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The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law...
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An efficient algorithm for some highly nonlinear fractional PDEs in mathematical physics.

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This study introduces a novel method using fractional complex transform (FCT) and Reduced Differential Transform Method (RDTM) to solve fractional partial differential equations (FPDEs). The approach proves efficient for complex nonlinear problems.

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Area of Science:

  • Applied Mathematics
  • Numerical Analysis
  • Fractional Calculus

Background:

  • Fractional partial differential equations (FPDEs) model complex phenomena but are challenging to solve.
  • Existing methods often lack efficiency or broad applicability for nonlinear FPDEs.

Purpose of the Study:

  • To develop and present an efficient algorithm for solving linear and nonlinear time-fractional PDEs.
  • To demonstrate the efficacy of the fractional complex transform (FCT) combined with the Reduced Differential Transform Method (RDTM).

Main Methods:

  • A fractional complex transform (FCT) is employed to convert FPDEs into standard PDEs.
  • The Reduced Differential Transform Method (RDTM) is applied to the transformed system.
  • Inverse transformation is used to obtain solutions in terms of original variables.

Main Results:

  • The combined FCT-RDTM method successfully solves time-fractional PDEs.
  • The algorithm demonstrates high efficiency and appropriateness for handling both linear and nonlinear FPDEs.
  • The method provides solutions in the original variables.

Conclusions:

  • The proposed FCT-RDTM algorithm is a powerful and efficient tool for solving FPDEs.
  • The methodology shows potential for extension to a wider range of complex nonlinear problems.
  • This approach offers a valuable contribution to the field of fractional calculus and numerical methods.