I am reading a book on mathamatical physics and in its integral expansion section, integral representation is introduced. However, my book only states the integral representations of solution to well studied functions and explains in detailed how to expand them, but wihout explaining how the integral representations are found in the firsr place.
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I think a good place to start would be to study Green’s Functions. – DaveNine Dec 21 '17 at 04:04
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Another option is to use integral transforms. In this answer I go through an example using the Laplace transform. – Antonio Vargas Dec 21 '17 at 04:54
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1Can you please elaborate a little more, e.g. by giving an example? Usually in physics we introduce the integral transforms (representations) of the solutions to the differential equation as follows: You study an equation on some normed functional space (usually Hilbert space), often it happens such that one can find a "nice" orthonormal frame (in case of the integral transform it will have an uncountable index set and the orthogonality is in some kind of distribution sence). So what you do, is search for the solution to your equation as an expansion in that basis. – Kiryl Pesotski Dec 21 '17 at 19:38
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1E.g. if you solve an equation for some $f(x)$ on $L^{2}(\mathbb{R})$, then we can expand it, say, in the orthonormal Fourier basis $f(x)=\frac{1}{(2\pi)^{1/2}}\int_{\mathbb{R}}\hat{f}(k)e^{ikx}dk$, which is onb in the sense $<k, q>=\frac{1}{(2\pi)}\int_{\mathbb{R}}e^{i(q-k)x}dx=\delta(q-k)$. If the support was $\mathbb{R}^{+}$, you can use a basis of spherical bessel functions $f(x)=\int_{0}^{\infty}\hat{f(k)}j_{l}(kx)dk$, where $\hat{f}(k)=\frac{2k^{2}}{\pi}\int_{0}^{\infty}f(x)j_{l}(kx)x^{2}dx$ as one has $\frac{2k^{2}}{\pi}\int_{0}^{\infty}j_{l}(qx)j_{l}(kx)x^{2}dx=\delta(q-k)$, and so on.. – Kiryl Pesotski Dec 21 '17 at 19:51
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Are you willing to find the series solution first? There is a way, sometimes, to render it there. – rrogers May 03 '18 at 20:54