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Derive the following formula

$u(t,x)={1 \over 3}t^3+{1 \over 2}t^2x^2$

for the one-dimensional non-homogeneous heat equation

$u_t=u_{xx}+tx^2, t>0, x\in (-\infty,\infty)$

$u(0,x)=0.$

(without plug in)

I tried Fourier Transform but nothing helped since

$\int e^{-iwx} tx^2 dx$ is not integrable.

Please help.

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Use laplace transform on t $U'=\frac{\partial U(s,x)}{\partial x}$

$$U''(x)-s U(x)=-\frac{x^2}{s^2}$$

$$U(s,x)=c_1 e^{\sqrt{s} x}+c_2 e^{-\sqrt{s} x}+\frac{s x^2+2}{s^4}$$

Applying initial conditions (https://en.wikipedia.org/wiki/Initial_value_theorem) using $s\rightarrow \infty$ and keeping in mind that we have both positive and negative $x$

$$U(s,x)=\frac{s x^2+2}{s^4}$$

Inverse laplace transform gives us

$$\frac{t^3}{3}+\frac{t^2 x^2}{2}$$

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  • $\begingroup$ Only Laplace Transformation can solve this problem? I don't know Laplace Transformation and this problem appeared in the chapter before learning Laplace Transformation. $\endgroup$ – SW PARK Jun 14 '15 at 14:06

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