Fokas method for linear boundary value problems involving mixed spatial derivatives

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Partial Differential Equations

Background:

• Initial boundary value problems (IBVPs) with mixed spatial derivatives pose significant analytical challenges.
• The unified transform method (UTM), or Fokas method, offers a powerful framework for solving such problems.
• Eliminating mixed derivatives is often difficult, especially in higher-order equations like biharmonic problems.

Purpose of the Study:

• To derive solution representation formulas for linear IBVPs with mixed spatial derivatives on a half space using the UTM.
• To demonstrate the applicability of the UTM to both second-order parabolic PDEs and biharmonic problems.
• To investigate the role and conditions for the well-definedness of invariant maps within the UTM framework.

Main Methods:

• Application of the unified transform method (UTM) to linear initial boundary value problems on a half space.
• Utilizing invariant maps, a key component of the UTM, to construct solution representations.
• Addressing analyticity issues to ensure the well-definedness of the invariant maps.

Main Results:

• Obtained explicit solution representation formulas for specific linear parabolic and biharmonic initial boundary value problems.
• Demonstrated that the UTM can be applied even when mixed derivatives cannot be eliminated by simple variable changes.
• Characterized the conditions under which the invariant maps used in the UTM are well-defined, involving careful handling of analyticity.

Conclusions:

• The unified transform method is a versatile and effective technique for solving linear PDEs with mixed derivatives on a half space.
• The method's success hinges on properly managing analyticity conditions related to invariant maps.
• The findings extend the applicability of the UTM to a broader class of problems, including biharmonic equations.