Numerical solution of differential equations using multiquadric radial basis functions networks

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Applied Mathematics

Background:

• Radial Basis Function Networks (RBFNs) are effective for function approximation.
• Solving linear ordinary differential equations (ODEs) and elliptic partial differential equations (PDEs) often requires mesh-based methods.
• Mesh-free methods offer an alternative approach, potentially simplifying problem setup and handling complex geometries.

Purpose of the Study:

• To develop and present novel mesh-free procedures for solving linear ODEs and elliptic PDEs.
• To introduce two types of RBFN procedures: direct (DRBFN) and indirect (IRBFN).
• To investigate the accuracy and performance of these methods under various conditions.

Main Methods:

• Utilizing multiquadric (MQ) radial basis function networks (RBFNs) for approximation.
• Developing direct (DRBFN) and indirect (IRBFN) RBFN procedures for solving differential equations.
• Adjusting the width of RBFs using the parameter beta, where alpha(i) = beta * d(i).
• Testing different combinations of RBF centers and collocation points on regular and irregular domains.

Main Results:

• The indirect RBFN (IRBFN) method demonstrates higher accuracy compared to the direct RBFN (DRBFN) method.
• For a 1D Poisson's equation, DRBFN achieved an error norm of O(10^-4) and IRBFN achieved O(10^-8).
• For a 2D Poisson's equation, DRBFN achieved an error norm of O(10^-3) and IRBFN achieved O(10^-6).
• Optimal performance for IRBFN was observed with beta values between 7 and 10.

Conclusions:

• Mesh-free RBFN procedures provide an effective approach for solving linear ODEs and elliptic PDEs.
• The indirect RBFN procedure offers superior accuracy over the direct method.
• The proposed methods are applicable to both regularly and irregularly shaped domains.