scholarly journals Conservation laws for second-order parabolic partial differential equations

2004 ◽  
Vol 45 (3) ◽  
pp. 333-348 ◽  
Author(s):  
B. Van Brunt ◽  
D. Pidgeon ◽  
M. Vlieg-Hulstman ◽  
W. D. Halford
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Time Derivative ◽  
Sufficient Conditions ◽  
Second Order ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Necessary And Sufficient ◽  
Derivatives Of

AbstractConservation laws for partial differential equations can be characterised by an operator, the characteristic and a condition involving the adjoint of the Fréchet derivatives of this operator and the operator defining the partial differential equation. This approach was developed by Anco and Bluman and we exploit it to derive conditions for second-order parabolic partial differential equations to admit conservation laws. We show that such partial differential equations admit conservation laws only if the time derivative appears in one of two ways. The adjoint condition, however, is a biconditional, and we use this to prove necessary and sufficient conditions for a certain class of partial differential equations to admit a conservation law.

scholarly journals Modification of balayage spaces by transitions with application to coupling of PDE’s

2003 ◽  
Vol 169 ◽  
pp. 77-118 ◽  
Author(s):  
Wolfhard Hansen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Second Order ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential

AbstractModifications of balayage spaces are studied which, in probabilistic terms, correspond to killing and transitions (creation of mass combined with jumps). This is achieved by a modification of harmonic kernels for sufficiently small open sets. Applications to coupling of elliptic and parabolic partial differential equations of second order are discussed.

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

1990 ◽  
Vol 1 (3) ◽  
pp. 189-216 ◽  
Author(s):  
G. W. Bluman ◽  
S. Kumei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Linear Partial Differential Equation ◽  
Variable Coefficients ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Flow Equations ◽  
Local Symmetries ◽  
Necessary And Sufficient

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV 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.

scholarly journals The Exponential Behaviour of Nonlinear Stochastic Functional Equations of Second Order in Time

2003 ◽  
Vol 03 (02) ◽  
pp. 169-186 ◽  
Author(s):  
Tomás Caraballo ◽  
María J. Garrido-Atienza ◽  
José Real
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Partial Differential ◽  
Mean Square Stability ◽  
Exponential Behaviour ◽  
Second Order In Time ◽  
The Stability ◽  
Exponential Mean Square Stability

Sufficient conditions for exponential mean square stability of solutions to delayed stochastic partial differential equations of second order in time are established. As a consequence of these results, some on the pathwise exponential stability of the system are proved. The stability results derived are applied also to partial differential equations without hereditary characteristics. The results are illustrated with several examples.

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