Q13Advanced Engineering Mathematics

We are presented with a continuous integral equation of the form ${\int }_0^{\infty }f(t)\cos (\alpha t)dt=g(\alpha )$, where the piecewise function $g(\alpha )$ is defined as $(1-\alpha )$ for $0\le \alpha \le 1$ and $0$ for $\alpha >1$. The objective is to rigorously determine the unknown function $f(t)$.

1. Recognizing the Transform Structure

A crucial step in solving this problem is recognizing the mathematical architecture of the left-hand side of the equation. The integral ${\int }_0^{\infty }f(t)\cos (\alpha t)dt$ is precisely the defining mathematical formula for the Fourier Cosine Transform, typically denoted as $F_c(\alpha )$. Therefore, the problem has essentially provided us with the frequency-domain Fourier Cosine Transform $F_c(\alpha )$ and is asking us to reconstruct the original time-domain function $f(t)$.

To recover the original function, we must apply the Inverse Fourier Cosine Transform formula, which is formally defined as:

2. Evaluating the Inverse Transform Integral

We substitute the given piecewise function $g(\alpha )$ for $F_c(\alpha )$. Because $F_c(\alpha )$ is identically zero for all values of $\alpha >1$, the infinite upper bound of integration securely drops to 1, significantly simplifying the problem:

To evaluate this definite integral, we employ the mathematical technique of integration by parts, $\int udv=uv-\int vdu$. We logically select $u=(1-\alpha )$ because its derivative is a constant, leading to simplification, and $dv=\cos (\alpha t)d\alpha$. This gives $du=-1d\alpha$ and $v=\frac{\sin (\alpha t)}{t}$.

Let us carefully evaluate the boundary conditions. At the upper bound $\alpha =1$, the term $(1-\alpha )$ becomes $(1-1)=0$. At the lower bound $\alpha =0$, the sine term $\sin (0)$ is 0. Therefore, the entire $uv$ boundary term evaluates to exactly zero.

We now perform the final integration with respect to $\alpha$. The variable $t$ acts as a constant during this integration process.

Finally, we must not forget to multiply by the scaling coefficient $\frac{2}{\pi }$ from the inverse transform formula. The final, mathematically proven expression for the unknown function is: