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 $$u(\mathbf{x},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{u}(\mathbf{k},t)e^{i\mathbf{k}\cdot\mathbf{x}}$$(adsbygoogle = window.adsbygoogle || []).push({});

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# I Solution to PDEs via Fourier transform

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