Determination of an Extremal in Two-Dimensional Variational Problems Based on the RBF Collocation Method

Area of Science:

• Applied Mathematics
• Numerical Analysis

Background:

• Variational problems are fundamental in various scientific fields.
• Solving these problems often requires complex numerical methods.

Purpose of the Study:

• To introduce a direct method for solving variational problems using radial basis function (RBF) interpolation.
• To parameterize solutions and transform variational problems into constrained optimization problems.

Main Methods:

• Global radial basis function (RBF) interpolation over arbitrary collocation nodes.
• Parameterization of solutions using arbitrary RBFs.
• Transformation of two-dimensional variational problems (2DVP) into constrained optimization problems.
• Application of Lagrange multiplier technique to convert optimization problems into algebraic equation systems.

Main Results:

• The proposed method offers flexibility in selecting RBFs and parameterizing nodal points.
• It effectively reduces constrained variation problems to constrained optimization.
• Numerical examples confirm the high efficiency and accuracy of the technique.

Conclusions:

• The direct RBF interpolation method provides an efficient and accurate approach for solving variational problems.
• The flexibility in RBF and nodal point selection enhances its applicability.