scholarly journals Singular integrals and partial differential equations of parabolic type

1966 ◽  
Vol 28 (1) ◽  
pp. 81-131 ◽  
Author(s):  
Eugene Fabes
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Singular Integrals ◽  
Parabolic Type ◽  
Partial Differential

Similarity and Generalized Finite-Difference Solutions of Parabolic Partial Differential Equations

1972 ◽  
Vol 39 (2) ◽  
pp. 584-590
Author(s):  
A. M. Clausing
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Parabolic Equations ◽  
Finite Difference Methods ◽  
Parabolic Type ◽  
Computational Effort ◽  
Partial Differential ◽  
Considerable Saving ◽  
Boundary Layer Equations

Techniques are presented for obtaining generalized finite-difference solutions to partial differential equations of the parabolic type. It is shown that the advantages of similarity in the solution of similar problems are generally not lost if the solution to the original partial differential equations is effected in the physical plane by finite-difference methods. The analysis results in a considerable saving in computational effort in the solution of both similar and nonsimilar problems. Several examples, including both the heat-conduction equation and the boundary-layer equations, are given. The analysis also provides a practical means of estimating the accuracy of finite-difference solutions to parabolic equations.

scholarly journals Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Sixian Jin ◽  
Henry Schellhorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Malliavin Calculus ◽  
Series Representation ◽  
Parabolic Type ◽  
Second Order ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Path Dependent

We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.

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