Chapter 12 Fourier Solutions of Partial Differential Equations

\(u(x,t)=\dst{8\over\pi^3}\sum_{n=1}^\infty {1\over(2n-1)^3} e^{-(2n-1)^2\pi^2t}\sin(2n-1)\pi x\)

\(u(x,t)=\dst{4\over\pi}\sum_{n=1}^\infty{1\over(2n-1)} e^{-9(2n-1)^2\pi^2t/16}\sin{(2n-1)\pi x\over 4}\)

\(u(x,t)=\dst{\pi\over2}e^{-3t}\sin x-{16\over\pi}\sum_{n=1}^\infty{n\over(4n^2-1)^2}e^{-12n^2t}\sin2nx\)

\(u(x,t)=\dst-{32\over\pi^3}\sum_{n=1}^\infty {(1+(-1)^n2)\over n^3} e^{-9n^2\pi^2t/4}\sin{n\pi x\over 2}\)

\(u(x,t)=\dst-{324\over\pi^3}\sum_{n=1}^\infty{(-1)^n\over n^3} e^{-4n^2\pi^2t/9}\sin{n\pi x\over 3}\)

\(u(x,t)=\dst{8\over\pi^2}\sum_{n=1}^\infty{(-1)^{n+1}\over(2n-1)^2} e^{-(2n-1)^2\pi^2t}\sin{(2n-1)\pi x\over 2}\)

\(u(x,t)=\dst-{720\over\pi^5}\sum_{n=1}^\infty{(-1)^n\over n^5}e^{-7n^2\pi^2t}\sin n\pi x\)

\(u(x,t)=\dst{96\over\pi^5}\sum_{n=1}^\infty{1\over(2n-1)^5} e^{-5(2n-1)^2\pi^2t}\sin(2n-1)\pi x\)

\(u(x,t)=\dst-{240\over\pi^5}\sum_{n=1}^\infty{1+(-1)^n2\over n^5}e^{-2n^2\pi^2t}\sin n\pi x\).

\(u(x,t)=\dst{16\over3}+{64\over\pi^2}\sum_{n=1}^\infty{(-1)^n\over n^2} e^{-9\pi^2n^2t/16}\cos{n\pi x\over 4}\)

\(u(x,t)=\dst-{8\over3}+{16\over\pi^2}\sum_{n=1}^\infty{1\over n^2}e^{-n^2\pi^2t}\cos{n\pi x\over 2}\)

\(u(x,t)=\dst{1\over6}-{1\over\pi^2}\sum_{n=1}^\infty{1\over n^2}e^{-36n^2\pi^2t}\cos 2n\pi x\)

\(u(x,t)=\dst4-{384\over\pi^4}\sum_{n=1}^\infty{1\over (2n-1)^4}e^{-3(2n-1)^2\pi^2t/4}\cos{(2n-1)\pi x\over 2}\)

\(u(x,y)=\dst-{28\over5}-{576\over\pi^4}\sum_{n=1}^\infty{(-1)^n\over n^4}e^{-5n^2\pi^2t/2}\cos{n\pi x\over\sqrt2}\)

\(u(x,t)=\dst-{2\over5}-{48\over\pi^4}\sum_{n=1}^\infty{1+(-1)^n2\over n^4}e^{-3n^2\pi^2t}\cos n\pi x\)

\(u(x,t)=\dst{3\over5}-{48\over\pi^4}\sum_{n=1}^\infty{2+(-1)^n\over n^4}e^{-n^2\pi^2t}\cos n\pi x\)

\(\dst u(x,t)={\pi^4\over30}-3\sum_{n=1}^\infty{1\over n^4}e^{-4n^2t}\cos2nx\)

\(u(x,t)=\dst{{8\over\pi} \sum^\infty_{n=1} {(-1)^n\over(2n+1)(2n-3)} e^{-(2n-1)^2\pi^2 t/4}\sin{(2n-1)\pi x\over 2}}\)

\(u(x,t)=\dst8\sum_{n=1}^\infty{1\over(2n-1)^2}\left[ (-1)^n+{4\over(2n-1)\pi}\right]e^{-3(2n-1)^2t/4}\sin{(2n-1) x\over 2}\)

\(u(x,t)=\dst {128\over\pi^3}\sum_{n=1}^\infty{1\over (2n-1)^3}e^{-5(2n-1)^2t/16}\sin{(2n-1)\pi x\over4}\)

\(u(x,t) =\dst-{96\over\pi^3}\sum_{n=1}^\infty {1\over(2n-1)^3}\left[1+(-1)^n{4\over(2n-1)\pi}\right] e^{-(2n-1)^2\pi^2t/4}\sin{(2n-1)\pi x\over2}\)

\(u(x,t) =\dst{96\over\pi^3}\sum_{n=1}^\infty {1\over(2n-1)^3}\left[1+(-1)^n{2\over(2n-1)\pi}\right] e^{-(2n-1)^2\pi^2t/4}\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{192\over\pi^4}\sum_{n=1}^\infty{(-1)^n\over(2n-1)^4} e^{-(2n-1)^2\pi^2t/4}\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{1536\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4} \left[(-1)^n+{3\over(2n-1)\pi}\right] e^{-(2n-1)^2\pi^2t/4}\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{384\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4} \left[(-1)^n+{4\over(2n-1)\pi}\right] e^{-(2n-1)^2\pi^2t/4}\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst64\sum_{n=1}^\infty{e^{-3(2n-1)^2t/4}\over(2n-1)^3}\left[ (-1)^n+{3\over(2n-1)\pi}\right]\cos{(2n-1) x\over2}\)

\(u(x,t)=-\dst{16\over\pi} \sum^\infty_{n=1} {(-1)^n\over2n-1} e^{-(2n-1)^2t}\cos{(2n-1)x\over4}\)

\(u(x,t)=\dst-{64\over\pi}\sum_{n=1}^\infty{(-1)^n\over2n-1}\left[ 1-{8\over(2n-1)^2\pi^2}\right]e^{-9(2n-1)^2\pi^2 t/64}\cos{(2n-1)\pi x\over8}\)

\(u(x,t)= \dst{8\over\pi^2}\sum_{n=1}^\infty{1\over(2n-1)^2} e^{-3(2n-1)^2\pi^2t/4} \cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst{96\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ (-1)^n+{2\over(2n-1)\pi}\right]e^{-(2n-1)^2\pi^2t/4}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst {32\over\pi}\sum_{n=1}^\infty{(-1)^n\over (2n-1)^3}e^{-7(2n-1)^2t/4}\cos{(2n-1) x\over2}\)

\(u(x,t)=\dst{96\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ (-1)^n5+{8\over(2n-1)\pi}\right]e^{-(2n-1)^2\pi^2t/4}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{96\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ (-1)^n3+{4\over(2n-1)\pi}\right]e^{-(2n-1)^2\pi^2t/4}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst{768\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4}\left[ 1+{(-1)^n2\over(2n-1)\pi}\right]e^{-(2n-1)^2\pi^2t/4}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst{384\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4}\left[ 1+{(-1)^n4\over(2n-1)\pi}\right]e^{-(2n-1)^2\pi^2t/4}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{1\over2}-{2\over\pi}\sum_{n=1}^\infty {(-1)^n\over2n-1}e^{-(2n-1)^2\pi^2a^2t/L^2}\cos{(2n-1)\pi x\over L}\)

\(u(x,t)=\dst{2\over\pi}\sum_{n=1}^\infty{1\over n} \left[1-\cos{n\pi\over2}\right]e^{-n^2\pi^2a^2t/L^2}\sin{n\pi x\over L}\)

\(u(x,t)=\dst{4\over\pi}\sum_{n=1}^\infty{1\over2n-1}\sin{(2n-1)\pi\over4} e^{-(2n-1)^2\pi^2a^2t/4L^2}\cos{(2n-1)\pi x\over 2L}\)

\(u(x,t)=\dst{4\over\pi}\sum_{n=1}^\infty{1\over2n-1} \left[1-\cos{(2n-1)\pi)\over4}\right] e^{-(2n-1)^2\pi^2a^2t/4L^2}\sin{(2n-1)\pi x\over 2L}\)

\(u(x,t)=1-x+x^3+\dst{4\over\pi}\sum_{n=1}^\infty{e^{-9\pi^2(2n-1)^2t/16}\over(2n-1)} \sin{(2n-1)\pi x\over 4}\)

\(u(x,t)=\dst1+x+x^2 -\dst{8\over\pi^3}\sum_{n=1}^\infty {e^{-(2n-1)^2\pi^2t}\over(2n-1)^3} \sin(2n-1)\pi x\)

\(u(x,t)=-1-x+x^3+ \dst{8\over\pi^2}\sum_{n=1}^\infty{1\over(2n-1)^2} e^{-3(2n-1)^2\pi^2t/4} \cos{(2n-1)\pi x\over2}\)

\(u(x,t)=x^2-x-2\dst-{64\over\pi}\sum_{n=1}^\infty{(-1)^n\over2n-1}\left[ 1-{8\over(2n-1)^2\pi^2}\right]e^{-9(2n-1)^2\pi^2 t/64}\cos{(2n-1)\pi x\over8}\)

\(u(x,t)=\sin\pi x+\dst{{8\over\pi} \sum^\infty_{n=1} {(-1)^n\over(2n+1)(2n-3)} e^{-(2n-1)^2\pi^2 t/4}\sin{(2n-1)\pi x\over 2}}\)

\(u(x,t)=x^3-x+3+\dst{32\over\pi^3}\sum_{n=1}^\infty{e^{-(2n-1)^2\pi^2t/4}\over (2n-1)^3}\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{4\over3\pi^3}\sum_{n=1}^\infty {(-1)^{n+1}\over(2n-1)^3}\sin3(2n-1)\pi t\sin(2n-1)\pi x\)

\(u(x,t)=\dst{8\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3} \cos3(2n-1)\pi t\sin(2n-1)\pi x\)

\(u(x,t)=\dst-{4\over\pi^3}\sum_{n=1}^\infty{(1+(-1)^n2)\over n^3} \cos n\sqrt7\,\pi t\,\sin n\pi x\)

\(u(x,t)=\dst{8\over3\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4} \sin3(2n-1)\pi t\sin(2n-1)\pi x\)

\(u(x,t)=\dst-{4\over\sqrt7\,\pi^4}\sum_{n=1}^\infty{(1+(-1)^n2)\over n^4} \sin n\sqrt7\,\pi t\,\sin n\pi x\)

\(u(x,t)=\dst{324\over\pi^3}\sum_{n=1}^\infty{(-1)^n\over n^3}\cos{8n\pi t\over3}\sin{n\pi x\over3}\)

\(u(x,t)=\dst{96\over\pi^5}\sum_{n=1}^\infty{1\over(2n-1)^5} \cos2(2n-1)\pi t\sin(2n-1)\pi x\)

\(u(x,t)=\dst{243\over2\pi^4}\sum_{n=1}^\infty{(-1)^n\over n^4}\sin{8n\pi t\over3}\sin{n\pi x\over3}\)

\(u(x,t)=\dst{48\over\pi^6}\sum_{n=1}^\infty{1\over(2n-1)^6} \sin2(2n-1)\pi t\sin(2n-1)\pi x\).

\(u(x,t)=\dst{\pi\over2}\cos\sqrt{5}\,t\sin x-{16\over\pi} \sum_{n=1}^\infty{n\over(4n^2-1)^2}\cos2n\sqrt5\,t\sin 2nx\)

\(u(x,t)=\dst-{240\over\pi^5}\sum_{n=1}^\infty{1+(-1)^n2\over n^5} \cos n\pi t\sin n\pi x\)

\(u(x,t)=\dst{\pi\over2\sqrt5}\sin\sqrt{5}\,t\sin x-{8\over\pi\sqrt5} \sum_{n=1}^\infty{1\over(4n^2-1)^2}\sin2n\sqrt5\,t\sin 2nx\)

\(u(x,t)=\dst-{240\over\pi^6}\sum_{n=1}^\infty{1+(-1)^n2\over n^6} \sin n\pi t\sin n\pi x\)

\(u(x,t)=\dst-{720\over\pi^5}\sum_{n=1}^\infty{(-1)^n\over n^5} \cos3n\pi t\sin n\pi x\)

\(u(x,t)=\dst-{240\over\pi^6}\sum_{n=1}^\infty{(-1)^n\over n^6} \sin3n\pi t\sin n\pi x\)

\(u(x,t)=-\dst {128\over\pi^3}\sum_{n=1}^\infty{(-1)^n\over (2n-1)^3}\cos{3(2n-1)\pi t\over 4}\,\cos{(2n-1)\pi x\over4}\)

\(u(x,t)=-\dst{64\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ (-1)^n+{3\over(2n-1)\pi}\right]\cos(2n-1)\pi t\,\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst {512\over3\pi^4}\sum_{n=1}^\infty{(-1)^n\over (2n-1)^4}\sin{3(2n-1)\pi t\over 4}\,\cos{(2n-1)\pi x\over4}\)

\(u(x,t)=-\dst{64\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4}\left[ (-1)^n+{3\over(2n-1)\pi}\right]\sin(2n-1)\pi t\,\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{96\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ (-1)^n3+{4\over(2n-1)\pi}\right]\cos{(2n-1)\sqrt5\,\pi t\over2}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst96\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ (-1)^n+{2\over(2n-1)\pi}\right]\cos{(2n-1)\sqrt3\, t\over2}\cos{(2n-1) x\over2}\)

\(u(x,t)=\dst{192\over\pi^4\sqrt5}\sum_{n=1}^\infty{1\over(2n-1)^4}\left[ (-1)^n3+{4\over(2n-1)\pi}\right]\sin{(2n-1)\sqrt5\,\pi t\over2}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst{192\over\sqrt3}\sum_{n=1}^\infty{1\over(2n-1)^4}\left[ (-1)^n+{2\over(2n-1)\pi}\right]\sin{(2n-1)\sqrt3 t\over2}\sin{(2n-1) x\over2}\)

\(u(x,t)=-\dst{384\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4}\left[ 1+{(-1)^n4\over(2n-1)\pi}\right]\cos{3(2n-1)\pi t\over2}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{96\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ (-1)^n5+{8\over(2n-1)\pi}\right]\cos{(2n-1)\sqrt7\,\pi t\over2}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst{768\over3\pi^5}\sum_{n=1}^\infty{1\over(2n-1)^5}\left[ 1+{(-1)^n4\over(2n-1)\pi}\right]\sin{3(2n-1)\pi t\over2}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{192\over\pi^4\sqrt7}\sum_{n=1}^\infty{1\over(2n-1)^4}\left[ (-1)^n5+{8\over(2n-1)\pi}\right]\sin{(2n-1)\sqrt7\,\pi t\over2}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst{768\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4}\left[ 1+{(-1)^n2\over(2n-1)\pi}\right]\cos{(2n-1)\pi t\over2}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=-\dst{1536\over\pi^5}\sum_{n=1}^\infty{1\over(2n-1)^5}\left[ 1+{(-1)^n2\over(2n-1)\pi}\right]\sin{(2n-1)\pi t\over2}\cos{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{1\over2}[C_{M\!f}(x+at)+C_{M\!f}(x-at)]+ {1\over2a}\int_{x-at}^{x+at}C_{M\!g}(\tau)\,d\tau\)

\(u(x,t)=\dst {32\over\pi}\sum_{n=1}^\infty{1\over (2n-1)^3}\cos4(2n-1)t\,\sin{(2n-1)x\over2}\)

\(u(x,t) =\dst-{96\over\pi^3}\sum_{n=1}^\infty {1\over(2n-1)^3}\left[1+(-1)^n{4\over(2n-1)\pi}\right] \cos{3(2n-1)\pi t\over2}\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst {8\over\pi}\sum_{n=1}^\infty{1\over (2n-1)^4}\sin4(2n-1)t\,\sin{(2n-1)x\over2}\)

\(u(x,t) =\dst-{64\over\pi^4}\sum_{n=1}^\infty {1\over(2n-1)^4}\left[1+(-1)^n{4\over(2n-1)\pi}\right] \sin{3(2n-1)\pi t\over2}\sin{(2n-1)\pi x\over2}\)

\(u(x,t) =\dst{96\over\pi^3}\sum_{n=1}^\infty {1\over(2n-1)^3}\left[1+(-1)^n{2\over(2n-1)\pi}\right] \cos{3(2n-1)\pi t\over2} \sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{192\over\pi}\sum_{n=1}^\infty{(-1)^n\over(2n-1)^4} \cos{(2n-1)\sqrt3\,t\over2}\sin{(2n-1) x\over2}\)

\(u(x,t) =\dst{64\over\pi^4}\sum_{n=1}^\infty {1\over(2n-1)^4}\left[1+(-1)^n{2\over(2n-1)\pi}\right] \sin{3(2n-1)\pi t\over2} \sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{384\over\sqrt3\,\pi}\sum_{n=1}^\infty{(-1)^n\over(2n-1)^5} \sin{(2n-1)\sqrt3\,t\over2}\sin{(2n-1) x\over2}\)

\(u(x,t)=\dst{1536\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4} \left[(-1)^n+{3\over(2n-1)\pi}\right] \cos{(2n-1)\sqrt5\,\pi t\over2}\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{384\over\pi^4}\sum_{n=1}^\infty{1\over(2n-1)^4} \left[(-1)^n+{4\over(2n-1)\pi}\right]\cos(2n-1)\pi t\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{3072\over\sqrt5\,\pi^5}\sum_{n=1}^\infty{1\over(2n-1)^5} \left[(-1)^n+{3\over(2n-1)\pi}\right] \sin{(2n-1)\sqrt5\,\pi t\over2}\sin{(2n-1)\pi x\over2}\)

\(u(x,t)=\dst{384\over\pi^5}\sum_{n=1}^\infty{1\over(2n-1)^5} \left[(-1)^n+{4\over(2n-1)\pi}\right]\sin(2n-1)\pi t\sin{(2n-1)\pi x\over2}\)

\(\dst u(x,t)={1\over2}[S_{M\!f}(x+at)+S_{M\!f}(x-at)]+ {1\over2a}\int_{x-at}^{x+at}S_{M\!g}(\tau)\,d\tau\)

\(u(x,t)=\dst4-{768\over\pi^4}\sum_{n=1}^\infty{1\over (2n-1)^4}\cos{\sqrt5(2n-1)\pi t\over2}\,\cos{(2n-1)\pi x\over 2}\)

\(u(x,t)=\dst4t-{1536\over\sqrt5\,\pi^5}\sum_{n=1}^\infty{1\over (2n-1)^5}\sin{\sqrt5(2n-1)\pi t\over2}\,\cos{(2n-1)\pi x\over 2}\)

\(u(x,t)=\dst-{2\pi^4\over5}-48\sum_{n=1}^\infty{1+(-1)^n2\over n^4}\cos2nt\cos nx\)

\(u(x,t)=\dst-{7\over5}-{144\over\pi^4}\sum_{n=1}^\infty{(-1)^n\over n^4}\cos n\sqrt7\,\pi t\cos n\pi x\)

\(u(x,t)=\dst-{2\pi^4t\over5}-24\sum_{n=1}^\infty{1+(-1)^n2\over n^5}\sin2n t\cos nx\)

\(u(x,t)=\dst-{7t\over5}-{144\over\pi^5\sqrt7}\sum_{n=1}^\infty{(-1)^n\over n^5}\sin n\sqrt7\,\pi t\cos n\pi x\)

\(u(x,t)=\dst{{\pi^4\over30}-3\sum_{n=1}^\infty{1\over n^4}\cos8nt\cos2nx}\)

\(u(x,t)=\dst{3\over5}-{48\over\pi^4}\sum_{n=1}^\infty{2+(-1)^n\over n^4}\cos n\pi t\cos n\pi x\)

\(u(x,t)=\dst{{\pi^4t\over30}-{3\over8}\sum_{n=1}^\infty{1\over n^5}\sin8nt\cos2nx}\)

\(u(x,t)=\dst{3t\over5}-{48\over\pi^5}\sum_{n=1}^\infty{2+(-1)^n\over n^5}\sin n\pi t\cos n\pi x\)

\(u(x,t)={1\over2}[C_f(x+at)+C_f(x-at)]+ {1\over2a}\int_{x-at}^{x+at}C_g(\tau)\,d\tau\)

(c) \(u(x,t)=\dst{f(x+at)+f(x-at)\over2}+ {1\over2a}\int_{x-at}^{x+at}g(u)\,du\)

\(u(x,t)=x(1+4at)\)

\(u(x,t)=x^2+a^2t^2+t\)

\(u(x,t)=\sin(x+at)\)

\(u(x,t)=x^3+6tx^2+3a^2t^2x+2a^2t^3\)

\(u(x,t)=\dst x\sin x\cos at+at\cos x\sin at+{\sin x\sin at\over a}\)

\(u(x,y)=\dst{8\over\pi^3}\sum_{n=1}^\infty{\sinh(2n-1)\pi(1-y) \over(2n-1)^3\sinh(2n-1)\pi} \sin(2n-1)\pi x\)

\(u(x,y)=\dst-{32\over\pi^3}\sum_{n=1}^\infty{(1+(-1)^n2)\sinh n\pi(3-y)/2\over n^3\sinh3n\pi/2} \sin{n\pi x\over 2}\)

\(u(x,y)=\dst{8\over\pi^2}\sum_{n=1}^\infty(-1)^{n+1}{ \sinh(2n-1)\pi(1-y/2)\over(2n-1)^2\sinh(2n-1)\pi} \sin{(2n-1)\pi x\over 2}\)

\(u(x,y)=\dst{\pi\over2}{\sinh(1-y)\over\sinh1}\sin x-{16\over\pi}\sum_{n=1}^\infty{n\sinh2n(1-y)\over(4n^2-1)^2\sinh2n}\sin2nx\)

\(u(x,y)=\dst3y+{108\over\pi^3}\sum_{n=1}^\infty(-1)^n{\sinh n\pi y/3\over n^3\cosh2n\pi/3}\cos{n\pi x\over 3}\)

\(u(x,y)=\dst{y\over2}+{4\over\pi^3}\sum_{n=1}^\infty{ \sinh(2n-1)\pi y\over(2n-1)^3\cosh2(2n-1)\pi} \cos(2n-1)\pi x\)

\(u(x,y)=\dst-{8y\over3}+{32\over\pi^3}\sum_{n=1}^\infty(-1)^n{ \sinh n\pi y/2\over n^3\cosh n\pi}\cos{n\pi x\over 2}\)

\(u(x,y)=\dst{y\over3}+{4\over\pi^3}\sum_{n=1}^\infty{\sinh n\pi y\over n^3\cosh n\pi}\cos n\pi x\)

\(u(x,y)=\dst {128\over\pi^3}\sum_{n=1}^\infty{\cosh(2n-1)\pi(x-3)/4\over (2n-1)^3\cosh3(2n-1)\pi/4} \sin{(2n-1)\pi y\over4}\)

\(u(x,y) =\dst-{96\over\pi^3}\sum_{n=1}^\infty \left[1+(-1)^n{4\over(2n-1)\pi}\right] {\cosh(2n-1)\pi(x-2)/2\over(2n-1)^3\cosh(2n-1)\pi} \sin{(2n-1)\pi y\over2}\)

\(u(x,y) =\dst{768\over\pi^3}\sum_{n=1}^\infty \left[1+(-1)^n{2\over(2n-1)\pi}\right] {\cosh(2n-1)\pi(x-2)/4\over(2n-1)^3\cosh(2n-1)\pi/2} \sin{(2n-1)\pi y\over4}\)

\(u(x,y) =\dst{96\over\pi^3}\sum_{n=1}^\infty \left[3+(-1)^n{4\over(2n-1)\pi}\right] {\cosh(2n-1)\pi(x-3)/2\over(2n-1)^3\cosh3(2n-1)\pi/2} \sin{(2n-1)\pi y\over2}\)

\(u(x,y)= -\dst{16\over\pi}\sum_{n=1}^\infty{\cosh(2n-1)x/2\over (2n-3)(2n+1)(2n-1)\sinh(2n-1)/2} \cos{(2n-1) y\over2}\)

\(u(x,y)=-\dst{432\over\pi^3}\sum_{n=1}^\infty \left[1+{4(-1)^n\over(2n-1)\pi}\right] {\cosh(2n-1)\pi x/6\over(2n-1)^3\sinh(2n-1)\pi/3} \cos{(2n-1)\pi y\over6}\)

\(u(x,y)=-\dst {64\over\pi}\sum_{n=1}^\infty(-1)^n{\cosh(2n-1)x/2\over (2n-1)^4\sinh(2n-1)/2}\cos{(2n-1) y\over2}\).

\(u(x,y)=-\dst{192\over\pi^4}\sum_{n=1}^\infty {\cosh(2n-1)\pi x/2\over(2n-1)^4\sinh(2n-1)\pi/2} \left[(-1)^n+{2\over(2n-1)\pi}\right] \cos{(2n-1)\pi y\over2}\)

\(\dst u(x,y)=\sum_{n=1}^\infty\alpha_n{\sinh n\pi y/a\over \sinh n\pi b/a}\sin {n\pi x\over a},\quad \alpha_n={2\over a}\int_0^a f(x)\sin {n\pi x\over a}\,dx\)

    \(u(x,y)=\dst{72\over\pi^3}\sum_{n=1}^\infty {\sinh(2n-1)\pi y/3\over(2n-1)^3\sinh2(2n-1)\pi/3} \sin{(2n-1)\pi x\over 3}\)

\(\dst u(x,y)=\alpha_0(1-y/b)+\sum_{n=1}^\infty\alpha_n{\sinh n\pi(b-y)/ a\over\sinh n\pi b/a}\cos {n\pi x\over a}\),   \(\dst\alpha_0={1\over a}\int_0^a f(x)\,dx\),

   \(\dst\alpha_n={2\over a}\int_0^a f(x)\cos {n\pi x\over a}\,dx,\quad n\ge1\)

   \(\dst u(x,y)={8(1-y)\over15}-{48\over\pi^4}\sum_{n=1}^\infty{1\over n^4}{\sinh n\pi(1-y)\over\sinh n\pi}\cos n\pi x\)

\(\dst u(x,y)=\sum_{n=1}^\infty\alpha_n{\sinh(2n-1)\pi(b-y)/2a \over\sinh(2n-1)\pi b/2a}\cos {(2n-1)\pi x\over2a}\),

   \(\dst\alpha_n={2\over a}\int_0^a f(x)\cos {(2n-1)\pi x\over2a}\,dx\)

   \(u(x,y)=\dst {288\over\pi^3}\sum_{n=1}^\infty{\sinh(2n-1)\pi(2-y)/6\over (2n-1)^3\sinh(2n-1)\pi/3}\sin{(2n-1)\pi x\over6}\)

\(\dst u(x,y)=\sum_{n=1}^\infty\alpha_n{\sinh(2n-1)\pi(b-y)/2a\over \sinh(2n-1)\pi b/2a}\sin {(2n-1)\pi x\over2a}\),

   \(\dst\alpha_n={2\over a}\int_0^a f(x)\sin {(2n-1)\pi x\over2a}\,dx\)

   \(u(x,y)=\dst{32\over\pi^3}\sum_{n=1}^\infty\left[ (-1)^n5+{18\over(2n-1)\pi}\right] {\sinh(2n-1)\pi(2-y)/2\over(2n-1)^3\sinh(2n-1)\pi} \cos{(2n-1)\pi x\over2}\).

\(\dst u(x,y)=\sum_{n=1}^\infty\alpha_n{\cosh n\pi(y-b)/a\over \cosh n\pi b/a}\sin {n\pi x\over a}\),   \(\dst\alpha_n={2\over a}\int_0^a f(x)\sin {n\pi x\over a}\,dx\)

   \(u(x,y)=\dst-12\sum_{n=1}^\infty(-1)^n{\cosh n(y-2)\over n^3\cosh2n} \sin nx\)

\(\dst u(x,y)=\alpha_0+\sum_{n=1}^\infty\alpha_n{\cosh n\pi y/a \over\cosh n\pi b/a}\cos {n\pi x\over a}\), \(\dst\alpha_0={1\over a}\int_0^a f(x)\,dx\),

   \(\dst\alpha_n={2\over a}\int_0^a f(x)\cos {n\pi x\over a}\,dx,\quad n\ge1\)

   \(\dst u(x,y)={\pi^4\over30}-3\sum_{n=1}^\infty{1\over n^4}{\cosh2ny\over\cos2n}\cos 2nx\)

\(\dst u(x,y)={a\over\pi}\sum_{n=1}^\infty\alpha_n{\sinh n\pi (y-b)/a\over n\cosh n\pi b/a}\sin {n\pi x\over a}\),   \(\dst\alpha_n={2\over a}\int_0^a f(x)\sin {n\pi x\over a}\,dx\)

   \(u(x,y)=\dst{4\over\pi}\sum_{n=1}^\infty(-1)^{n+1} {\sinh(2n-1)(y-1)\over(2n-1)^3\cosh(2n-1)} \sin(2n-1)x\)

\(\dst u(x,y)=\sum_{n=1}^\infty\alpha_n{\cosh n\pi x/b\over \cosh n\pi a/b} \sin{n\pi y\over b}\),  \(\dst\alpha_n={2\over b}\int_0^b g(y)\sin {n\pi y\over b}\,dy\)

   \(u(x,y)=\dst{96\over\pi^5}\sum_{n=1}^\infty{\cosh(2n-1)\pi x \over(2n-1)^5\cosh(2n-1)\pi}\sin(2n-1)\pi y\).

\(\dst u(x,y)=\sum_{n=1}^\infty\alpha_n{\cosh(2n-1)\pi x/2b\over \cosh(2n-1)\pi a/2b} \cos{(2n-1)\pi y\over2b}\), 

   \(\dst\alpha_n={2\over b}\int_0^b g(y)\cos {(2n-1)\pi y\over2b}\,dy\)

   \(u(x,y)=-\dst {128\over\pi^3}\sum_{n=1}^\infty(-1)^n{\cosh(2n-1)\pi x/4\over (2n-1)^3\cosh(2n-1)\pi/2}\cos{(2n-1)\pi y\over4}\).

\(\dst u(x,y)={b\over \pi}\sum_{n=1}^\infty\alpha_n {\cosh n\pi x/b\over n \sinh n\pi a/b} \sin{n\pi y\over b}\),  \(\dst\alpha_n={2\over b}\int_0^b g(y)\sin {n\pi y\over b}\,dy\)

   \(u(x,y)=\dst{64\over\pi^3}\sum_{n=1}^\infty(-1)^{n+1} {\cosh(2n-1)\pi x/4\over(2n-1)^3\sinh(2n-1)\pi/4} \sin{(2n-1)\pi y\over 4}\)

\(\dst u(x,y)=-{2b\over\pi}\sum_{n=1}^\infty\alpha_n{\cosh(2n-1)\pi(x-a)/2b \over(2n-1)\sinh(2n-1)\pi a/2b} \sin{(2n-1) y\over2b}\), 

   \(\dst\alpha_n={2\over b}\int_0^b g(y)\sin {(2n-1)\pi y\over2b}\,dy\)

   \(u(x,y)=\dst192\sum_{n=1}^\infty \left[1+(-1)^n{4\over(2n-1)\pi}\right] {\cosh(2n-1)(x-1)/2\over(2n-1)^4\sinh(2n-1)/2} \sin{(2n-1)y\over2}\).

\(\dst u(x,y)=\alpha_0(x-a)+{b\over\pi}\sum_{n=1}^\infty\alpha_n{\sinh n\pi(x-a)/b \over n\cosh n\pi a/b} \cos{n\pi y\over b}\),  \(\dst\alpha_0={1\over b}\int_0^bg(y)\cos{n\pi y\over b}\,dy\),

   \(\dst\alpha_n={2\over b}\int_0^b g(y)\cos {n\pi y\over b}\,dy\)

   \(u(x,y)=\dst{\pi(x-2)\over2}-{4\over\pi}\sum_{n=1}^\infty {\sinh(2n-1)(x-2)\over(2n-1)^3\cosh2(2n-1)} \cos(2n-1)y\).

\(u(x,y)=\dst\alpha_0+\sum_{n=1}^\infty\alpha_n e^{-n\pi y/a}\cos {n\pi x\over a}\),  \(\alpha_0=\dst{1\over a}\int_0^a f(x)\,dx\),

   \(\alpha_n=\dst{2\over a}\int_0^a f(x)\cos {n\pi x\over a}\,dx,\quad n\ge1\)

   \(u(x,y)={\pi^3\over2}-{48\over\pi}\sum_{n=1}^\infty{1\over (2n-1)^4}e^{-(2n-1)y}\cos(2n-1)x\)

\(u(x,y)=\dst\sum_{n=1}^\infty\alpha_n e^{-(2n-1)\pi y/2a}\cos {(2n-1)\pi x\over2a}\),  \(\alpha_n=\dst{2\over a}\int_0^a f(x)\cos {(2n-1)\pi x\over2a}\,dx\)

   \(u(x,y)=-\dst {288\over\pi^3}\sum_{n=1}^\infty{(-1)^n\over (2n-1)^3}e^{-(2n-1)\pi y/6}\cos{(2n-1)\pi x\over6}\)

\(u(x,y)=\dst\sum_{n=1}^\infty\alpha_ne^{-(2n-1)\pi y/2a}\sin {(2n-1)\pi x\over2a}\),  \(\alpha_n=\dst{2\over a}\int_0^a f(x)\sin {(2n-1)\pi x\over2a}\,dx\)

   \(u(x,y)=\dst {32\over\pi}\sum_{n=1}^\infty{1\over (2n-1)^3}e^{-(2n-1)y/2}\sin{(2n-1) x\over2}\).

\(u(x,y)=-\dst{a\over \pi}\sum_{n=1}^\infty{\alpha_n\over n} e^{-n\pi y/a}\sin {n\pi x\over a}\),   \(\alpha_n=\dst{2\over a}\int_0^a f(x)\sin {n\pi x\over a}\,dx\)

   \(u(x)=\dst4\sum_{n=1}^\infty{(1+(-1)^n2)\over n^4} e^{-ny}\sin nx\)

\(u(x,y)=-\dst{2a\over\pi}\sum_{n=1}^\infty{\alpha_n\over2n-1} e^{-(2n-1)\pi y/2a}\cos {(2n-1)\pi x\over2a}\),  \(\alpha_n=\dst{2\over a}\int_0^a f(x)\cos {(2n-1)\pi x\over2a}\,dx\)

   \(u(x,y)=\dst{5488\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ 1+{4(-1)^n\over(2n-1)\pi}\right]e^{-(2n-1)\pi y/14}\cos{(2n-1)\pi x\over14}\)

\(u(x,y)=\dst-{2a\over\pi}\sum_{n=1}^\infty{\alpha_n\over2n-1}e^{-(2n-1)\pi y/2a}\sin {(2n-1)\pi x\over2a}\),  \(\alpha_n=\dst{2\over a}\int_0^a f(x)\sin {(2n-1)\pi x\over2a}\,dx\)

   \(u(x,y)=-\dst{2000\over\pi^3}\sum_{n=1}^\infty{1\over(2n-1)^3}\left[ (-1)^n+{4\over(2n-1)\pi}\right]e^{-(2n-1)\pi y/10}\sin{(2n-1)\pi x\over10}\)

\(u(x,y)=\dst \sum_{n=1}^\infty{A_n\sinh n\pi(b-y)/a+B_n\sinh n\pi y/a\over \sinh n\pi b/a}\sin{n\pi x\over a}\\ \hspace*{6.25em}+\sum_{n=1}^\infty {C_n\sinh n\pi(a-x)/b+D_n\sinh n\pi x/b \over \sinh n\pi a/b}\sin{n\pi y\over b}\)

\(\dst u(x,y)=C+ {a\over\pi}\sum_{n=1}^\infty{B_n\cosh n\pi y/a-A_n\cosh n\pi(y-b)/a\over n\sinh n\pi b/a}\cos{n\pi x\over a}\\ \hspace*{6.5em}+{b\over\pi}\sum_{n=1}^\infty{D_n\cosh n\pi x/b-C_n\cosh n\pi(x-a)/b\over n\sinh n\pi a/b}\cos{n\pi y\over b}\)

\(u(r,\theta)=\dst\alpha_0\frac{\ln r/\rho}{\ln\rho_0/\rho}+ \sum_{n=1}^\infty \frac{r^n\rho^{-n}-\rho^nr^{-n}} {\rho_0^n\rho^{-n}-\rho^n\rho_0^{-n}} (\alpha_n\cos n\theta+\beta_n\sin n\theta)\) \(\dst\alpha_0=\frac{1}{2\pi}\int_{-\pi}^\pi f(\theta)\,d\theta\),

   and \(\dst\alpha_n=\frac{1}{\pi}\int_{-\pi}^\pi f(\theta)\cos n\theta\,d\theta\),  \(\dst\beta_n=\frac{1}{\pi}\int_{-\pi}^\pi f(\theta)\sin n\theta\,d\theta\),  \(n=1\), \(2\), \(3\), …

\(\dst u(r,\theta)=\sum_{n=1}^\infty\alpha_n \dst\frac{\rho_0^{-n\pi/\gamma}r^{n\pi/\gamma}-\rho_0^{n\pi/\gamma} r^{-n\pi/\gamma}}{\rho_0^{-n\pi/\gamma}\rho^{n\pi/\gamma}-\rho_0^{n\pi/\gamma} \rho^{-n\pi/\gamma}} \sin\frac{n\pi\theta}{\gamma}\)  

    \(\alpha_n=\dst\frac{1}{\gamma}\int_0^\gamma f(\theta)\sin\frac{n\pi\theta}{\gamma}\,d\theta\), \(n=1\), \(2\), \(3\),…

\(\dst u(r,\theta)= \rho\alpha_0\dst\ln\frac{r}{\rho_0}+\frac{\rho\gamma}{\pi}\sum_{n=1}^\infty\frac{\alpha_n}{n} \dst\frac{\rho_0^{-n\pi/\gamma}r^{n\pi/\gamma}-\rho_0^{n\pi/\gamma} r^{-n\pi/\gamma}} {\rho_0^{-n\pi/\gamma}\rho^{n\pi/\gamma}+\rho_0^{n\pi/\gamma} \rho^{-n\pi/\gamma}} \dst\cos\frac {n\pi\theta}{\gamma}\)

    \(\alpha_0=\dst\frac{1}{\gamma}\int_0^\gamma f(\theta)\,d\theta\),  \(\alpha_n=\dst\frac{2}{\gamma}\int_0^\gamma f(\theta)\cos\frac{n\pi\theta}{\gamma}\,d\theta\), \(n=1\), \(2\), \(3\),…

\(u(r,\theta)=\dst\sum_{n=1}^\infty\alpha_n \dst\frac{r^{(2n-1)\pi/2\gamma}}{\rho^{(2n-1)\pi/2\gamma}} \cos\frac {(2n-1)\pi\theta}{2\gamma}\)  

    \(\alpha_n=\dst\frac{2}{\gamma}\int_0^\gamma f(\theta)\cos\frac{(2n-1)\pi\theta}{2\gamma}\,d\theta\),  \(n=1\), \(2\), \(3\),…

\(u(r,\theta)=\dst\frac{2\gamma\rho_0}{\pi}\sum_{n=1}^\infty \frac{\alpha_n}{2n-1} \frac{\rho^{-(2n-1)\pi/2\gamma}r^{(2n-1)\pi/2\gamma}+ \rho^{(2n-1)\pi/2\gamma}r^{-(2n-1)\pi/2\gamma}} {\rho^{-(2n-1)\pi/2\gamma}\rho_0^{(2n-1)\pi/2\gamma}- \rho^{(2n-1)\pi/2\gamma}\rho_0^{-(2n-1)\pi/2\gamma}} \dst\sin\frac {(2n-1)\pi\theta}{2\gamma}\),

    \(\alpha_n=\dst\frac{2}{\gamma}\int_0^\gamma g(\theta)\sin\frac{(2n-1)\pi\theta}{2\gamma}\,d\theta\),  \(n=1\), \(2\), \(3\),…

\(\dst{u(r,\theta)=\alpha_0+\sum_{n=1}^\infty\alpha_n \dst\frac{r^{n\pi/\gamma}}{\rho^{n\pi/\gamma}}\cos\frac{n\pi\theta}{\gamma}}\)   \(\alpha_0=\dst\frac{1}{\gamma}\int_0^\gamma f(\theta)\,d\theta\), 

    \(\alpha_n=\dst\frac{2}{\gamma}\int_0^\gamma f(\theta)\cos\frac{n\pi\theta}{\gamma}\,d\theta\),  \(n=1\), \(2\), \(3\),…

\(\dst{ v_n(r,\theta)=\frac{r^n}{n\rho^{n-1}}(\alpha_n\cos n\theta+\sin n\theta)}\)

    \(u(r,\theta)=\dst{ c+\sum_{n=1}^\infty\frac{r^n}{n\rho^{n-1}}(\alpha_n\cos n\theta+\beta_n\sin n\theta)}\) \(\alpha_n=\dst{\frac{1}{\pi}\int_{-\pi}^\pi f(\theta)\cos n\theta\,d\theta}\), 

    \(\dst{\beta_n=\frac{1}{\pi}\int_{-\pi}^\pi f(\theta)\sin n\theta\,d\theta}\), \(n=1\), \(2\), \(3\),…