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Besse relaxation difference scheme for a nonlinear integro-differential equation.

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New Besse relaxation difference schemes improve accuracy and stability for nonlinear integro-differential equations, crucial for modeling complex systems with memory and nonlocal effects.

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Applied mathematics

Background:

  • Nonlinear integro-differential equations are essential for modeling complex systems with memory and nonlocal effects.
  • Existing numerical methods may face challenges with accuracy and stability when dealing with nonlinear terms in these equations.

Purpose of the Study:

  • To propose novel Besse relaxation difference and compact difference schemes for nonlinear integro-differential equations.
  • To enhance the accuracy and stability of numerical solutions for these complex equations.
  • To verify the effectiveness and convergence properties of the proposed schemes.

Main Methods:

  • Developing a Besse relaxation difference scheme using Besse relaxation time discretization and second-order spatial discretization.
  • Constructing a Besse relaxation compact difference scheme incorporating a fourth-order compact finite difference approximation for improved spatial accuracy.
  • Establishing unconditional stability and optimal convergence in discrete [Formula: see text] norms.

Main Results:

  • The proposed Besse relaxation schemes demonstrate enhanced accuracy and stability in handling nonlinear terms.
  • Numerical experiments confirm that the schemes achieve predicted convergence rates.
  • The methods are effective for various solution types, including smooth, singular, and unbounded derivative solutions.

Conclusions:

  • The Besse relaxation difference and compact difference schemes offer effective and accurate numerical solutions for nonlinear integro-differential equations.
  • These methods provide a robust approach for simulating complex systems with memory and nonlocal characteristics.
  • The established stability and convergence properties support their practical application in scientific modeling.