Fractal-Based Methods and Inverse Problems for Differential Equations: Current State of the Art

We illustrate, in this short survey, the current state of the art of fractal-based techniques and their application to the solution of inverse problems for ordinary and partial differential equations. We review several methods based on the Collage Theorem and its extensions. We also discuss two innovative applications: the first one is related to a vibrating string model while the second one considers a collage-based approach for solving inverse problems for partial differential equations on a perforated domain.

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EXPLICIT FAST AND STABLE METHOD FOR SOLUTION OF SOME COEFFICIENT INVERSE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS

Eurasian Journal of Mathematical and Computer Applications ◽

10.32523/2306-3172-2013-1-2-92-101 ◽

2013 ◽

Vol 1 (1) ◽

pp. 92-101

Author(s):

Alexandre Grebennikov ◽

Keyword(s):

Partial Differential Equations ◽

Inverse Problems ◽

Differential Equations ◽

Stable Method ◽

Partial Differential ◽

Coefficient Inverse Problems

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The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations

Mathematics ◽

10.3390/math8081303 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1303 ◽

Cited By ~ 1

Author(s):

Carl Leake ◽

Hunter Johnston ◽

Daniele Mortari

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

State Of The Art ◽

Linear Constraints ◽

Prior Work ◽

Partial Differential ◽

Seminal Paper ◽

Functional Connections ◽

Mathematical Proofs ◽

Multivariate Theory

This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals—called constrained expressions—and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expressions. Furthermore, the extension of the technique to and proofs in n-dimensions is immediate through a recursive application of the univariate formulation. In all, the results of this reformulation are compared to prior work to highlight the novelty and mathematical convenience of using this approach. Finally, the methodology presented in this paper is applied to two partial differential equations with different boundary conditions, and, when data is available, the results are compared to state-of-the-art methods.

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