The state of the art in the numerical solution of time-varying partial differential equations

1977 ◽  
Vol 81 (25) ◽  
pp. 2335-2339 ◽  
Author(s):  
N. L. Schryer
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
State Of The Art ◽  
The State ◽  
Time Varying ◽  
Partial Differential

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

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1303 ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document