# Splitting Decomposition Homotopy Perturbation Method To Solve One -Dimensional Navier -Stokes Equation

## DOI:

https://doi.org/10.24297/jam.v13i2.5965## Keywords:

Splitting scheme, Adomian decomposition, homotopy perturbation method, Navier-Stokes equation, convergence analysis## Abstract

We have proposed in this research a new scheme to find analytical approximating solutions for Navier-Stokes equation of one dimension. The new methodology depends on combining Adomian decomposition and Homotopy perturbation methods with the splitting time scheme for differential operators . The new methodology is applied on two problems of the test: The first has an exact solution while the other one has no exact solution. The numerical results we obtained from solutions of two problems, have good convergent and high accuracy in comparison with the two traditional Adomian decomposition and Homotopy perturbationmethods .

### Downloads

## References

[2] Chorin A. J. , Numerical solution of the Navier â€“Stokes equation , Mathematics of Computation , 22(104):745-762, 1968.

[3] Jane B. ,'' Exact solution to one dimensional non-homogenous parabolic problems by the homogenous Adomian decomposition method '', Appl. Math. Comp. ,186: 969-979, 2007.

[4] Momani S. and Odibat Z. , Analytic solution of a time-fractional Navier-Stokes equation by Adomian decomposition method , Applied Mathematics and Computation ,177:488-494, 2006.

[5] Musa A. A. ,Numerical solution of Navier Stokes equation using control volume and finite element method, International Journal of App. Math. Res. ,5 (1) :63-68, 2006.

[6] Al-Saif A. S. J., Analytical approximate solutions for two-dimensional incompressible Navier-Stokes equations, Advances in Physics Theories and Application , 49,2015.

[7] Zhilin Li and Cheng Wang, A fast finite difference method for solving Navier- Stokes equations on irregular domains ,Communication in Math. Sci.1(1):181-197,2003.

[8] Donea J., Giuliani S., Laval H. and Quartapelle L., Finite element solution of the unsteady Navier-Stokes equations by a fractional step method, Computer methods in Appl. Mech. and Eng. ,30(1):53-73, 1982.

[9] Stelian Ion and Anca Veroncia Ion, A finite volume method for solving generalized Navier-Stokes equations, In Annals of Academy of Romanian Scientists Series on Math. and Appl. ,3(1):145-163,2011.

[10] He J. H., ''Homotopy perturbation technique '' Com. Math. App. Mech. Engineering ,178(3-4) : 257-262,1999.

[11] He J. H., ''Homotopy perturbation method: a new nonlinear analytic technique''. App. Math. and Comp. , 135: 37-79, 2003.

[12] He J. H. , ''Application of Homotopy perturbation method to nonlinear wave equation'' ,Chaos ,Solitons Fractals, 26(3):695-700,2005.

[13] He J. H., ''Homotopy perturbation method for solving boundary value problem '', physics letters A , 250, (1-2): 87-88, 2006.

[14] Zhang, X., ''A modification of the decomposition method for a class of non-linear singular boundary value problems,'' J. Comp. and Appl. Math. , 180 : 377-389, 2005.

[15] Jafari H. and Daftardar-Gerjji V., ''Revised ADM for solving a system of non-linear equations,'' Accepted for Pub. in Appl. Math. and Comput., 181:598-608,2006.

[16] Luo , X. G. ,''A two-step Adomian decomposition method ''Appl. Math. and Comput. , 170: 570-583, 2005.

[17] Liu C. M., ''On the study of oscillating viscous flows by using the Adomian-Pade' approximation'', Journal of Applied Mathematics, 2015:1-5, 2015.

[18] Ali A. H. and Al-Saif A. S. J. ''Adomian decomposition method for solving some models of nonlinear partial differential equations'' Basrah J. Sci. A, 26(1): 1-11, 2008.

[19] Celik E. , Bayram M. and Yeloglu T. ''Solution of differential-algebra equations by Adomian decomposition method'', Int. J. Pure Appl. Math. Sci. , 3( 1): 93-100, 2006.

[20] Javidi M. and Golbabai A. ,''Adomian decomposition method for approximating the solution of parabolic equations,'' J. Appl. Math. Sci., 1(5): 219-225, 2007.

[21] Ganji D. D., Afrouzi G. A., Hosseinzadeh H. and Talarposhit R. A. , Application of Homotopy perturbation method to the second kind of nonlinear integral equations, physics letters A, 371(1): 20-25, 2007.

[22] Ganji D. D. and Sadighi A., Application of He's Homotopy perturbation method to nonlinear coupled system of reaction-diffusion equation, international journal of nonlinear Sciences and Numerical Simulation , 7 (3): 411-418, 2006.

[23] Ganji D. D. and Rajabi A., Assessment Homotopy perturbation and perturbation methods in heat radiation equations, International Communication in heat and mass transfer, 33: 391-400, 2006.

[24] Ganji D. D. and Rafei M. , Solitary wave solutions for generalized Hirota-Satsuma coupled KdV equation by Homotopy perturbation method physics letters A ,356 : 131-137, 2006.

[25] He J. H. ,A coupling method of homotopy technique for nonlinear problem , Int. J. Nonlinear Mech. 35(1): 37-43, 2000.

[26] He J. H., Limit cycle and bifurcation of nonlinear problems , Chaos Soliton. Fract. 26 , 827-833, 2006.

[27] He J. H., Homotopy perturbation method for bifurcation of nonlinear problems , Int. J. Nonlinear Sci. Numer. Simul. 6(2) : 207-208, 2005.

[28] Mohyud-Din S. T. and Noor M. A. ''Homotopy perturbation method for solving fourth-order boundary value problems'' Math. Problems Eng. , 2007: 1-15 ,2007.

[39] Biazar J. and Ghazvini H. ''Exact solution for non-linear SchrÃ¶dinger equation by He's homotopy perturbation method'' Phys. Letters, 366: 79-84,2007.

[30] Jin L.'' Application of the modified homotopy perturbation method to the two dimensional sine-Gordon equation'' Int. J. Contemp. Math. Sci. , 5(20) : 985-990,2010.

[31] Seng V., Abbaoui k. and Cherruault Y. ,'' Adomian's polynomial for nonlinear operators'', J. Math. Comput. Modeling, 24 (1) : 59-65, 1996.

[32] Liao S. J., An approximate solution technique not depending on small parameters : a special example, Int. J. Non-Linear Mechaincs , 30(3) : 371-380, 1995.

[33] Liao S. J. , Boundary element method for general nonlinear differential operators, Engineering Analysis with Boundary Element , 20(2): 91-99, 1995.

[34] Biazar J. and Aminikhah H. ''Study of convergence of homotopy perturbation method for system of partial differential equations'' Comp. and Math. with Appl. ,58: 2221-2230, 2007.

[35] Alkalla I. L., Abd-Elmonem R. A. and Gomaa A. M. ''Convergence of discrete Adomian method for solving a class of nonlinear Fredholm integral equations'' Appl. Math., 4: 217-222, 2013.

[36] Inc M.''On numerical solutions of one-dimensional nonlinear Burger's equation and of the decomposition method '' Appl. Math. Comput. , 170: 76-85, 2005.

[37] Jane B.''Solutions to the non-homogeneous parabolic problems by the extended HADM'' Appl. Comput. , 191: 466-483, 2007.

[38] Ammar Al-Salih ,''Generating exact solution for some nonlinear system of partial differential equations by using FIM'', Thesis of Master, University of Basrah, College of Education for Pure Sciences, Department of Mathematics, October,2012.

## Downloads

## Published

## How to Cite

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*13*(2), 7123–7134. https://doi.org/10.24297/jam.v13i2.5965

## Issue

## Section

## License

All articles published in *Journal of Advances in Linguistics* are licensed under a Creative Commons Attribution 4.0 International License.