MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

View Equation

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

Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.1. Second-Order Parabolic Equations
Equation(s):$\displaystyle x^a\frac{\partial w}{\partial t}+b\frac{\partial^2w}{\partial x^2}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,t)=1$,\\
$\displaystyle w_1(x,t)=t-\frac {x^{a+2}}{b(a+1)(a+2)}$,\\
$\displaystyle w_2(x,y)=y^2-\frac {2yx^{a+2}}{b(a+1)(a+2)}+\frac {2x^{2a+4}}{b^22!\left(\frac{a+1}{a+2}\right)_2(a+2)^4}$,\\
$\displaystyle w_3(x,y)=y^3-\frac {3y^2x^{a+2}}{b(a+1)(a+2)}+\frac
{6yx^{2a+4}}{b^22!\left(\frac{a+1}{a+2}\right)_2(a+2)^4}-
\frac {6x^{3a+6}}{b^33!\left(\frac{a+1}{a+2}\right)_3(a+2)^6}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=y^n+\sum_{k=1}^n \frac {n!(-1)^ky^{n-k}x^{ka+2k}}{b^k(n-k)!k!\left(\frac{a+1}{a+2}\right)_k(a+2)^{2k}}$.
\medskip

2. Particular solutions:\\
$w_0(x,t)=x$,\\
$\displaystyle w_1(x,t)=xt-\frac {x^{a+3}}{b(a+2)(a+3)}$,\\
$\displaystyle w_2(x,y)=xy^2-\frac {2yx^{a+3}}{b(a+2)(a+3)}+\frac {2x^{2a+5}}{b^22!\left(\frac{a+3}{a+2}\right)_2(a+2)^4}$,\\
$\displaystyle w_3(x,y)=xy^3-\frac {3y^2x^{a+3}}{b(a+2)(a+3)}+\frac
{6yx^{2a+5}}{b^22!\left(\frac{a+3}{a+2}\right)_2(a+2)^4}-
\frac {6x^{3a+7}}{b^33!\left(\frac{a+3}{a+2}\right)_3(a+2)^6}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=xy^n+\sum_{k=1}^n \frac {n!(-1)^ky^{n-k}x^{ka+2k+1}}{b^k(n-k)!k!\left(\frac{a+3}{a+2}\right)_k(a+2)^{2k}}$.
Remarks:Here $a\neq-2$, $\displaystyle a\neq-2-\frac 1k$, $\displaystyle a\neq-2+\frac 1k$, where $k=1,\,2,\,\dots$
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latviya
City:Sigulda
Statistic information
Submission date:Tue 29 Apr 2008 19:48
Edits by author:1
Last edit by author:Mon 05 Oct 2015 19:42

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin