Numerical Solution of the Singularly Perturbed Problem With Nonlocal Boundary Condition

G. M. Amiraliyev; Musa ÇakIr

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2002 > 23(7): 673-681

G. M. Amiraliyev, Musa ÇakIr, . Numerical Solution of the Singularly Perturbed Problem With Nonlocal Boundary Condition[J]. Applied Mathematics and Mechanics, 2002, 23(7): 673-681.

Citation:

G. M. Amiraliyev, Musa ÇakIr, . Numerical Solution of the Singularly Perturbed Problem With Nonlocal Boundary Condition[J]. Applied Mathematics and Mechanics, 2002, 23(7): 673-681.

Citation:

G. M. Amiraliyev, Musa ÇakIr, . Numerical Solution of the Singularly Perturbed Problem With Nonlocal Boundary Condition[J]. Applied Mathematics and Mechanics, 2002, 23(7): 673-681.

PDF( 351 KB)

Numerical Solution of the Singularly Perturbed Problem With Nonlocal Boundary Condition

Department of Mathematics, Y Y University, 65080 Van, Turkey

Received Date: 2001-09-10
Publish Date: 2002-07-15

Abstract

Abstract

Singularly perturbed boundary value problem with nonlocal conditions is examined. The appopriate solution exhibits boundary layer behavior for small positive values of the perturbative parameter. An exponentially fitted finite difference scheme on a non-equidistant mesh is constructed for solving this problem. The uniform convergence analysis in small parameter is given. Numerical example is provided, too.
- exponentially fitted difference scheme,
- singular perturbation,
- nonlocal boundary condition

FullText(HTML)

References(25)

References

[1]	Doolan E P,Miller J J H,Schilders W H A.Uniform Numerica l Methods for Problems With Initial and Boundary Layers[M].Dublin:Boole Press,1980.
[2]	Kadalbajoo M K,Reddy Y N.Asymptotic and numerical analysis of singular perturbation problems:a Survey[J].Appl Math Comput,1989,30:223-259.
[3]	Weekman V W,Gorring R L.Influence of volume change on gas-phase reactions in porous catalysts[J].J Catalysis,1965,4[STBZ]:260-270.
[4]	Nayfeh A H.Introduction to Perturbation Techniques[M].New York:Wiley,1993.
[5]	O'Malley R E.Sigular Perturbation Methods for Ordinary Differential Equation[M].New York:Springer-Verlag,1991.
[6]	Farrell P A,Hegarty A F,Milier J J H,et al.Robust Computational Techniques for Boundary Layers[M].Boca Raton:Chapman & Hall/CRC,2000.
[7]	Miller J J H,O'Riordan E,Shishkin G I.Fitted Numerical Methods for Singular Perturbation Problems[M].Singapore:World Scientific,1996.
[8]	Roos H G,Stynes M,Tobiska L.Numerical Methods for Singu larly Perturbed Differential Equations:Convection-Diffusion and Flow Problems [M].Berlin:Springer-Verlag,1996.
[9]	Cziegis R.The numerical solution of singularly perturbed nonlocal problem[J].Lietuvas Matem Rink,1988,28:144-152.(in Russian)
[10]	Cziegis R.The difference schemes for problems with nonlocal conditions[J].Informatica (Lietuva),1991,2:155-170.
[11]	Bitsadze A B,Samarskii A A.Of some simple generalization the linear elliptic boundary value problems[J].Soviet Math Dokl,1969,185:739-740.(in Russian)
[12]	Nahushev A M.On nonlocal boundary value problems[J].Differential Equations,1985,21:92-101.(in Russian)
[13]	HE Ji-huan.Variationaliteration method:a kind of nonlinear analytical technique:some examples[J].International Journal of Nonlinear Mechanics,1999,34(4):699-708.
[14]	HE Ji-huan.Homotopy perturbation technique[J].Computer Methods in Applied Mechanics and Engineering,1999,178:257-262.
[15]	HE Ji-huan.A new perturbation technique which is also valid for large parameters[J].Journal of Sound and Vibration,1999,229(5):1257-1263.
[16]	HE Ji-huan.A coupling method of homotopy technique and pertur bation technique for nonlinear problems[J].International Journal of Nonlinear Mechanics,2000,35(1):37-43.
[17]	HE Ji-huan.A review on some new recently developed nonlinear anal ytical techniques[J].Int J Nonlinear Sciences & Numerical Simulation,2000,1(1):51-70.
[18]	HE Ji-huan.A modified perturbation technique depending uponanartificial parameter[J].Meccanica,2000,35:299-311.
[19]	HE Ji-huan.Iteration perturbation method for strongly nonlinea roscillations[J].Journal of Vibration & Control,2001,7(5):631-642.
[20]	Amiraliyev G M.Difference method for the solution of one problem of the theory of dispersive waves[J].Differential Equations,1990,26:2146-2154.(in Russian)
[21]	艾米雷利耶弗GM,哈基杜柔.奇异摄动初值问题的一致收敛有限差分法[J].应用数学和力学,1999,20(4):363-370.
[22]	Amiraliyev G M,Mamedov Y D.Difference scheme on the uniform mesh for singularly perturbed pseudo-parabolic equation[J].Turkish J Math,1995,19:207-222.
[23]	Samarskii A A.Theory of Difference Schemes[M].2nd Ed.Nauka:Moscow,1983;German Transl.Leibzig:Geest Portig,1984.
[24]	Dorr F W,Parter S V,Shampine L F.Shampine,applications of the maximum principle to singular perturbation problems[J].SIAM Rev,1973,15:43-88.
[25]	Protter M H,Weinberger H F.Maximum Principles in Differential Equations[M].Englewood Cliffs N J:Prentice-Hall,1967