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The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameter

This paper is concerned with

Singularly perturbed differential equations are characterized by the presence of a small parameter

It is known that these problems depend on a small positive parameter

There is also an increasing interest in the application of Shishkin meshes to singularly perturbed convection-diffusion problems (see [

In the present paper, we analyse a fitted finite-difference scheme on a Shishkin type mesh for the numerical solution of the semilinear nonlocal boundary value problem (

The numerical method presented here comprises a fitted difference scheme on a piecewise uniform mesh. We have derived this approach on the basis of the method of integral identities using interpolating quadrature rules with the weight and remainder terms in integral form. This results in a local truncation error containing only first-order derivatives of exact solution and hence facilitates examination of the convergence. A summary of paper is as follows. Section

In this section, we give uniform bounds for the solution of the BVP (

Let

We rewrite the problem (

After establishing (

Let

The difference scheme we will construct follows from the identity

It then follows from (

To define an approximation for the boundary condition (

Now, it remains to define an approximation for the second boundary condition (

Based on (

The difference scheme (

We note that on this mesh the coefficient

Let

The solution

The problem (

Since the discrete maximum principle is valid here, we have the proof of (

Under the above assumptions of Section

From explicit expression (

We consider first the case

We now consider the case

The same estimate is obtained in the layer region

It remains to estimate

It remains to estimate

The same estimate is obtained for

The same estimate is valid when only one of the values

Next, we estimate the remainder term

Finally, we estimate the remainder term

Thus Lemma

Combining the two previous lemmas gives us the following convergence result.

Let

In this section, we present some numerical results which illustrate the present method.

Consider the test problem:

The exact solution of our test problem is unknown. Therefore, we use a double-mesh method [

To solve the nonlinear problem (

To solve (

The computed maximum pointwise errors

For the case of

10^{-2} | 0.02233220 | 0.01350608 | 0.00954256 | 0.00428337 | 0.00310038 | 0.00176533 |

0.77 | 0.82 | 0.86 | 0.93 | 0.98 | 0.99 | |

10^{-4} | 0.02233215 | 0.01376835 | 0.00850615 | 0.00538213 | 0.00315446 | 0.00160520 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

10^{-6} | 0.02233210 | 0.01376065 | 0.00851454 | 0.00540203 | 0.00314356 | 0.00160544 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

10^{-8} | 0.02233205 | 0.01376074 | 0.00851405 | 0.00540203 | 0.00314356 | 0.00160522 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

10^{-10} | 0.02233209 | 0.01376065 | 0.00851405 | 0.00540203 | 0.00314356 | 0.00160522 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

10^{-12} | 0.02233209 | 0.01376065 | 0.00851405 | 0.00540203 | 0.00314356 | 0.00160522 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

10^{-14} | 0.02233209 | 0.01376065 | 0.00851405 | 0.00540203 | 0.00314356 | 0.00160522 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

10^{-16} | 0.02233213 | 0.01376074 | 0.00851405 | 0.00540203 | 0.00314356 | 0.00160522 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

10^{-18} | 0.02233213 | 0.01376074 | 0.00851405 | 0.00540203 | 0.00314356 | 0.00160522 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

10^{-20} | 0.02233213 | 0.01376074 | 0.00851405 | 0.00540203 | 0.00314356 | 0.00160522 |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 | |

0.02233220 | 0.01376835 | 0.00954256 | 0.00540203 | 0.00315446 | 0.00176533 | |

0.75 | 0.80 | 0.85 | 0.92 | 0.97 | 0.99 |

For the case

10^{-2} | 0.02453225 | 0.01721110 | 0.01123085 | 0.00652485 | 0.00439674 | 0.00189543 |

0.68 | 0.79 | 0.83 | 0.88 | 0.96 | 0.99 | |

10^{-4} | 0.02453220 | 0.01716585 | 0.01120973 | 0.00651194 | 0.00439686 | 0.00188515 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

10^{-6} | 0.02453221 | 0.01706582 | 0.01120403 | 0.00651089 | 0.0043 9225 | 0.00188523 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

10^{-8} | 0.02453215 | 0.01706553 | 0.01120282 | 0.00651046 | 0.00439214 | 0.00188517 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

10^{-10} | 0.02453215 | 0.01706593 | 0.01120275 | 0.00651093 | 0.00439214 | 0.00188517 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

10^{-12} | 0.02453216 | 0.01706452 | 0.01120270 | 0.00651025 | 0.00439214 | 0.00188517 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

10^{-14} | 0.02453209 | 0.01706425 | 0.01120256 | 0.00651025 | 0.00439214 | 0.00188517 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

10^{-16} | 0.02453209 | 0.01706425 | 0.01120256 | 0.00651025 | 0.00439214 | 0.00188517 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

10^{-18} | 0.02453209 | 0.01706425 | 0.01120256 | 0.00651025 | 0.00439214 | 0.00188517 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

10^{-20} | 0.02453209 | 0.01706425 | 0.01120256 | 0.00651025 | 0.00439214 | 0.00188517 |

0.67 | 0.76 | 0.80 | 0.85 | 0.94 | 0.99 | |

0.02453225 | 0.01716585 | 0.01123085 | 0.00652485 | 0.00439686 | 0.00189543 | |

0.68 | 0.79 | 0.83 | 0.88 | 0.96 | 0.99 |

Consider the test problem:

The authors wish to thank the referees for their suggestions and comments which helped improve the quality of manuscript.