scholarly journals PROLONGATIONS OF GEOMETRIC OVERDETERMINED SYSTEMS

2006 ◽  
Vol 17 (06) ◽  
pp. 641-664 ◽  
Author(s):  
THOMAS BRANSON ◽  
ANDREAS ČAP ◽  
MICHAEL EASTWOOD ◽  
A. ROD GOVER
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Wide Class ◽  
Linear Equations ◽  
Solution Space ◽  
Partial Differential ◽  
Sharp Bounds ◽  
Overdetermined Systems ◽  
Closed Systems

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.

Some Partial Differential Equations of Multivariable Vector Functions and the Integral Equations of Multivariable Vector Functions

2010 ◽  
Vol 159 ◽  
pp. 205-209
Author(s):  
Han Zhang Qu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Vector Function ◽  
Weak Topology ◽  
Wavelet Transforms ◽  
Linear Equations ◽  
Continuous Wavelet ◽  
Partial Differential ◽  
Continuous Wavelet Transforms

The relations between the partial differential equations of multivariable vector functions and the integral equations of multivariable vector functions which are correspond to them are discussed. The partial differential linear equations of multivariable vector functions can be transformed into the integral linear equations of multivariable vector functions by using the continuous wavelet transforms of multivariable vector function spaces. The result that in the weak topology the partial differential equations of multivariable vector functions are equivalent to the integral equations of multivariable vector functions which are correspond to them is obtained.

Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II

2016 ◽  
Vol 19 (3) ◽  
pp. 632-647 ◽  
Author(s):  
John C. Morrison ◽  
Kyle Steffen ◽  
Blake Pantoja ◽  
Asha Nagaiya ◽  
Jacek Kobus ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Domain Decomposition ◽  
Diatomic Molecule ◽  
Linear Equations ◽  
Finite Difference Approximation ◽  
Partial Differential ◽  
Hartree Fock ◽  
Finite Difference Approach

AbstractIn order to solve the partial differential equations that arise in the Hartree- Fock theory for diatomicmolecules and inmolecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variablemodel problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9- point difference formula produces the same level of accuracy with fewer points. The domain decompositionmethod has the strength that it can be applied to problemswith a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which – for a diatomic molecule – involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments that are calculated independently. Since the domain decomposition approachmakes it possible to decompose the variable space into separate regions in which the equations are solved independently, this approach is well-suited to parallel computing.

scholarly journals ON ASYMPTOTIC PROPERTIES OF SOLUTIONS, DEFINED ON THE HALF OF AXIS OF ONE SEMILINEAR ODE

2017 ◽  
Vol 21 (6) ◽  
pp. 130-134
Author(s):  
I.V. Filimonova ◽  
T.S. Khachlaev
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unknown Function ◽  
Linear Equations ◽  
Unbounded Domains ◽  
Elliptic Partial Differential Equations ◽  
Order Derivative ◽  
Partial Differential ◽  
First Order ◽  
Fowler Equation

The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential. The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.

Stochastic Differential Equations

2019 ◽  
pp. 67-84
Author(s):  
Tomas Björk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Representation Theorem ◽  
Existence And Uniqueness ◽  
Linear Equations ◽  
Expected Value ◽  
Partial Differential ◽  
Parabolic Pde

In this chapter we introduce stochastic differential equations (SDEs) and discuss existence and uniqueness questions. The geometric and linear equations are studied in some detail and their most important properties are derived. We then discuss the connection between SDEs and partial differential equations (PDEs). In particular we prove the Feynman–Kač representation theorem which provides the solution to a parabolic PDE in terms of an expected value connected to a certain SDE. We also discuss and derive the Kolmogorov forward and backward equations.

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