thus,

where

We use in a similar way as above for

Next, we use estimate for the remainder term r:

From the Equation (41) we can write

from the Equation (8)

Lemma 4.3. Let z_{i} be solution of the Equations (45)-(47). Then there is the following inequality,

Proof. Rearranging the Equation (45) gives

where

according to the proof of Lemma 2.1, we can use the maximum principle, and so it is easy to obtain,

Conclusion 4.1. We know that the solution of the Equations (1)-(3) is

In this section, an example of nonlinear singularly perturbed multi-point boundary value problem is given to illustrate the efficiency of the numerical method described above. The example is computed using maple 10. Results obtained by the method are compared with the exact solution of example and found to be good agreement with each other. We compute approximate errors

Example 5.1.

We solve the difference scheme the Equations (42), (44) using the following iteration technique,

where

The system of the Equations (68)-(70) is solved by the following procedure,

It is easy to verify that

For this reason, the described procedure above is stable. Also, the Equations (42)-(44) has only one solution.

Now, we consider the following test problem,

which has the exact solution,

In the computations in this section, we will take

The error estimates are denoted by

and

The convergence rates are

The numerical results obtained from the problem of the difference scheme by comparison, the error and uniform rates of convergence were found and these are shown in

The results point out that the convergence rate of the established scheme is really in unision with theoretical analysis.

From the graps it is show that the error is maximum near the boundary layer and it is almost zero in outer region in the

ε | N değerleri | ||||||||
---|---|---|---|---|---|---|---|---|---|

8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | ||

2^{−10} | 0.1350816058 p = 0.826 | 0.0761888740 p = 0.806 | 0.00435607918 p = 0.821 | 0.00246466665 p = 0.884 | 0.00133500352 p = 1.02 | 0.0065502193 p = 1.300 | 0.0026594582 p = 1.72 | 0.0008027767 | |

2^{−12} | 0.1365365902 p = 0.811 | 0.0778089349 p = 0.777 | 0.0454015276 p = 0.763 | 0.0267503827 p = 0.768 | 0.0157072143 p = 0.797 | 0.0090342052 p = 0.869 | 0.0049452671 p = 1.016 | 0.0024447312 | |

2^{−14} | 0.1369028476 p = 0.807 | 0.0782173980 p = 0.769 | 0.0458717849 p = 0.748 | 0.0273010175 p = 0.739 | 0.0163561085 p = 0.740 | 0.0097927781 p = 0.753 | 0.0058067718 p = 0.789 | 0.0033601955 | |

2^{−16} | 0.1369926017 p = 0.806 | 0.0783189688 p = 0.768 | 0.0459894145 p = 0.745 | 0.0274402097 p = 0.731 | 0.0165223717 p = 0.725 | 0.0099918880 p = 0.725 | 0.0060447308 p = 0.731 | 0.0036407030 | |

2^{−18} | 0.1370138364 p = 0.806 | 0.07834719995 p = 0.767 | 0.0460165306 p = 0.744 | 0.0274742487 p = 0.730 | 0.0165639692 p = 0.721 | 0.0100425784 p = 0.717 | 0.0061060714 p = 0.717 | 0.0037143751 | |

2^{−20} | 0.1370841205 p = 0.806 | 0.0783970437 p = 0.768 | 0.0460092097 p = 0.743 | 0.0274772611 p = 0.729 | 0.0165727285 p = 0.720 | 0.0100552118 p = 0.715 | 0.0061220254 p = 0.713 | 0.0037341617 | |

2^{−22} | 0.1369241824 p = 0.802 | 0.0784855980 p = 0.770 | 0.0460110587 p = 0.743 | 0.0274794446 p = 0.727 | 0.0165923346 p = 0.722 | 0.0100583725 p = 0.715 | 0.0061258653 p = 0.712 | 0.0037388392 | |

2^{−24} | 0.1370859052 p = 0.806 | 0.0783990459 p = 0.767 | 0.0460605903 p = 743 | 0.0275085873 p = 0.729 | 0.0165929886 p = 0.720 | 0.0100693779 p = 0.715 | 0.0061330123 p = 0.712 | 0.0037437729 | |

2^{−26} | 0.1370859946 p = 0.806 | 0.0783991464 p = 0.767 | 0.0460607056 p = 0.743 | 0.0275087238 p = 0.729 | 0.0165931517 p = 0.720 | 0.0100695762 p = 0.715 | 0.00613332523 p = 0.712 | 0.0037440664 |

Consequently, the aim of this paper was to give uniform finite difference method for numerical solution of nonlinear singularly perturbed problem with nonlocal boundary conditions. The numerical method was constructed on Shishkin mesh. The method was pointed out to be convergent, uniformly in the ε-parameter, of first order in the discrete maximum norm. The numerical example illustrated in practice the result of convergence proved theoretically.

Musa Çakır,Derya Arslan, (2016) A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem. Journal of Applied Mathematics and Physics,04,1143-1156. doi: 10.4236/jamp.2016.46119