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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.6. Systems of Equations
Equation(s):\noindent
$\displaystyle \frac{\partial w_i}{\partial t}=\sum_{j=1}^n {\Large L}_{i,j}(t,\vec{x})\cdot w_j+\phi_i(t,\vec{x}),\qquad i=1,\dots,n$,\hfill\break
where $t$ and $\vec{x}=(x_1,\dots,x_m)$ are independent variables, $w_i=w_i(t,\vec{x})$ are dependent variables, 
${\Large L}_{i,j}(t,\vec{x})$ are arbitrary \emph{linear differential operators}, 
which do not depend on $\frac{\partial}{\partial t}$, and $\phi_i(t,\vec{x})$ are arbitrary functions.

%\noindent
%$\displaystyle \frac{\partial}{\partial t}w_i(t,\vec{x})=\sum_{j=1}^n {\Large \hat{D}}_{i,j}(t,\vec{x})\cdot w_j(t,\vec{x})+\phi_i(t,\vec{x})\qquad (i=1,\dots,n)$,\hfill\break
%where $t$ and $\vec{x}=(x_1,\dots,x_m)$ are independent variables, ${\Large \hat{D}}_{i,j}(t,\vec{x})$ are arbitrary \emph{linear differential operators}, which do not depend on $\frac{\partial}{\partial t}$ explicitely, $\phi_i(t,\vec{x})$ are arbitrary functions.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
1. The formal exact general solution in an operator form $(i=1,\dots,n)$:\hfill\break
$\displaystyle w_i(t,\vec{x})=\left[\exp\left\{(t-a)\left[\sum_{i,j=1}^n v_i{\Large L}_{i,j}(s,\vec{x})\frac{\partial}{\partial v_j}-\frac{\partial}{\partial s}\right]\right\}\sum_{j=1}^n v_j F_j(\vec{x})+\right.\\
\left.+\int_a^t\,\exp\left\{(t-\tau)\left[\sum_{i,j=1}^n v_i{\Large L}_{i,j}(s,\vec{x})\frac{\partial}{\partial v_j}-\frac{\partial}{\partial s}\right]\right\}\sum_{j=1}^n v_j \phi_j(\tau,\vec{x}) \,d\tau\right]|_{\Large {s=t,v_i=1,v_{j\neq i}=0}}$,\hfill\break
where $F_j(\vec{x})$ are arbitrary functions and $a$ is an arbitrary constant.
\medskip

\noindent
2. If expand the operator exponents into Taylor series and execute operations, one can obtain some sort of formal power series to solutions of given system of differential equations.
\medskip

\noindent
3. There are other operator representations of formal exact general solution, which lead to another series type. 
%\noindent
%The formal exact general solution in an operator form $(i=1,\dots,n)$:\hfill\break
%$\displaystyle w_i(t,\vec{x})=\left[\exp\left\{(t-a)\left[\sum_{i,j=1}^n v_i{\Large \hat{D}}_{i,j}(s,\vec{x})\frac{\partial}{\partial v_j}-\frac{\partial}{\partial s}\right]\right\}\sum_{j=1}^n v_j F_j(\vec{x})+\right.\\
%\left.+\int_a^t\,\exp\left\{(t-\tau)\left[\sum_{i,j=1}^n v_i{\Large \hat{D}}_{i,j}(s,\vec{x})\frac{\partial}{\partial v_j}-\frac{\partial}{\partial s}\right]\right\}\sum_{j=1}^n v_j \phi_j(\tau,\vec{x}) \,d\tau\right]|_{\Large {s=t,v_i=1,v_{j\neq i}=0}}$,\hfill\break
%where $F_j(\vec{x})$ are arbitrary functions and $a$ is an arbitrary constant.
Remarks:\noindent
The operators ${\Large L}_{i,j}(t,\vec{x})$ can be the linear differential operators of \emph{any} order with respect to $x_i$.

%\noindent
%The operators ${\Large \hat{D}}_{i,j}(t,\vec{x})$ can be the linear differential operators of \emph{any} order with respect to $x_i$.
Novelty:Material has been fully published elsewhere
References:Yu.N. Kosovtsov, The Chronological Operator Algebra and Formal Solutions of Differential Equations, 2004, http://arxiv.org/abs/math-ph/0409035. See also Maple implementations of the operator method at http://www.maplesoft.com/applications/app_center_advanced_search.aspx?ABA=325
Author/Contributor's Details
Last name:Kosovtsov
First name:Yurii
Country:Ukraine
City:Lvov
Statistic information
Submission date:Tue 12 Dec 2006 09:50
Edits by author:2
Last edit by author:Fri 15 Dec 2006 11:08

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