Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I don't know if there is a method to solve this following integro - differential equation:

$$\partial_t{u(x,t)}\int_{0}^{x}u(\zeta,t)d\zeta=\partial_{xx}u(x,t)$$ Can someone give me some hint? Thanks.

share|improve this question
    
One solution is zero. What answer would you like to have on the same question say for the heat equation? –  Andrew Apr 26 '12 at 14:06
    
@Andrew: zero is a trivial solution. I would like to know if there is a method to solve this kind equations. –  Riccardo.Alestra Apr 26 '12 at 15:05
add comment

1 Answer

up vote 4 down vote accepted

Let $v(x,t)=\int_0^xu(\zeta,t)\,d\zeta$. Then $0=v_x$ and the original equation is equivalent to $$ \tag{1}v\,v_{tx}=v_{xxx}. $$ This is a non-linear equation for which I doubt that a general solution can be found. However, you can find spcial solutions by reducing it to an ODE.

Solutions in separated variables. Looking for solutions of the form $v(x,t)=X(x)\,T(t)$ leads to $$ \frac{X'''}{X\,X'}=T'=a,\quad\text{$a$ constant.} $$ The solution of the equation $T'=\lambda$ is $T(t)=a\,t+b$. The equation for $X$ is $$ X'''=a\,X\,X'=\frac{a}{2}\,(X^2)'\implies X''=\frac{a}{2}\,X^2+b. $$ Multiplying the last equation by $X'$ and integrating we get the first order equation $$ (X')^2=3\,a\,X^3+2\,b\,X+c, $$ whose solution is $$ \int\frac{dx}{\sqrt{3\,a\,X^3+2\,b\,X+c}}\,dx=\pm\,x+d. $$ Taking $a=0$ gives $u(x,t)=A\,x+B$, $A,B\in\mathbb{R}$. Taking $a\ne0$ and $b=c=0$ gives the family of solutions $$ u(x,t)=\frac{a\,(a\,t+B)}{12\,(C\pm\,x)^{3}}. $$

Self-similar solutions. For any $\beta\in\mathbb{R}$, the function $v(x,t)=t^{1+2\beta}w(x\,t^\beta)$ is a solution of (1) if $w(\xi)$ is a solution of the ODE $$ (1+2\,\beta)\,w\,w'+\beta\,\xi\,w''=w'''. $$

Travelling wave solutions. $v(x,t)=\phi(x+c\,t)$ leads to the ODE $$ c\,\phi\,\phi''=\phi'''. $$

I have not tried to solve the last two ODE's, and I do not think it will be easy, if at all feasible. But may be some properties of their solutions can be found.

share|improve this answer
    
I think there are not any traveling wave solutions because $u_{xx}(0,t)=0$, which implies $\phi'''=0$, etc. But I am curious what is the purpose for this equation? –  Bob Terrell Apr 27 '12 at 19:07
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.