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Convolution computations can be simplified by utilizing their inherent properties.
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Trigonometric identities are equations that relate trigonometric functions and hold for all angles within their domains. A fundamental identity among these is the Pythagorean identity, which arises directly from the geometry of the unit circle. For any angle θ, a point on the unit circle has coordinates (cos⁡ θ, sin ⁡θ), and since the radius of the circle is one, the Pythagorean Theorem gives:This identity serves as the basis for deriving additional identities. Dividing the Pythagorean...
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Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and

Nikolay K Vitanov1, Zlatinka I Dimitrova2, Kaloyan N Vitanov1

  • 1Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, Bulgaria.

Entropy (Basel, Switzerland)
|December 30, 2020
PubMed
Summary

The Simple Equations Method (SEsM) unifies various techniques for solving nonlinear partial differential equations. This study demonstrates SEsM

Keywords:
Hirota methodJacobi elliptic function expansion methodauxiliary equation methodexact solutionsextended homogeneous balance methodf-expansion methodfirst integral methodhomogeneous balance methodinverse scattering transform methodmodified simple equation methodnonlinear partial differential equationssimple equations method (sesm)trial function method

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Computational Mathematics

Background:

  • Nonlinear partial differential equations (NPDEs) model complex phenomena across science and engineering.
  • Numerous analytical methods exist for finding exact solutions to NPDEs.
  • A unified framework for these methods is lacking.

Purpose of the Study:

  • To present the Simple Equations Method (SEsM) as a unified approach for solving NPDEs.
  • To demonstrate the connection between SEsM and established methods like Hirota's method and the inverse scattering transform.
  • To illustrate the versatility of SEsM through specific applications.

Main Methods:

  • The Simple Equations Method (SEsM) is detailed, involving specific steps for solution construction.
  • Connections are established between SEsM and the Hirota method, particularly for exponential function-based equations.
  • SEsM is shown to reproduce the inverse scattering transform method for Burgers' and Korteweg-de Vries equations via power series expansions and Fourier analysis.

Main Results:

  • The Hirota method is shown to be a special case of SEsM for specific equation types.
  • SEsM successfully yields multi-soliton solutions for Korteweg-de Vries, nonlinear Schrödinger, and Ishimori equations.
  • SEsM's ability to reproduce the inverse scattering transform method is confirmed for Burgers' and Korteweg-de Vries equations.
  • Numerous other methods, including homogeneous balance and Jacobi elliptic function expansion, are shown to be particular cases of SEsM.

Conclusions:

  • SEsM provides a unifying framework for a wide array of techniques used to solve NPDEs.
  • The method offers a systematic way to derive exact solutions, including multi-soliton solutions.
  • SEsM enhances the understanding of the relationships between different analytical approaches in nonlinear science.