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Summary

This study simplifies the variational iteration method-II (VIM) and Adomian decomposition methods by reducing unnecessary calculations. The new approach offers a more efficient and reliable solution for mathematical problems.

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Area of Science:

  • Numerical Analysis
  • Applied Mathematics

Background:

  • The standard variational iteration algorithm II and Adomian decomposition methods are widely used for solving differential equations.
  • These methods, however, involve complex calculations, including the Lagrange multiplier and repeated iterative steps.

Purpose of the Study:

  • To propose an alternative and simplified formulation of the variational iteration method-II (VIM).
  • To present a more straightforward approach to the Adomian decomposition method (ADM) and its modified version (MADM).
  • To reduce computational complexity in existing iterative methods.

Main Methods:

  • Reconstruction of the variational iteration algorithm II.
  • Development of a simplified formulation for Adomian decomposition and modified Adomian decomposition methods.
  • Utilizing the newly proposed VIM-II as a basis for simplification.

Main Results:

  • Identified and eliminated unnecessary calculations of the Lagrange multiplier in VIM.
  • Reduced repeated calculations within each iteration of ADM and MADM.
  • Demonstrated the reliability and efficiency of the proposed simplified methods.

Conclusions:

  • The proposed alternative approach offers a more efficient and simpler formulation of VIM, ADM, and MADM.
  • The simplified methods maintain reliability and efficiency, as verified by several examples.
  • This work contributes to more accessible and computationally feasible numerical solutions.