Numerical integration of bivariate fractal interpolation functions on rectangular domains

Area of Science:

• Numerical Analysis
• Fractal Geometry
• Scientific Computing

Background:

• Traditional numerical integration methods can be computationally intensive.
• Fractal interpolation functions offer a data-driven approach to function approximation.

Purpose of the Study:

• To derive a fractal numerical integration method for bivariate data over rectangular regions.
• To enhance accuracy and reduce computational effort in numerical integration.
• To analyze the convergence properties of the proposed fractal integration method.

Main Methods:

• Formulation of fractal numerical integration using recursive relations of bivariate fractal interpolation functions.
• Evaluation of iterated function system coefficients from data points.
• Derivation of an integration formula based on these coefficients.
• Correlation with bilinear interpolation functions.
• Derivation of a vertical scaling factor formula to minimize approximation error.

Main Results:

• A novel fractal numerical integration formula is proposed.
• The method demonstrates accurate results with minimal computation.
• Convergence of the fractal method to traditional double integration is established.
• Numerical analysis of benchmark functions validates the proposed technique.

Conclusions:

• The proposed fractal numerical integration method is effective for bivariate data.
• The method provides an efficient and accurate alternative to traditional techniques.
• The derived vertical scaling factor aids in error reduction and convergence proof.