Exam 14: Integral Transform Method

Consider the problem of a vibrating string, tightly-stretched between $x = 0$ and $x = 1$ , with a fixed initial position, $f ( x )$ , and zero initial velocity. The mathematical problem for the deflection, $u ( x , t )$ , is $( \text { with } u ( 0 , t ) = 0 , u ( 1 , t ) = 0 )$

The value of $\lim _ { x \rightarrow \infty } \operatorname { erfc } ( x )$ is

In the previous problem, apply the Laplace transform. The resulting equation for $U ( x , s ) = \mathcal { L } \{ u ( x , t ) \}$ is

The Fourier integral representation of a function f is given by

In the previous problem, the solution for $U ( \alpha , t )$ is

The value of $\mathcal { L } \left\{ \mathrm { e } ^ { t } \operatorname { erf } ( \sqrt { t } ) \right\}$ is

In the previous two problems, the integral converges for $x < 0$ to the function

The solution of the integral equation $\int _ { 0 } ^ { - \infty } f ( x ) \cos ( a x ) d x = F ( \alpha )$ is

In the previous problem, apply the Laplace transform. The resulting equation for $U ( x , s ) = \mathcal { L } \{ u ( x , t ) \}$ is

The Fourier cosine integral of $f ( x ) = e ^ { - x } , x \geq 0$ is

The complex form of the Fourier integral of a function f is

The value of $\mathcal { L } \{ \operatorname { erf } ( \sqrt { t } ) \}$ is

In the three previous problems, the solution for $u ( x , t )$ is

In the previous three problems, the solution for $u ( x , t )$ is

In the previous problem, if $F ( \alpha ) = e ^ { - \alpha } \text {, then } f ( x )$ is

Consider the problem of finding the temperature in a semi-infinite rod with zero initial temperature and a fixed constant temperature, $u _ { 0 }$ , at $x = 0$ . The mathematical model for this problem is

In the previous problem, the integral converges at $x = 0$ to the value

The value of $\lim _ { x \rightarrow \infty } \operatorname { erf } ( x )$ is

In the previous problem, apply a Fourier sine transform in x. The new problem for the transform $U ( a , t )$ is

Consider the temperature, $u ( x , t )$ , in an infinite rod $( - \infty < x < \infty )$ , with an initial temperature of $f ( x ) = e ^ { - | x | }$ . The mathematical model for this is