I have the following PDE:
$$A_z=A_x+\int A dt\cdot A$$
with $A$ a complex function. How can I separate the variables, such that the equation can be solved? Simple integration would lead to a double integral on the right hand side. Or should I rather differentiate on both sides?
In the original form it is
$$A_z=A_x+f\cdot A$$
with
$$\frac{df}{dt}=c\cdot \vert A\vert^4$$
and $c$ as a constant. Thus I thought that I can rewrite it as following:
$$\begin{split}
\frac{df}{dt}&=c\cdot A\\
df&=c\cdot A\cdot dt\\
\int df &= c\cdot \int A(t)dt\\
f&=c\cdot\int A(t)dt\end{split}$$
Is that correct?
Best approach for a PDE containing an integral?
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2So $A = A(x,z,t)$? And does $$\int A dt \cdot A = A \int A dt$$? – 2017-01-12
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0Added explanation, I hope that helps. – 2017-01-12
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2You should have limits of integration for your integral, otherwise this does not really make any sense. – 2017-01-12
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2I think you are missing some things. For example, what is $f$ a function of? If $f = f(t)$, then the solution is just $A = g(z+x) \exp(-x f(t))$ for arbitrary $g$, but I don't think $f$ is just a function of $t$. – 2017-01-12
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0@Mattos: I added how I got to $f(t)$, and how the derivation of $f$ is created. – 2017-01-12
1 Answers
1
$$A_z=A_x+A\int A dt$$ I will not answer to the textual question : Best approach for a PDE containing an integral?
I will only answer to the question : An approach to simplify a PDE containing an integral ?
Let $\quad A(x,z,t)=e^{F(x,z,t)} \quad\to\quad A_x=A\:F_x \quad;\quad A_z=A\:F_z $
$$F_z=F_x+\int e^F dt$$ Then, differentiating with respect to $t$ : $$F_{z,t}=F_{x,t}+e^F$$ which is a non-linear second order PDE, on a more usual form than a PDE containing an integral.
By the way, if the question was to solve the PDE, this would be a different kettle of fish.
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0If I extend the question to "How to solve it?", what would the answer be in that case? – 2017-01-12
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0As I already said : This would be a different kettle of fish. Now, I add : I am not fisherman and the fish is too big. – 2017-01-12