${\mathrm{pde}}_1≔\frac{\partial }{\partial t}u\left(x\,y\,t\right)=\frac{1}{10}\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\,t\right)+\frac{{\partial }^2}{\partial y^2}u\left(x\,y\,t\right)\right)\:$

${\mathrm{iv}}_1≔u\left(0\,y\,t\right)=0,u\left(1\,y\,t\right)=0,u\left(x\,0\,t\right)=0,u\left(x\,1\,t\right)=0,u\left(x\,y\,0\right)=x\left(1-x\right)\left(1-y\right)y\:$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_1\,{\mathrm{iv}}_1\right]\,u\left(x\,y\,t\right)\right)$

$u\left(x\,y\,t\right)=\sum _{\mathrm{n1}=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}-\frac{16\left(-{\left(\mathrm{-1}\right)}^{\mathrm{n1}+n}+{\left(\mathrm{-1}\right)}^{\mathrm{n1}}+{\left(\mathrm{-1}\right)}^n-1\right)\sin \left(n\mathrm{\pi }x\right)\sin \left(\mathrm{n1}\mathrm{\pi }y\right){\ⅇ}^{-\frac{t{\mathrm{\pi }}^2\left(n^2+{\mathrm{n1}}^2\right)}{10}}}{n^3{\mathrm{\pi }}^6{\mathrm{n1}}^3}$

$\mathrm{map}\left(\mathrm{plot3d}\,\left[\mathrm{subs}\left(t=0\,\mathrm{infinity}=100\,\mathrm{rhs}\left(\right)\right)\,x\left(1-x\right)\left(1-y\right)y\right]\,x=0..1\,y=0..1\right)$

$\left[\mathrm{PLOT3D}\left(\mathrm{...}\right)\,\mathrm{PLOT3D}\left(\mathrm{...}\right)\right]$

${\mathrm{pde}}_2≔\frac{\partial }{\partial t}u\left(x\,y\,t\right)=\frac{1}{10}\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\,t\right)\right)+\frac{1}{10}\left(\frac{{\partial }^2}{\partial y^2}u\left(x\,y\,t\right)\right)-\frac{1}{5}u\left(x\,y\,t\right)\:$

${\mathrm{iv}}_2≔{\mathrm{D}}_1\left(u\right)\left(0\,y\,t\right)=0,u\left(1\,y\,t\right)=0,u\left(x\,0\,t\right)=0,{\mathrm{D}}_2\left(u\right)\left(x\,1\,t\right)=0,u\left(x\,y\,0\right)=\left(-x^2+1\right)\left(1-\frac{1}{2}y\right)y\:$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_2\,{\mathrm{iv}}_2\right]\,u\left(x\,y\,t\right)\right)$

$u\left(x\,y\,t\right)=\sum _{\mathrm{n1}=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{512{\left(\mathrm{-1}\right)}^n\cos \left(\frac{\left(1+2n\right)\mathrm{\pi }x}{2}\right)\sin \left(\frac{\left(1+2\mathrm{n1}\right)\mathrm{\pi }y}{2}\right){\ⅇ}^{-\frac{\left(2+\left(n^2+{\mathrm{n1}}^2+n+\mathrm{n1}+\frac{1}{2}\right){\mathrm{\pi }}^2\right)t}{10}}}{{\left(1+2n\right)}^3{\mathrm{\pi }}^6{\left(1+2\mathrm{n1}\right)}^3}$

$\mathrm{map}\left(\mathrm{plot3d}\,\left[\mathrm{subs}\left(t=0\,\mathrm{infinity}=100\,\mathrm{rhs}\left(\right)\right)\,\left(-x^2+1\right)\left(1-\frac{1}{2}y\right)y\right]\,x=0..1\,y=0..1\right)$

$\left[\mathrm{PLOT3D}\left(\mathrm{...}\right)\,\mathrm{PLOT3D}\left(\mathrm{...}\right)\right]$

${\mathrm{pde}}_3≔\frac{{\partial }^2}{\partial t^2}u\left(x\,y\,t\right)=\frac{1}{4}\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\,t\right)\right)+\frac{1}{4}\left(\frac{{\partial }^2}{\partial y^2}u\left(x\,y\,t\right)\right)\:$

${\mathrm{iv}}_3≔{\mathrm{D}}_1\left(u\right)\left(0\,y\,t\right)=0,{\mathrm{D}}_1\left(u\right)\left(\mathrm{\pi }\,y\,t\right)=0,u\left(x\,0\,t\right)=0,u\left(x\,\mathrm{\pi }\,t\right)=0,{\mathrm{D}}_3\left(u\right)\left(x\,y\,0\right)=0,u\left(x\,y\,0\right)=xy\left(\mathrm{\pi }-y\right)\:$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_3\,{\mathrm{iv}}_3\right]\,u\left(x\,y\,t\right)\right)$

$u\left(x\,y\,t\right)=\left(\sum _{n=1}^{\mathrm{\infty }}-\frac{2\left({\left(\mathrm{-1}\right)}^n-1\right)\sin \left(ny\right)\cos \left(\frac{nt}{2}\right)}{n^3}\right)+\left(\sum _{n=1}^{\mathrm{\infty }}\sum _{\mathrm{n1}=1}^{\mathrm{\infty }}\frac{8\left(-{\left(\mathrm{-1}\right)}^{\mathrm{n1}+n}+{\left(\mathrm{-1}\right)}^{\mathrm{n1}}+{\left(\mathrm{-1}\right)}^n-1\right)\cos \left(\mathrm{n1}x\right)\sin \left(ny\right)\cos \left(\frac{\sqrt{n^2+{\mathrm{n1}}^2}t}{2}\right)}{{\mathrm{\pi }}^2{\mathrm{n1}}^2{n}^3}\right)$

$\mathrm{map}\left(\mathrm{plot3d}\,\left[\mathrm{subs}\left(t=0\,\mathrm{infinity}=100\,\mathrm{rhs}\left(\right)\right)\,xy\left(\mathrm{\pi }-y\right)\right]\,x=0..\mathrm{\pi }\,y=0..\mathrm{\pi }\right)$

$\left[\mathrm{PLOT3D}\left(\mathrm{...}\right)\,\mathrm{PLOT3D}\left(\mathrm{...}\right)\right]$

${\mathrm{pde}}_4≔\frac{{\partial }^2}{\partial t^2}u\left(x\,y\,t\right)=\frac{1}{4}\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\,t\right)+\frac{{\partial }^2}{\partial y^2}u\left(x\,y\,t\right)\right)-\frac{1}{10}\left(\frac{\partial }{\partial t}u\left(x\,y\,t\right)\right)\:$

${\mathrm{iv}}_4≔u\left(0\,y\,t\right)=0,{\mathrm{D}}_1\left(u\right)\left(1\,y\,t\right)=0,u\left(x\,0\,t\right)=0,{\mathrm{D}}_2\left(u\right)\left(x\,1\,t\right)=0,u\left(x\,y\,0\right)=0,{\mathrm{D}}_3\left(u\right)\left(x\,y\,0\right)=x\left(1-\frac{1}{2}x\right)\left(1-\frac{1}{2}y\right)y\:$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_4\,{\mathrm{iv}}_4\right]\,u\left(x\,y\,t\right)\right)$

$u\left(x\,y\,t\right)=\sum _{\mathrm{n1}=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{5120\sin \left(\frac{\left(1+2n\right)\mathrm{\pi }x}{2}\right)\sin \left(\frac{\left(1+2\mathrm{n1}\right)\mathrm{\pi }y}{2}\right){\ⅇ}^{-\frac{t}{20}}\sin \left(\frac{t\sqrt{-1+\left(100n^2+100{\mathrm{n1}}^2+100n+100\mathrm{n1}+50\right){\mathrm{\pi }}^2}}{20}\right)}{\sqrt{-1+\left(100n^2+100{\mathrm{n1}}^2+100n+100\mathrm{n1}+50\right){\mathrm{\pi }}^2}{\mathrm{\pi }}^6{\left(1+2\mathrm{n1}\right)}^3{\left(1+2n\right)}^3}$

$\mathrm{map}\left(\mathrm{plot3d}\,\left[\mathrm{subs}\left(t=0\,\mathrm{infinity}=100\,\frac{\partial }{\partial t}\mathrm{rhs}\left(\right)\right)\,x\left(1-\frac{1}{2}x\right)\left(1-\frac{1}{2}y\right)y\right]\,x=0..1\,y=0..1\right)$

$\left[\mathrm{PLOT3D}\left(\mathrm{...}\right)\,\mathrm{PLOT3D}\left(\mathrm{...}\right)\right]$

${\mathrm{pde}}_5≔\frac{\partial }{\partial t}u\left(x\,t\right)=k\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right)\:$

${\mathrm{iv}}_5≔u\left(x\,0\right)=f\left(x\right),u\left(-l\,t\right)=u\left(l\,t\right),{\mathrm{D}}_1\left(u\right)\left(-l\,t\right)={\mathrm{D}}_1\left(u\right)\left(l\,t\right)\:$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_5\&comma;{\mathrm{iv}}_5\right]\&comma;u\left(x\&comma;t\right)\right)\mathbf{assuming}0<l$

$u\left(x\&comma;t\right)=\frac{2\left(\sum _{n=1}^{\mathrm{\infty }}\frac{\left(\left({\int }_{-l}^{l}f\left(x\right)\sin \left(\frac{n\mathrm{\pi }x}{l}\right)\&DifferentialD;x\right)\sin \left(\frac{n\mathrm{\pi }x}{l}\right)+\left({\int }_{-l}^{l}f\left(x\right)\cos \left(\frac{n\mathrm{\pi }x}{l}\right)\&DifferentialD;x\right)\cos \left(\frac{n\mathrm{\pi }x}{l}\right)\right){\&ExponentialE;}^{-\frac{k{\mathrm{\pi }}^2{n}^{2}t}{l^2}}}{l}\right)l+{\int }_{-l}^{l}f\left(x\right)\&DifferentialD;x}{2l}$

${\mathrm{pde}}_6≔\frac{\frac{\partial }{\partial r}\left(r\left(\frac{\partial }{\partial r}u\left(r\&comma;\mathrm{\theta }\right)\right)\right)}{r}+\frac{\frac{{\partial }^2}{\partial {\mathrm{\theta }}^2}u\left(r\&comma;\mathrm{\theta }\right)}{r^2}=0\&colon;$

${\mathrm{iv}}_6≔u\left(a\&comma;\mathrm{\theta }\right)=f\left(\mathrm{\theta }\right),u\left(r\&comma;-\mathrm{\pi }\right)=u\left(r\&comma;\mathrm{\pi }\right),{\mathrm{D}}_2\left(u\right)\left(r\&comma;-\mathrm{\pi }\right)={\mathrm{D}}_2\left(u\right)\left(r\&comma;\mathrm{\pi }\right)\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_6\&comma;{\mathrm{iv}}_6\right]\&comma;u\left(r\&comma;\mathrm{\theta }\right)\&comma;\mathrm{HINT}=\mathrm{boundedseries}\right)\mathbf{assuming}0<a$

$u\left(r\&comma;\mathrm{\theta }\right)=\frac{2\left(\sum _{n=1}^{\mathrm{\infty }}\frac{r^n\left(\left({\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}f\left(\mathrm{\theta }\right)\sin \left(n\mathrm{\theta }\right)\&DifferentialD;\mathrm{\theta }\right)\sin \left(n\mathrm{\theta }\right)+\left({\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}f\left(\mathrm{\theta }\right)\cos \left(n\mathrm{\theta }\right)\&DifferentialD;\mathrm{\theta }\right)\cos \left(n\mathrm{\theta }\right)\right)a^{-n}}{\mathrm{\pi }}\right)\mathrm{\pi }+{\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}f\left(\mathrm{\theta }\right)\&DifferentialD;\mathrm{\theta }}{2\mathrm{\pi }}$

${\mathrm{pde}}_7≔\frac{\partial }{\partial t}u\left(x\&comma;t\right)=\frac{{\partial }^2}{\partial x^2}u\left(x\&comma;t\right)\&colon;$

${\mathrm{iv}}_7≔u\left(x\&comma;0\right)=f\left(x\right),u\left(-1\&comma;t\right)=0,u\left(1\&comma;t\right)=0\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_7\&comma;{\mathrm{iv}}_7\right]\&comma;u\left(x\&comma;t\right)\right)$

$u\left(x\&comma;t\right)=\sum _{n=1}^{\mathrm{\infty }}\left(\left({\int }_{\mathrm{-1}}^{1}f\left(x\right)\sin \left(n\mathrm{\pi }x\right)\&DifferentialD;x\right)\sin \left(n\mathrm{\pi }x\right){\&ExponentialE;}^{\frac{t{\mathrm{\pi }}^2{\left(2n-1\right)}^2}{4}}+\left({\int }_{\mathrm{-1}}^{1}f\left(x\right)\cos \left(\frac{\mathrm{\pi }x\left(2n-1\right)}{2}\right)\&DifferentialD;x\right)\cos \left(\frac{\mathrm{\pi }x\left(2n-1\right)}{2}\right){\&ExponentialE;}^{t{\mathrm{\pi }}^2{n}^2}\right){\&ExponentialE;}^{-\frac{t{\mathrm{\pi }}^2\left(8n^2-4n+1\right)}{4}}$

${\mathrm{pde}}_8≔\frac{\partial }{\partial t}u\left(x\&comma;t\right)=\frac{{\partial }^2}{\partial x^2}u\left(x\&comma;t\right)\&colon;$

${\mathrm{iv}}_8≔u\left(0\&comma;t\right)=20,u\left(1\&comma;t\right)=50,u\left(x\&comma;0\right)=0\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_8\&comma;{\mathrm{iv}}_8\right]\&comma;u\left(x\&comma;t\right)\right)$

$u\left(x\&comma;t\right)=20+30x+\left(\sum _{n=1}^{\mathrm{\infty }}\frac{\left(-40+100{\left(\mathrm{-1}\right)}^n\right)\sin \left(n\mathrm{\pi }x\right){\&ExponentialE;}^{-t{\mathrm{\pi }}^2{n}^2}}{n\mathrm{\pi }}\right)$

${\mathrm{pde}}_9≔\frac{\partial }{\partial t}u\left(x\&comma;t\right)+k\left(\frac{{\partial }^2}{\partial x^2}u\left(x\&comma;t\right)\right)+k\&colon;$

${\mathrm{iv}}_9≔u\left(x\&comma;0\right)=f\left(x\right),u\left(0\&comma;t\right)=A,u\left(L\&comma;t\right)=B\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_9\&comma;{\mathrm{iv}}_9\right]\&comma;u\left(x\&comma;t\right)\right)$

$u\left(x\&comma;t\right)=\frac{2\left(\sum _{n=1}^{\mathrm{\infty }}\left(-\frac{\left({\int }_0^{L}2\left(-f\left(x\right)L+\frac{L^{2}x}{2}+\left(-\frac{x^2}{2}+A\right)L-x\left(A-B\right)\right)\sin \left(\frac{\mathrm{\pi }nx}{L}\right)\&DifferentialD;x\right)\sin \left(\frac{\mathrm{\pi }nx}{L}\right){\&ExponentialE;}^{\frac{k{\mathrm{\pi }}^2{n}^{2}t}{L^2}}}{L^2}\right)\right)L+L^{2}x+\left(-x^2+2A\right)L-2x\left(A-B\right)}{2L}$

${\mathrm{pde}}_{10}≔\frac{\partial }{\partial t}u\left(x\&comma;t\right)=k\left(\frac{{\partial }^2}{\partial x^2}u\left(x\&comma;t\right)\right)+f\left(x\&comma;t\right)\&colon;$

${\mathrm{iv}}_{10}≔u\left(0\&comma;t\right)=0,u\left(l\&comma;t\right)=0,u\left(x\&comma;0\right)=g\left(x\right)\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{10}\&comma;{\mathrm{iv}}_{10}\right]\&comma;u\left(x\&comma;t\right)\right)\mathbf{assuming} 0\le x \le l\&semi;$

$u\left(x\&comma;t\right)=\left(\sum _{\mathrm{n1}=1}^{\mathrm{\infty }}\frac{2\left({\int }_0^{l}g\left(x\right)\sin \left(\frac{\mathrm{n1}\mathrm{\pi }x}{l}\right)\&DifferentialD;x\right)\sin \left(\frac{\mathrm{n1}\mathrm{\pi }x}{l}\right){\&ExponentialE;}^{-\frac{k{\mathrm{\pi }}^2{\mathrm{n1}}^{2}t}{l^2}}}{l}\right)+{\int }_0^t\left(\sum _{n=1}^{\mathrm{\infty }}\frac{2\left({\int }_0^{l}f\left(x\&comma;\mathrm{\&tau;1}\right)\sin \left(\frac{n\mathrm{\pi }x}{l}\right)\&DifferentialD;x\right)\sin \left(\frac{n\mathrm{\pi }x}{l}\right){\&ExponentialE;}^{-\frac{k{\mathrm{\pi }}^2{n}^2\left(t-\mathrm{\&tau;1}\right)}{l^2}}}{l}\right)\&DifferentialD;\mathrm{\&tau;1}$

${\mathrm{pde}}_{11}≔\frac{\frac{\partial }{\partial r}u\left(r\&comma;\mathrm{\theta }\right)+r\left(\frac{{\partial }^2}{\partial r^2}u\left(r\&comma;\mathrm{\theta }\right)\right)}{r}+\frac{\frac{{\partial }^2}{\partial {\mathrm{\theta }}^2}u\left(r\&comma;\mathrm{\theta }\right)}{r^2}=0\&colon;$

${\mathrm{iv}}_{11}≔u\left(0\&comma;\mathrm{\theta }\right)=0,u\left(1\&comma;\mathrm{\theta }\right)=\mathrm{\theta }\left(\frac{1}{3}\mathrm{\pi }-\mathrm{\theta }\right),u\left(r\&comma;0\right)=0,u\left(r\&comma;\frac{1}{3}\mathrm{\pi }\right)=0\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{11}\&comma;{\mathrm{iv}}_{11}\right]\&comma;u\left(r\&comma;\mathrm{\theta }\right)\right)\mathbf{assuming} 0<r,0< \mathrm{\&theta;}<\frac{1}{3}\mathrm{\pi }$

$u\left(r\&comma;\mathrm{\theta }\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{\left(\left(-\mathrm{\_C5}\left(n\right)\mathrm{\pi }{n}^3-\frac{4{\left(\mathrm{-1}\right)}^n}{9}+\frac{4}{9}\right)r^{3n}+\mathrm{\_C5}\left(n\right)r^{-3n}\mathrm{\pi }{n}^3\right)\sin \left(3n\mathrm{\theta }\right)}{\mathrm{\pi }{n}^3}$

${\mathrm{pde}}_{12}≔\frac{\frac{\partial }{\partial r}u\left(r\&comma;\mathrm{\theta }\right)}{r}+\frac{{\partial }^2}{\partial r^2}u\left(r\&comma;\mathrm{\theta }\right)+\frac{\frac{{\partial }^2}{\partial {\mathrm{\theta }}^2}u\left(r\&comma;\mathrm{\theta }\right)}{r^2}=0\&colon;$

${\mathrm{iv}}_{12}≔u\left(1\&comma;\mathrm{\theta }\right)=f\left(\mathrm{\theta }\right),u\left(r\&comma;0\right)=0,u\left(r\&comma;\frac{1}{3}\mathrm{\pi }\right)=0\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{12}\&comma;{\mathrm{iv}}_{12}\right]\right)\mathbf{assuming}0<r\mathit{}\le 1\&comma;0< \mathrm{\&theta;}\le \frac{1}{3}\mathrm{\pi }$

$u\left(r\&comma;\mathrm{\theta }\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{\sin \left(3n\mathrm{\theta }\right)\left(6r^{3n}\left({\int }_0^{\frac{\mathrm{\pi }}{3}}f\left(\mathrm{\theta }\right)\sin \left(3n\mathrm{\theta }\right)\&DifferentialD;\mathrm{\theta }\right)+\mathrm{\pi }\mathrm{\_C5}\left(n\right)\left(r^{-3n}-r^{3n}\right)\right)}{\mathrm{\pi }}$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{12}\&comma;{\mathrm{iv}}_{12}\right]\&comma;\mathrm{HINT}=\mathrm{boundedseries}\right)\mathbf{assuming} 0<r\le 1\&comma;0<\mathrm{\&theta;}\le \frac{1}{3}\mathrm{\pi }$

$u\left(r\&comma;\mathrm{\theta }\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{6\left({\int }_0^{\frac{\mathrm{\pi }}{3}}f\left(\mathrm{\theta }\right)\sin \left(3n\mathrm{\theta }\right)\&DifferentialD;\mathrm{\theta }\right)r^{3n}\sin \left(3n\mathrm{\theta }\right)}{\mathrm{\pi }}$

${\mathrm{pde}}_{13}≔\frac{\frac{\partial }{\partial r}u\left(r\&comma;\mathrm{\theta }\right)+r\left(\frac{{\partial }^2}{\partial r^2}u\left(r\&comma;\mathrm{\theta }\right)\right)}{r}+\frac{\frac{{\partial }^2}{\partial {\mathrm{\theta }}^2}u\left(r\&comma;\mathrm{\theta }\right)}{r^2}=0\&colon;$

${\mathrm{iv}}_{13}≔u\left(1\&comma;\mathrm{\theta }\right)=\mathrm{\theta }\left(1-\frac{\frac{3}{2}\mathrm{\theta }}{\mathrm{\pi }}\right),u\left(r\&comma;0\right)=0,{\mathrm{D}}_2\left(u\right)\left(r\&comma;\frac{1}{3}\mathrm{\pi }\right)=0\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{13}\&comma;{\mathrm{iv}}_{13}\right]\&comma;u\left(r\&comma;\mathrm{\theta }\right)\&comma;\mathrm{HINT}=\mathrm{boundedseries}\right)$

$u\left(r\&comma;\mathrm{\theta }\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{16r^{\frac{3}{2}+3n}\sin \left(3n\mathrm{\theta }+\frac{3}{2}\mathrm{\theta }\right)}{3{\mathrm{\pi }}^2{\left(1+2n\right)}^3}$

${\mathrm{pde}}_{14}≔\frac{{\partial }^2}{\partial x^2}u\left(x\&comma;y\right)+\frac{{\partial }^2}{\partial y^2}u\left(x\&comma;y\right)=0\&colon;$

${\mathrm{iv}}_{14}≔u\left(0\&comma;y\right)=-y^2+y,{\mathrm{D}}_1\left(u\right)\left(1\&comma;y\right)=0,u\left(x\&comma;0\right)=0,u\left(x\&comma;1\right)=x\left(1-\frac{1}{2}x\right)\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{14}\&comma;{\mathrm{iv}}_{14}\right]\&comma;u\left(x\&comma;y\right)\right)\mathbf{assuming} 0< x\le 1$

$u\left(x\&comma;y\right)=\left(\sum _{n=1}^{\mathrm{\infty }}-\frac{4\left({\left(\mathrm{-1}\right)}^n-1\right)\sin \left(\mathrm{\pi }yn\right)\left({\&ExponentialE;}^{-\mathrm{\pi }n\left(x-2\right)}+{\&ExponentialE;}^{n\mathrm{\pi }x}\right)}{n^3{\mathrm{\pi }}^3\left({\&ExponentialE;}^{2n\mathrm{\pi }}+1\right)}\right)+\left(\sum _{\mathrm{n1}=0}^{\mathrm{\infty }}\frac{16\sin \left(\frac{\mathrm{\pi }\left(1+2\mathrm{n1}\right)x}{2}\right)\left({\&ExponentialE;}^{\left(1+2\mathrm{n1}\right)\mathrm{\pi }y}-1\right){\&ExponentialE;}^{-\frac{\left(1+2\mathrm{n1}\right)\mathrm{\pi }\left(y-1\right)}{2}}}{\left({\&ExponentialE;}^{\mathrm{\pi }\left(1+2\mathrm{n1}\right)}-1\right){\left(1+2\mathrm{n1}\right)}^3{\mathrm{\pi }}^3}\right)$

${\mathrm{pde}}_{15}≔\frac{\partial }{\partial x}u\left(x\&comma;y\right)+u\left(x\&comma;y\right)\left(\frac{\partial }{\partial y}u\left(x\&comma;y\right)\right)=0\&colon;$

${\mathrm{iv}}_{15}≔u\left(x\&comma;0\right)=\frac{1}{x+1}\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{15}\&comma;{\mathrm{iv}}_{15}\right]\&comma;u\left(x\&comma;y\right)\right)$

$u\left(x\&comma;y\right)=\frac{y+1}{x+1}$

${\mathrm{pde}}_{16}≔100\left(\frac{{\partial }^2}{\partial x^2}u\left(x\&comma;t\right)\right)=\frac{{\partial }^2}{\partial t^2}u\left(x\&comma;t\right)\&colon;$

${\mathrm{iv}}_{16}≔u\left(0\&comma;t\right)=0,u\left(2\&comma;t\right)=0,u\left(x\&comma;0\right)=32\sin \left(\mathrm{\pi }x\right)+e^2\sin \left(3\mathrm{\pi }x\right)+25\sin \left(6\mathrm{\pi }x\right),{\mathrm{D}}_2\left(u\right)\left(x\&comma;0\right)=6\sin \left(2\mathrm{\pi }x\right)-16\sin \left(5\mathrm{\pi }x\left(\frac{1}{2}\right)\right)\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{16}\&comma;{\mathrm{iv}}_{16}\right]\&comma;u\left(x\&comma;t\right)\right)\mathbf{assuming} 0< x\le 2$

$u\left(x\&comma;t\right)=\frac{50\sin \left(3\mathrm{\pi }x\right)e^2\cos \left(30\mathrm{\pi }t\right)\mathrm{\pi }+1600\sin \left(\mathrm{\pi }x\right)\cos \left(10\mathrm{\pi }t\right)\mathrm{\pi }+1250\sin \left(6\mathrm{\pi }x\right)\cos \left(60\mathrm{\pi }t\right)\mathrm{\pi }-32\sin \left(\frac{5\mathrm{\pi }x}{2}\right)\sin \left(25\mathrm{\pi }t\right)+15\sin \left(2\mathrm{\pi }x\right)\sin \left(20\mathrm{\pi }t\right)}{50\mathrm{\pi }}$

${\mathrm{pde}}_{17}≔4\left(\frac{{\partial }^2}{\partial x^2}u\left(x\&comma;t\right)\right)=\frac{{\partial }^2}{\partial t^2}u\left(x\&comma;t\right)\&colon;$

${\mathrm{iv}}_{17}≔u\left(0\&comma;t\right)=0,u\left(\mathrm{\pi }\&comma;t\right)=0,u\left(x\&comma;0\right)=0,{\mathrm{D}}_2\left(u\right)\left(x\&comma;0\right)=6\&colon;$

$\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{17}\&comma;{\mathrm{iv}}_{17}\right]\&comma;u\left(x\&comma;t\right)\right)\mathbf{assuming} 0< x\le \mathrm{\&pi;}$

$u\left(x\&comma;t\right)=\sum _{n=1}^{\mathrm{\infty }}-\frac{6\left({\left(\mathrm{-1}\right)}^n-1\right)\sin \left(nx\right)\sin \left(2nt\right)}{\mathrm{\pi }{n}^2}$