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Explicit finite difference methods for the delay pseudoparabolic equations.

I Amirali1, G M Amiraliyev1, M Cakir2

  • 1Department of Mathematics, Faculty of Art and Science, Sinop University, 57000 Sinop, Turkey.

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Summary
This summary is machine-generated.

A new finite difference method provides a stable and accurate numerical solution for semilinear delay Sobolev equations. This approach achieves second-order accuracy in space and first-order accuracy in time for complex pseudoparabolic problems.

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Area of Science:

  • Numerical Analysis
  • Partial Differential Equations
  • Computational Mathematics

Background:

  • Semilinear delay Sobolev (or pseudoparabolic) equations present significant challenges in numerical solutions.
  • Existing methods may struggle with stability and accuracy for these complex initial-boundary value problems.
  • Efficient and reliable numerical techniques are crucial for understanding phenomena modeled by such equations.

Purpose of the Study:

  • To develop and analyze a novel numerical method for solving semilinear delay Sobolev equations.
  • To establish the stability and convergence properties of the proposed numerical scheme.
  • To provide accurate error estimates for the developed method.

Main Methods:

  • Application of the finite difference technique to discretize the spatial domain.
  • Construction of a two-level difference scheme using the method of integral identities.
  • Implicit rule employed for time integration, ensuring stability.
  • Energy estimates utilized to rigorously prove absolute stability and convergence.

Main Results:

  • A fully discrete difference scheme is constructed and proven to be absolutely stable.
  • The scheme demonstrates second-order accuracy in the spatial discretization.
  • First-order accuracy in time integration is achieved.
  • Error estimates are derived in the discrete norm, quantifying the method's precision.

Conclusions:

  • The proposed finite difference method offers a robust and accurate approach for semilinear delay pseudoparabolic equations.
  • The method's stability and convergence properties are theoretically established.
  • Numerical results validate the theoretical findings and confirm the method's effectiveness.