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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:3.1. Second-Order Parabolic Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial t}=a\frac{\partial^2w}{\partial
x^2}+\frac bx \frac{\partial w}{\partial x}$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,t)=1$,\\
$\displaystyle w_1(x,t)=t+\frac {x^2}{2(a+b)}$,\\
$\displaystyle w_2(x,t)=t^2+\frac {tx^2}{a+b}+\frac {x^4}{4(a+b)(3a+b)}$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^n+\sum_{k=1}^n\frac {n!t^{n-k}x^{2k}}{(n-k)!(4a)^k\left(\frac{a+b}{2a}\right)_k}$.
\medskip

2. Particular solutions:\\
$w_0(x,t)=x^{1-\frac ba}$,\\
$\displaystyle w_1(x,t)=tx^{1-\frac ba}+\frac{x^{3-\frac ba}}{2(3a-b)}$,\\
$\displaystyle w_2(x,t)=t^2x^{1-\frac ba}+\frac{x^{3-\frac ba}}{3a-b}+\frac {x^{5-\frac ba}}{4(3a-b)(5a-b)}$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^nx^{1-\frac ba}+x^{1-\frac ba}\sum_{k=1}^{n}\frac {n!t^{n-k}x^{2k}}{(n-k)!(4a)^k\left(\frac{3a-b}{2a}\right)_k}$.\\
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Sat 05 Jul 2008 11:18
Edits by author:1
Last edit by author:Wed 07 Oct 2015 18:41

Edit (Only for author/contributor)


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