# Why we study fundamental solution of PDE?

Why we study fundamental solution of PDE?

I just started learning Laplace equation and Heat Equation.I studied theorem related to fundamental solution. I do not understand why it is used Even it has singularity at origin.

I am new at pde Please Any help will be appreciated Thanks a lot

We want to have a solution $$\Phi$$ of the PDE $$-\Delta \Phi = \delta_0 \\$$ in $$\mathbb{R}^n$$. The reason this is natural is that this then lets us solve $$-\Delta u = f$$ for any (nice enough) $$f$$ by convolving $$\Phi$$ with $$f$$. Indeed, the solution to $$-\Delta u = f$$ in $$\mathbb{R}^n$$ is $$u = \Phi * f$$. This is justified rigorously in Evans chapter 2, for instance, but informally we have \begin{align*} -\Delta u(x_0) &= -\Delta \int_{\mathbb{R}^n} \Phi(x_0 - y)f(y)\, dy \\ &= \int_{\mathbb{R}^n} -\Delta \Phi(x_0 - y)f(y)\, dy \\ &= \int_{\mathbb{R}^n}\delta_0(x_0 - y) f(y) \, dy = f(x_0). \end{align*} Here we really just used that the "fundamental solution" had minus Laplacian equal to the Dirac mass at zero. The point is that convolution with this fundamental solution allows us to solve Poisson's equation on $$\mathbb{R}^n$$ for any nice enough $$f$$. (Evans assumes $$f \in C_c^2(\mathbb{R}^n)$$, but then refers to Gilbarg-Trudinger for a proof of this fact with less requirements on $$f$$.)