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An optimal and modified homotopy perturbation method for strongly nonlinear differential equations

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Abstract

Homotopy perturbation method (HPM) is one of the most popular semi-analytical methods to solve a nonlinear differential equation. However, in HPM, there is no strict rule for the choice of its linear operator, and its series solution may not always converges. In this study, firstly, we define the linear operator as an auxiliary linear operator (\({\mathcal {L}}_a\)), in the frame of homotopy analysis method (HAM). Then we generalize this \({\mathcal {L}}_a\) based on the auxiliary roots of \({\mathcal {L}}_a=0\). Finally, using the optimization technique (on minimization of the residual error) we determine the best-fitted optimal \({\mathcal {L}}_a\) for a problem. By doing this we ensure and accelerate the convergence of our semi-analytical homotopy perturbation series solution. Thereby we rename the HPM as Optimal and Modified Homotopy Perturbation Method (OMHPM). We consider three strongly nonlinear differential equations of nonlinear dynamical phenomena associated with the fluid dynamics to certify our technique. The dependencies of the form of the optimal \({\mathcal {L}}_a\) and the convergence of the solution (obtained by HPM, Optimal HAM and OMHPM) on the values of parameters (involved in the scale transformation), initial/boundary conditions and artificial controlling parameters (involved in optimal HAM) are explored here. It is reported that our OMHPM is highly accurate and efficient than HPM, optimal HAM and Domain decomposition optimal HAM. Moreover, OMHPM is simple and can be applied to directly to any singular/non-singular highly nonlinear ordinary differential equations without any decomposition, special/scale transformation, linearization, artificial controlling parameters and discretization. An attempt is made to apply our optimal auxiliary linear operator onto the optimal HAM for possible fastest convergence.

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Acknowledgements

Authors are very grateful to anonymous referees for their valuable suggestions and comments which improved the paper. Also thankfully acknowledge support given by the DST-FIST, INDIA (Sanction Order No.: SR/FST/MS1/2018/21(C) dated-13/12/2019) for upgradation of research facility at the departmental level.

Funding

Govt. of West Bengal, INDIA supports scholarship to Mr. Roy for Ph.D. through the Swami Vivekananda Merit Cum Means Scholarship Scheme.

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  1. Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapur, WB, 721102, India

    Tapas Roy & Dilip K. Maiti

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TR has done mathematical calculation, developed the numerical code and partly written the manuscript, while DK Maiti conceived the problem, verified the results, and supervised the overall numerical computation and finalize the whole manuscript.

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Correspondence to Dilip K. Maiti.

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Roy, T., Maiti, D.K. An optimal and modified homotopy perturbation method for strongly nonlinear differential equations. Nonlinear Dyn 111, 15215–15231 (2023). https://doi.org/10.1007/s11071-023-08662-w

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