Manuscript Topics

The study of nonlocal/nonlinear evolution equations are concerned in the field of natural science and even social science since many phenomena in natural and engineering field are essentially nonlocal or nonlinear. They have aroused the interest and concern of engineers, physicists, mathematicians and many others. A large part of nonlocal/nonlinear evolutionary phenomena can be described by nonlinear/nonlocal differential equations or delay differential equations. However, it is commonly very difficult to find general solutions from these equations directly. Hence it is necessary to study the numerical theory and numerical simulation of the nonlinear/nonlocal differential equation. How to establish efficient, high-precision and robust numerical methods for fractional differential equations is still a very challenging problem.
The main aim of this Special Issue is to focus on some recent developments in efficient solutions of nonlocal/nonlinear evolution equations and phase-field model including numerical and theoretical results. All the articles and reviews devoted to the above theme on numerical methods of such fractional differential equations and nonlinear evolution equations, delay differential equations and their applications are welcome.

Topics of interest include, but are not limited to:
• Finite element, finite difference, finite volume methods
• Spline method, wavelet method
• Stochastic Localization Methods
• Galerkin Methods
• Runge-Kutta methods
• Two-grid method; iterative method
• Physics-informed neural networks (PINN)
• Machine learning algorithm
• Error estimate and stability analysis
• maximum principle, positivity preservation
• Spectral/collocation method
• Multigrid method
• alternating direction implicit (ADI)
• Numerical methods for ordinary/stochastic differential equations
• Numerical methods for fractional differential equations
• Numerical methods nonlocal nonlinear evolution equation
• Numerical methods for nonlocal differential equations
• Numerical methods for delay differential equations

 Keywords:
Fractional differential equations, nonlocal differential equations, partial differential equations, evolution equation, singular perturbation problem, stability, convergence, nonlinear, high-order, numerical and approximation methods, Deep learning, applications of fractional calculus.

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