- #1
Frank Castle
- 580
- 23
Suppose a PDE for a function of that depends on position, ##\mathbf{x}## and time, ##t##, for example the wave equation $$\nabla^{2}u(\mathbf{x},t)=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)$$ If I wanted to solve such an equation via a Fourier transform, can I Fourier transform with respect to ##\mathbf{x}##, but not ##t##? That is, can I assume an ansatz of the form $$u(\mathbf{x},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{u}(\mathbf{k},t)e^{i\mathbf{k}\cdot\mathbf{x}}$$