<p>The proposed analysis contribute an essential analysis towards comprehensive and theoretical grasp the how the existing gradient descent algorithm (GDA) beneficial efficiently on homotopy analysis technique. To illustrate the idea, we consider the fundamental Burgers’ equation which is one the simplest nonlinear model. The majority of PDEs has been evaluating by considering either an approximate analytical approach or numerical owing to the occurrence of severe nonlinearity. Further, among several approximation techniques, the HAM, shows better results and regulate the region of convergence of computed closed form solution due to presence of auxiliary parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40010_2025_922_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\hslash$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℏ</mi> </math></EquationSource> </InlineEquation>. In HAM, choosing <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40010_2025_922_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\hslash$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℏ</mi> </math></EquationSource> </InlineEquation> values is based on trial and error approach and due to this solution might be diverges. Therefore, it is necessary to determine convergence to the correct solution and ensure correctness. To do so, the well established GDA is the proper choice for obtaining the precise solution by establish the accurate values of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40010_2025_922_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\hslash$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℏ</mi> </math></EquationSource> </InlineEquation>. The results of obtained from GDA are validated with already-reported analytical and numerical results from the literature in order to confirm the efficacy of the proposed method. The work proposed here would be advantageous for solving the different kinds of nonlinear models.</p>

Homotopy Analysis Method for Burgers’ Equation: Application of Gradient Descent Approach

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发表日期：

2025年4月22日

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Mahesh Kumar,
Ranjan Kumar Jana

摘要

The proposed analysis contribute an essential analysis towards comprehensive and theoretical grasp the how the existing gradient descent algorithm (GDA) beneficial efficiently on homotopy analysis technique. To illustrate the idea, we consider the fundamental Burgers’ equation which is one the simplest nonlinear model. The majority of PDEs has been evaluating by considering either an approximate analytical approach or numerical owing to the occurrence of severe nonlinearity. Further, among several approximation techniques, the HAM, shows better results and regulate the region of convergence of computed closed form solution due to presence of auxiliary parameter $\hslash$ $ℏ$ . In HAM, choosing $\hslash$ $ℏ$ values is based on trial and error approach and due to this solution might be diverges. Therefore, it is necessary to determine convergence to the correct solution and ensure correctness. To do so, the well established GDA is the proper choice for obtaining the precise solution by establish the accurate values of $\hslash$ $ℏ$ . The results of obtained from GDA are validated with already-reported analytical and numerical results from the literature in order to confirm the efficacy of the proposed method. The work proposed here would be advantageous for solving the different kinds of nonlinear models.