Question

Continue with Problem 4 as in Problem 6.

Short answer

The solution is derived to be$u(x,t)=100\mathrm{erf}\left(\frac{x}{2\alpha \sqrt{t}}\right)-50\mathrm{erf}\left(\frac{x-1}{2\alpha \sqrt{t}}\right)-50\mathrm{erf}\left(\frac{x+1}{2\alpha \sqrt{t}}\right)$.

Step-by-step solution

Given Information:

It has been given to continue with problem 4.

Step 2:Uses of Laplace equation:

The Laplace transform of an ordinary differential equation converts it into an algebraic equation. Taking the Laplace transform of a partial differential equation reduces the number of independent variables by one, and so converts a two-variable partial differential equation into an ordinary differential equation.

Write the equation and solve further:

Write the expression for B(k) in the integral form.

$B\left(k\right)=\frac{200}{\pi }{\int }_0^1\sin \left(kz\right)dz$

Start solving.

$\begin{array}{c}u(x,t)=\frac{200}{\pi }{\int }_0^{\infty }B\left(k\right)e^{-{\alpha }^2{k}^{2}t}\sin \left(kx\right)dk\\ =\frac{200}{\pi }{\int }_0^{\infty }{e}^{-{\alpha }^2{k}^{2}t}\sin \left(kz\right)\sin \left(kx\right)dk{\int }_0^{1}dz\end{array}$

Use the following trigonometric formula.

$\sin \left(kx\right)\sin \left(kz\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.[\cos (kx-kz)-\cos (kx+kz)]$

It can written as mentioned below.

$u(x,t)=\frac{200}{\pi }{\int }_0^{\infty }{e}^{-{\alpha }^2{k}^{2}t}/2[\cos (kx-kz)-\cos (kx+kz)]dk{\int }_0^{1}dz$

If table of integrals is checked the equation mentioned below is found.

${\int }_0^{\infty }{e}^{-a{x}^2}\cos \left(bx\right)dx=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\sqrt{\pi /a}{e}^{-b^2/\left(4a\right)}$

Thus, the equation changes.

$u(x,t)=\frac{100}{\pi }{\int }_0^1\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\frac{\sqrt{\pi }}{{\alpha }^{2}t}(e^{\frac{{(x-z)}^2}{4{\alpha }^{2}t}}-e^{\frac{{(x+z)}^2}{4{\alpha }^{2}t}})dz$

For the final step, check on a table of integrals again.

${\int }_0^{\infty }{e}^{\frac{{(a-x)}^2}{b}}dx=-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\sqrt{\pi b}erf\left(\frac{a-x}{\sqrt{b}}\right)$

Thus, identifying every parameter with our integral.

$u(x,t)=100\mathrm{erf}\left(\frac{x}{2\alpha \sqrt{t}}\right)-50\mathrm{erf}\left(\frac{x-1}{2\alpha \sqrt{t}}\right)-50\mathrm{erf}\left(\frac{x+1}{2\alpha \sqrt{t}}\right)$

Hence, the final answer is$\mathrm{u}(\mathrm{x},\mathrm{t})=100\mathrm{erf}\left(\frac{\mathrm{x}}{2\mathrm{\alpha }\sqrt{\mathrm{t}}}\right)-50\mathrm{erf}\left(\frac{\mathrm{x}-1}{2\mathrm{\alpha }\sqrt{\mathrm{t}}}\right)-50\mathrm{erf}\left(\frac{\mathrm{x}+1}{2\mathrm{\alpha }\sqrt{\mathrm{t}}}\right)$.