Q18Advanced Engineering Mathematics

When solving partial differential equations, we often apply Fourier transforms to spatial derivatives. The property for the Fourier Cosine Transform of the second partial derivative of a function $f(x,t)$ with respect to $x$ is given by: $F_c\left[\frac{{\partial }^{2}f(x,t)}{\partial x^2}\right]=-s^2\bar{f}(s,t)-f^{\prime }(0,t)$, where $\bar{f}(s,t)$ is the Fourier cosine transform of $f(x,t)$.