Analytic solutions of Lorentz-invariant linear equations

1962 ◽  
Vol 270 (1342) ◽  
pp. 326-328 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Wave Packets ◽  
Linear Equations ◽  
Analytic Solutions ◽  
Regular Wave ◽  
Einstein’S Equations ◽  
Partial Differential ◽  
Einstein's Equations

All the algebraically special, wave-like solutions of Einstein’s equations so far discovered admit hypersurface-orthogonal propagation vectors. Little is known about metrics with curling propagation vectors. Even in electrodynamics, few solutions of this type have been exhibited. This note presents a method of constructing classes of new solutions to linear, special relativistic partial differential equations. In particular, the method may be used to produce null, curling solutions of Maxwell’s and linearized Einstein’s equations. It consists in a generalization of a procedure used by Synge to obtain regular wave-packets from the fundamental solution ( t 2 - x 2 - y 2 - z 2 ) -1 (Synge 1960 a, b )

Fundamental Solution via Invariant Approach for a Brain Tumor Model and its Extensions

2014 ◽  
Vol 69 (12) ◽  
pp. 725-732 ◽  
Author(s):  
Andrew G. Johnpillai ◽  
Fazal M. Mahomed ◽  
Saeid Abbasbandy
Keyword(s):  
Brain Tumor ◽  
Partial Differential Equations ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Closed Form Solution ◽  
Form Solution ◽  
Tumor Models ◽  
Partial Differential ◽  
Equivalence Transformations ◽  
Extended Class

AbstractWe firstly show how one can use the invariant criteria for a scalar linear (1+1) parabolic partial differential equations to perform reduction under equivalence transformations to the first Lie canonical form for a class of brain tumor models. Fundamental solution for the underlying class of models via these transformations is thereby found by making use of the well-known fundamental solution of the classical heat equation. The closed-form solution of the Cauchy initial value problem of the model equations is then obtained as well. We also demonstrate the utility of the invariant method for the extended form of the class of brain tumor models and find in a simple and elegant way the possible forms of the arbitrary functions appearing in the extended class of partial differential equations. We also derive the equivalence transformations which completely classify the underlying extended class of partial differential equations into the Lie canonical forms. Examples are provided as illustration of the results.

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.

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