Jacobi spectral collocation method for the approximate solution of multidimensional nonlinear Volterra integral...

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Scientific Computing

Background:

• Multidimensional nonlinear Volterra integral equations pose significant challenges in numerical approximation.
• Spectral methods offer high accuracy for smooth solutions but their application in higher dimensions requires careful consideration of collocation points and weight functions.

Purpose of the Study:

• To analyze the convergence properties of the Jacobi spectral collocation method for multidimensional nonlinear Volterra integral equations.
• To theoretically justify the exponential convergence of this spectral method in higher dimensions.
• To demonstrate the method's practical validity through numerical examples.

Main Methods:

• The Jacobi spectral collocation method is employed, utilizing Jacobi-Gauss points associated with the multidimensional Jacobi weight function as collocation points.
• Error analysis is performed in both L2-norm and L-infinity norm to establish convergence rates.
• The smoothness of the solution, source function, and kernel function are assumed.

Main Results:

• The Jacobi spectral collocation method exhibits exponential convergence for the approximate solution of the considered equations.
• Theoretical error analysis in specified norms supports the findings on convergence rates.
• The method is shown to be effective for smooth problems in multidimensional spaces.

Conclusions:

• The Jacobi spectral collocation method provides an efficient and accurate approach for solving multidimensional nonlinear Volterra integral equations.
• The theoretical convergence properties are well-supported by numerical evidence.
• This method holds promise for applications requiring high-precision solutions in computational science and engineering.