Abstract
Since late twentieth century, homotopy perturbation and homotopy analysis methods are widely used to obtain analytical approximations for nonlinear differential equations. If the operator of the nonlinear differential equation contains a linear part and a nonlinear part with a small parameter, then the original nonlinear problem can be transformed into an infinite number of linear sub-problems. Here, inverse of the linear operator should be calculated to find unknown functions. To overcome this obstacle in 2016, Liao introduced the Method of Directly Defining inverse Mapping (MDDiM) to solve a nonlinear ordinary differential equation with the freedom of choosing the inverse linear mapping directly, instead of calculating the inverse of an operator. Very recently, Vajravelu and his research group extended the Method of Directly Defining the inverse Mapping (MDDiM) to systems of nonlinear ordinary differential equations. In this study, we extend further this novel method to a nonlinear partial differential equation (Transitional Korteweg–de Vries equation) and to systems of nonlinear partial differential equations. The results that we obtained here agree very well with the results available in the literature, which are obtained by several numerical methods. In addition, with MDDiM, we save computational time. Furthermore, this novel method can be used to analyze complicated models arising in science and engineering applications: As an example, we applied this so-called MDDiM to solve a real-world application problem namely, the “Fingering phenomenon model in oil industry”.
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References
Baxter M, Dewasurendra M, Vajravelu K (2017) A method of directly defining the inverse mapping for solutions of coupled systems of nonlinear differential equations. Numer Algorithm 77:1199–1211
Biazar J, Eslami M (2007) Homotopy Perturbation Method for Systems of Partial Differential Equations. Int J Nonlinear Sci Numer Simul 8(3):413–8
Dewasurendra M, Baxter M, Vajravelu K (2018) A method of directly defining the inverse mapping for solutions of non-linear coupled systems arising in convection heat transfer in a second grade fluid. Appl Math Comput Elsevier 339:758–767
Liao S (1992) Proposed homotopy analysis techniques for the solutions of nonlinear problems. Shanghai Jiao Tong University (Ph.D. thesis)
Liao SJ (2003) Beyond perturbation: introduction to the homotopy analysis method. Chapman and Hall, Boca Raton
Liao SJ (2012) Homotopy analysis method in nonlinear differential equations. Springer and Higher Education Press, Heidelberg
Liao S, Zhao Y (2016) On the method of directly defining the inverse mapping for nonlinear differential equations. Numer Algorithm 72:989–1020
Prajapati DJ, Desai NB (2016) Application of the basic optimal homotopy analysis method to fingering phenomenon. Global J Pure Appl Math 12:2011–2022
Vajravelu K, Van Gorder RA (2013) Nonlinear flow phenomena and homotopy analysis: fluid flow and heat transfer. Springer, Heidelberg
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Communicated by Jose Alberto Cuminato.
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Sahabandu, C.W., Karunarathna, D., Sewvandi, P. et al. A Method of Directly Defining the inverse Mapping for a nonlinear partial differential equation and for systems of nonlinear partial differential equations. Comp. Appl. Math. 40, 234 (2021). https://doi.org/10.1007/s40314-021-01627-y
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DOI: https://doi.org/10.1007/s40314-021-01627-y
Keywords
- Directly defining the inverse mapping
- Coupled nonlinear partial differential equations
- Transitional Korteweg–de Vries equation
- Fingering phenomenon model in oil industry