### 1.4 Result lookup table on one web page

Miscellaneous PDE’s

 # PDE description Mathematica Maple hand solved? Animated? result time result time 1 Transport equation $$u_t+ u_x = 0$$ ✓ 0.024 ✓ 0.089 Yes 2 Transport equation $$u_t-3 u_x = 0$$ IC $$u(0,x)=e^{-x^2}$$. Peter Olver textbook, 2.2.2 (a) ✓ 0.004 ✓ 0.04 Yes Yes 3 Transport equation $$u_t+2 u_x = 0$$ IC $$u(-1,x)=\frac {x}{1+x^2}$$. Peter Olver textbook, 2.2.2 (b) ✓ 0.003 ✓ 0.017 Yes Yes 4 Transport equation $$u_t+u_x+\frac {1}{2}u = 0$$ IC $$u(0,x)=\arctan (x)$$. Peter Olver textbook, 2.2.2 (c) ✓ 0.003 ✓ 0.055 Yes Yes 5 Transport equation $$u_t-4u_x+u = 0$$ IC $$u(0,x)=\frac {1}{1+x^2}$$. Peter Olver textbook, 2.2.2 (d) ✓ 0.003 ✓ 0.016 Yes Yes 6 Transport equation $$u_t+2 u_x= \sin x$$ IC $$u(0,x)=\sin x$$. Peter Olver textbook, 2.2.5 ✓ 0.049 ✓ 0.049 Yes Yes 7 Transport equation $$u_t+\frac {1}{1+x^2} u_x= 0$$ IC $$u(x,0)=\frac {1}{1+(3+x)^2}$$. Peter Olver textbook, page 27 ✓ 0.014 ✓ 0.073 Yes Yes 8 Transport equation $$u_t-x u_x= 0$$ IC $$u(x,0)=\frac {1}{1+x^2}$$. Peter Olver textbook, problem 2.2.17 ✓ 0.004 ✓ 0.045 Yes Yes 9 Transport equation $$u_t+(1-2 t) u_x= 0$$ IC $$u(x,0)=\frac {1}{1+x^2}$$. Peter Olver textbook, problem 2.2.29 ✓ 0.013 ✓ 0.055 Yes Yes 10 Transport equation $$u_t+\frac {1}{x^2+4} u_x= 0$$ IC $$u(x,0)=e^{x^3+12 x}$$ ✓ 0.011 ✓ 0.065 Yes Yes 11 $$3 u_x + 5 u_y = x$$ ✓ 0.004 ✓ 0.007 Yes 12 $$x u_y + y u_x = -4 x y u$$ and $$u(x,0)=e^{-x^2}$$ ✓ 0.01 ✓ 0.049 Yes 13 $$u_t + u_x = 0$$ and $$u(x,0)=\sin x$$ and $$u(0,t)=0$$ ✓ 0.052 ✓ 0.371 Yes 14 $$u_t+ c u_x = 0$$ and $$u(x,0)=e^{-x^2}$$ ✓ 0.003 ✓ 0.019 Yes 15 (Haberman 12.2.2) $$\omega _t -3 \omega _x = 0$$ and $$\omega (x,0)=\cos x$$ ✓ 0.003 ✓ 0.016 Yes 16 (Haberman 12.2.4) $$\omega _t +c \omega _x = 0$$ and $$\omega (x,0)=f(x)$$ and $$\omega (0,t)=h(t)$$ ✗ 10.387 ✓Solution contains unresolved invlaplace calls 0.243 Yes 17 (Haberman 12.2.5 (a)) $$\omega _t +c \omega _x = e^{2 x}$$ and $$\omega (x,0)=f(x)$$ ✓ 0.027 ✓ 0.037 Yes 18 (Haberman 12.2.5 (d)) $$\omega _t +3 t \omega _x = \omega (x,t)$$ and $$\omega (x,0)=f(x)$$ ✓ 0.023 ✓ 0.05 Yes 19 $$2 u_x + 5 u_y = u^2(x,y) + 1$$ ✓ 0.083 ✓ 0.03 Yes 20 Clairaut equation $$x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0$$ ✓ 0.012 ✓ 0.196 Yes 21 Clairaut equation. $$x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0$$ with $$u(x,0)= \frac {1}{2} (1-x^2)$$ ✓ 0.009 ✓ 0.611 22 Clairaut equation. $$u = x u_x+ y u_y + \sin ( u_x + u_y )$$ ✓ 0.016 ✓ 0.013 23 Recover a function from its gradient vector ✓ 0.021 ✓ 0.059 24 $$x f_y - f_x = \frac {g(x)}{h(y)} f^2$$ ✓ 0.038 ✓ 0.036 Yes 25 $$f_x + (f_y)^2 = f(x,y,z)+z$$ ✓ 0.047 ✓ 0.464 26 $$x u_x+y u_y=u$$ (Example 3.5.1 in Lokenath Debnath) ✓ 0.007 ✓ 0.01 Yes 27 $$x u_x+y u_y=n u$$ Example 3.5.2 in Lokenath Debnath ✓ 0.007 ✓ 0.009 Yes 28 $$x^2 u_x+y^2 u_y=(x+y) u$$ Example 3.5.3 in Lokenath Debnath ✓ 0.082 ✓ 0.023 Yes 29 $$(y-z) u_x + (z-x) u_y + (x-y) u_z = 0$$ (Example 3.5.4 in Lokenath Debnath) ✗ (Timed out) 600. ✓ 1.899 Yes 30 $$u(x+y) u_x+u(x-y) u_y=x^2+y^2$$ (Example 3.5.5 in Lokenath Debnath) ✓ 0.361 ✓Maple does not accept this form of Cauchy data 0.154 Yes 31 $$u_x-u_y=1$$ with $$u(x,0)=x^2$$ Example 3.5.6 in Lokenath Debnath ✓ 0.003 ✓ 0.014 32 $$(y-x)u_x+(u-x)u_y=x-y$$ with $$u=0$$ on $$x y=1$$ Example 3.5.7 in Lokenath Debnath ✗Kernel error 120.019 ✗ (Timed out) 600. 33 $$y u_x+x u_y=u$$ with $$u(x,0)=x^3$$ and $$u(0,y)=y^3$$ Example 3.5.8 in Lokenath Debnath ✗ 1.297 ✗ 0.429 34 $$x u_x+y u_y=x e^{-u}$$ with $$u=0$$ on $$y=x^2$$ Example 3.5.10 in Lokenath Debnath ✓ 0.1 ✓Maple does not accept this form of Cauchy data 0.057 35 $$u_t+u u_x=x$$ with $$u(x,0)=f(x)$$ Example 3.5.11 in Lokenath Debnath. ✗Kernel error 0.645 ✓ 0.184 36 $$u_x=0$$ Problem 3.3(a) Lokenath Debnath ✓ 0.003 ✓ 0.002 37 $$a u_x+b u_y=0$$ Problem 3.3(b) Lokenath Debnath ✓ 0.004 ✓ 0.007 38 $$u_x+y u_y=0$$ Problem 3.3(c) Lokenath Debnath ✓ 0.013 ✓ 0.009 39 $$(1+x^2) u_x+ u_y=0$$ Problem 3.3(d) Lokenath Debnath ✓ 0.006 ✓ 0.009 40 $$2 x y u_x+(x^2+y^2)u_y=0$$ Problem 3.3(e) Lokenath Debnath ✓ 0.102 ✓ 0.023 41 $$(y+u) u_x+y u_y=x-y$$ Problem 3.3(f) Lokenath Debnath ✗ 4.092 ✓ 0.319 42 $$y^2 u_x- x y u_y=x(u-2 y)$$ Problem 3.3(g) Lokenath Debnath ✓ 0.029 ✓ 0.038 43 $$y u_y - x u_x = 1$$ Problem 3.3(h) Lokenath Debnath ✓ 0.007 ✓ 0.009 44 $$u_x+2 x y^2 u_y=0$$ Problem 3.4 Lokenath Debnath ✓ 0.074 ✓ 0.01 45 $$3 u_x+2 u_y=0$$ with $$u(x,0)=\sin x$$. Problem 3.5(a) Lokenath Debnath ✓ 0.003 ✓ 0.015 46 $$y u_x+x u_y=0$$ with $$u(0,y)=e^{-y^2}$$. Problem 3.5(b) Lokenath Debnath ✓ 0.01 ✓ 0.024 47 $$x u_x+y u_y=2 x y$$ with $$u=2$$ on $$y=x^2$$. Problem 3.5(c) Lokenath Debnath ✓ 0.006 ✓Maple does not accept this form of Cauchy data as I.C. 0.008 48 $$u_x+x u_y=0$$ with $$u(0,y)=\sin y$$. Problem 3.5(d) Lokenath Debnath ✓ 0.003 ✓ 0.149 49 $$y u_x+x u_y=x y$$ with $$u(0,y)=e^{-y^2},u(x,0)=e^{-x^2}$$. Problem 3.5(e) Lokenath Debnath ✗ 1.138 ✗ 0.371 50 $$u_x+x u_y=(y-\frac {1}{2}x^2)^2$$ with $$u(0,y)=e^{y}$$. Problem 3.5(f) Lokenath Debnath ✓ 0.005 ✓ 0.075 51 $$x u_x+y u_y=u+1$$ with $$u=x^2$$ on $$y=x^2$$ Problem 3.5(g) Lokenath Debnath ✓ 0.007 ✓Maple does not accept this form of Cauchy data as I.C. 0.01 52 $$u u_x - u u_y= u^2 + (x+y)^2$$ with $$u(x,0)=1$$ Problem 3.5(h) Lokenath Debnath ✓ 0.04 ✓ 0.057 53 $$x u_x+(x+y)u_y=u+1$$ with $$u(x,0)=x^2$$ Problem 3.5(i) Lokenath Debnath ✓ 0.011 ✓ 0.039 54 $$x u_x+y u_y+z u_z=0$$ Problem 3.8(a) .Lokenath Debnath ✓ 0.01 ✓ 0.011 55 $$x^2 u_x+y^2 u_y+z(x+y)u_z=0$$ Problem 3.8(b) Lokenath Debnath ✓ 0.089 ✓ 0.016 56 $$x(y-z)u_x+y(z-x)u_y+z(x-y)u_z=0$$ Problem 3.8(c) Lokenath Debnath ✓ 0.027 ✓ 0.68 57 $$y z u_x - x z u_y+ x y (x^2+y^2) u_z=0$$ Problem 3.8(d) Lokenath Debnath ✓ 0.068 ✓ 0.034 58 $$x(y^2-z^2) u_x + y(z^2-y^2) u_y+ z (x^2-y^2) u_z=0$$ Problem 3.8(e) Lokenath Debnath ✗ 44.854 ✗ 0.368 59 $$u_x+x u_y=y$$ with $$u(0,y)=y^2$$ Problem 3.9(a) Lokenath Debnath ✓ 0.004 ✓ 0.016 60 $$u_x+x u_y=y$$ with $$u(1,y)=2 y$$ Problem 3.9(b) Lokenath Debnath ✓ 0.005 ✓ 0.007 61 $$(u_x+u_y)^2-u^2=0$$. Problem 3.10 Lokenath Debnath ✓ 0.007 ✓ 0.012 62 $$(y+u)u_x+y u_y=x-y$$ with $$u(x,1)=1+x$$. Problem 3.11 Lokenath Debnath ✗ 7.967 ✗ 0.708 63 $$2 x u_x+(x+1) u_y=y$$ with $$u(1,y)=2 y$$. Problem 3.14(d) Lokenath Debnath ✓ 0.009 ✓ 0.124 64 $$x u_x+y u_y=x^2+y^2$$ with $$u(x,1)=x^2$$. Problem 3.14(e) Lokenath Debnath ✓ 0.01 ✓ 0.033 65 $$y^2 u_x+(x y) u_y=x$$ with $$u(x,1)=x^2$$. Problem 3.14(f) Lokenath Debnath ✓ 0.016 ✓ 0.037 66 $$x u_x+y u_y=x y$$ with $$u=\frac {x^2}{2}$$ at $$y=x$$. Problem 3.14(g) Lokenath Debnath ✓Mathematica does not support this Cauchy data I.C. 0.007 ✓Maple does not support this Cauchy data I.C. 0.009 67 $$u_x+u u_y=1$$ with $$u(0,y)=a y$$. Problem 3.16(a) Lokenath Debnath ✓ 0.018 ✓ 0.02 68 $$(y+u)u_x+(x+u)u_y=x+y$$. Problem 3.17(a) Lokenath Debnath ✗ 30.116 ✓ 20.194 69 $$x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4$$. Problem 3.17(b) Lokenath Debnath ✓ 0.027 ✓ 0.037 70 $$(x+y) u_x + (x-y)u_y =0$$. Problem 3.17(c) Lokenath Debnath ✓ 0.028 ✓ 0.045 71 $$y u_x - x u_y = e^u$$ with $$u(0,y)=y^2-1$$ ✓ 0.12 ✓ 0.085 Yes 72 $$y u_x - x u_y = e^u$$ ✓ 0.054 ✓ 0.002 Yes 73 $$u_t + x u_x = 0$$ with $$u(x,0)=x^2$$. Math 5587 ✓ 0.005 ✓ 0.02 Yes 74 $$u_t + t u_x = 0$$ with $$u(x,0)=e^x$$ ✓ 0.009 ✓ 0.03 Yes 75 $$2 u_x + 3 u_y = 1$$ ✓ 0.004 ✓ 0.007 Yes 76 $$x u_t - t u_x = 0$$ ✓ 0.009 ✓ 0.013 Yes 77 $$u_t + u_x = 0$$ with $$u(x,1)=\frac {x}{1+x^2}$$ ✓ 0.003 ✓ 0.008 Yes 78 $$u_x u_y = 1$$ ✓ 0.002 ✓ 0.013 Yes 79 $$u_x u_y = u$$ with $$u(x,0)=0,u(0,y)=0$$ ✗ 1.074 ✓ 0.556 Yes 80 $$u_{xx} + u_{xt} - 6 u_{tt} = 0$$ ✓ 0.514 ✓ 0.082 Yes 81 $$u_{xx} - u_{xt} - 12 u_{tt} = 0$$ ✗ 1.554 ✓ 0.246 Yes 82 $$u_{xx} - 3 u_{xt} - 4 u_{tt} = 0$$ ✗ 0.467 ✓ 2.034 Yes 83 $$u_{tt} - 2 u_{xt} - 3 u_{xx} = 0$$ with $$u(0,x)=x^2, u_t(x,0)=e^x$$ ✗ 0.392 ✓ 1.312 84 Logan textbook, page 30 ✓ 0.377 ✓ 0.487 85 In a square, zero potential ✓ 0.259 ✓ 2.934 86 From Mathematica help pages ✓ 0.66 ✓ 0.68 87 From Mathematica help pages ✓ 0.005 ✗Trivial solution. Maple does not support $$\infty$$ in boundary conditions 3.897 88 David Griﬃths, page 47 ✓ 35.841 ✓ 1.52 89 David Griﬃths, page 47 ✓ 0.011 ✓ 0.782 90 In a square ✓ 0.011 ✓ 2.282 91 Beam PDE $$u_{tt} + u_{xxxx} = 0$$ ✓ 0.206 ✓ 0.229 92 Inviscid Burgers $$u_x + u u_y = 0$$ ✓Implicit solution 0.027 ✓ 0.019 Yes 93 Inviscid Burgers with I.C. $$u_x+ u u_y = 0$$ and $$u(x,0)=\frac {1}{x+1}$$ ✓ 0.008 ✓ 0.047 Yes 94 $$u_t+ u u_x = \mu u_{xx}$$ ✓ 0.026 ✓ 0.064 95 $$u_t + u u_x + \mu u_{xx}$$ with IC ✓ 21.53 ✗ 0.655 96 $$u_t + u u_x + \mu u_{xx}$$ IC as UnitBox ✓ 24.232 ✗ 0.788 97 classic Black Scholes model from ﬁnance, European call version ✓ 3.046 ✓ 0.931 98 Boundary value problem for the Black Scholes equation ✓ 2.766 ✓ 2.334 99 $$u_{xxx} + u_t -6 u u_x = 0$$ ✓ 0.029 ✓ 0.177 100 $$u_{xx} + y u_{yy} = 0$$ with $$u(x,0)=0,u_y(x,0)=x^2$$ ✓ 4.196 ✓ 3.212 101 $$u_{xx} + x u_{yy} = 0$$ ✗ 0.005 ✓ 1.635 102 $$x u_{xx} + u_{yy} = 0$$ ✗ 0.004 ✓ 1.7 103 $$u_{xx} + u_{yy} + \frac {\beta }{x} u_x = 0$$ ✗ 0.007 ✓ 0.082 104 $$u_{xx} - u_{yy} + \frac {\beta }{x} u_x = 0$$ ✗ 0.006 ✓ 0.26 105 $$u_{tt} - u_{xx} - \frac {2}{x} u_x = 0$$ with $$u(x,0)=0,u_t(x,0)=g(x)$$ ✗ 1.963 ✓ 3.855 106 $$u_{\theta \theta }+\frac {v^2}{1-\frac {v^2}{c^2}} u_{vv} + v u_v=0$$ ✗ 0.021 ✓ 0.889 107 Cauchy Riemann PDE with Prescribe the values of $$u$$ and $$v$$ on the $$x$$ axis ✓ 0.005 ✓ 0.13 108 Cauchy Riemann PDE With extra term on right side ✗ 0.001 ✓ 0.052 109 Hamilton-Jacobi type PDE ✗ 0.007 ✓ 0.157 110 $$u_t + u_{xxx} = 0$$ ✗ 0.002 ✓ 0.064 Yes 111 Bateman-Burgers $$u_t+u u_x = \nu u_{xx}$$ ✓ 0.02 ✓ 0.06 112 Benjamin Bona Mahony $$u_t+u_x + u u+x - u_{xxt} = 0$$ ✓ 0.031 ✓ 0.075 113 Benjamin Ono $$u_t+H u_{xx} +u u_x = 0$$ ✓ 0.02 ✓ 0.062 114 Born Infeld $$(1-u_t^2) u_{xx} + 2 u_x u_t u_{xt} - (1+ u_x^2) u_{tt}=0$$ ✓ 0.008 ✓ 0.135 115 Boussinesq $$u_{tt}-u_{xx}-u_{xxxx} - 3 (u^2)_{xx} = 0$$ ✓ 0.042 ✓ 0.086 116 Boussinesq type $$u_{tt}-u_{xx}-2 \alpha (u u_x)_x - \beta u_{xxtt} = 0$$ ✓ 0.041 ✓ 0.106 117 Buckmaster $$u_t = (u^4)_{xx} + (u^3)_x$$ ✗ 0.07 ✓Answer in terms of RootOf. 0.352 118 Camassa Holm $$u_t + 2 k u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx}+ u u_{xxx}$$ ✗ 0.174 ✓Answer in terms of RootOf. 1.52 119 Chaﬀee Infante $$u_t = u_{xx} + \lambda (u^3 - u) = 0$$ ✗ 0.093 ✓ 0.176 120 Clarke. $$\left ( \theta _t - \gamma e^\theta \right )_{tt} = \left ( \theta _t - e^\theta \right )_{xx}$$ ✗ 0.01 ✗ 0.025 121 Degasperis Procesi $$u_t - u_{xxt} + 4 u u_x = 3 u_x u_xx + u u_{xxx}$$ ✗ 0.169 ✓But still has unresolved ODE’s in solution 0.52 122 Dym equation $$u_t =u^3 u_{xxx}$$ ✗ 0.083 ✓has RootOf 0.375 123 Estevez Mansﬁeld Clarkson $$u_{tyyy} + \beta u_y u_{yt} + \beta u_{yy} u_t + u_{tt} = 0$$ ✓ 0.033 ✓ 0.121 124 Fisher’s $$u_t = u(1-u)+u_{xx}$$ ✓ 0.05 ✓ 0.165 125 Hunter Saxton $$\left ( u_t + u u_x) \right )_x = \frac {1}{2} (u_x)^2$$ ✗ 0.044 ✓with RootOf 0.107 126 Kadomtsev Petviashvili $$\left ( u_t + u u_x + \epsilon ^2 u_{xxx} \right )_x + \lambda u_{yy} = 0$$ ✓ 0.071 ✓ 0.125 127 Klein Gordon $$u_{xx}+u_{yy}+ \lambda u^p=0$$ ✗ 0.004 ✗ 0.017 128 Klein Gordon $$u_{xx}+u_{yy}+ u^2=0$$ ✗ 0.228 ✓ 0.346 129 Khokhlov Zabolotskaya $$u_{x t} - (u u_x)_x = u_{yy}$$ ✗ 0.057 ✓ 0.237 130 Korteweg de Vries (KdV) $$u_t + (u_x)^3+ 6 u u_x = 0$$ ✓ 0.029 ✓ 0.092 131 Lin Tsien $$2 u_{tx} + u_x u_{xx} - u_{yy} = 0$$ ✗ 0.072 ✓ 0.243 132 Liouville $$u_{xx} + u_{yy} +e^{\lambda u} = 0$$ ✗ 0.004 ✗ 0.017 133 Plateau $$(1+u_y^2)u_{xx} - 2 u_x u_y y_{xy} + (1+u_x^2) u_{yy} = 0$$ ✗ 0.032 ✓ 0.156 134 Rayleigh $$u_{tt} - u_{xx} = \epsilon (u_t - u_t^3)$$ ✗ 0.078 ✓Has RootOf 0.127 135 Sawada Kotera $$u_t + 45 u^2 u_x + 15 u_x u_{xx} + 15 u u_{xxx} + u_{xxxxx} = 0$$ ✓ 0.085 ✓ 0.187 136 Sine Gordon $$\phi _{tt} - \phi _{xx} + \sin \phi = 0$$ ✗ 0.007 ✗ 0.013 137 Sinh Gordon $$u_{xt} = \sinh u$$ ✗ 0.007 ✗ 0.014 138 Sinh Poisson $$u_{xx}+u_{yy} + \sinh u=0$$ ✗ 0.006 ✗ 0.011 139 Thomas equation $$u_{xy} + \alpha u_x + \beta u_y+ \nu u_x u_y =0$$ ✗ 0.06 ✓ 0.339 140 phi equation $$\phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0$$ ✓ 0.043 ✓ 0.472 141 $$S S_{xy} + S_x S_y = 1$$ ✗ 0.031 ✓ 0.023 142 $$u_{rr} + u_{\theta \theta } = 0$$ ✗ 0.012 ✓ 0.513 143 $$u_{xx} + y u_{yy} = 0$$ ✓ 3.551 ✓ 2.448 144 $$u_t + u_{xxx} = 0$$ ✗ 0.413 ✓ 5.663 145 $$u_{xy} = \sin (x) \sin (y)$$ ✗ 0.271 ✓ 0.426 146 $$w_t = w_{x_1 x_1} + w_{x_2 x_2} + w_{x_3 x_3}$$ ✗ 0.003 ✓ 0.621 147 Linear PDE, initial conditions at $$t=t_0$$ ✗ 0.003 ✓ 0.616 148 second order in time, Linear PDE, initial conditions at $$t=t_0$$ ✗ 0.003 ✓ 2.179 149 Einstein-Weiner $$u_t = -\beta u_x + D u_{xx}$$ ✗ 0.004 ✓ 0.194 150 Using integral transforms. $${x}^{2}{\frac {\partial ^{2}}{\partial {x}^{2}}}u \left ( x,y \right ) +x{\frac {\partial }{\partial x}}u \left ( x,y \right ) +{\frac {\partial ^{2}}{\partial {y}^{2}}}u \left ( x,y \right ) =0$$ ✗ 0.982 ✓ 2.129

Parabolic PDE’s (Diﬀusion)

 # PDE description Mathematica Maple hand solved? Animated? result time result time 151 Finite domain (bar), Both ends homogeneous BC General initial conditions ✓ 30.383 ✓ 1.527 Yes 152 Finite domain (bar), Both ends homogeneous BC Speciﬁc initial condition ✓ 8.837 ✓ 0.577 Yes Yes 153 Finite domain (bar), Both ends homogeneous BC IC $$u=x(1-x)$$ ✓ 14.949 ✓ 0.467 Yes Yes 154 Finite domain (bar), Both ends homogeneous BC Haberman 2.3.3 (a) ✓ 0.416 ✓ 2.448 Yes 155 Finite domain (bar), Both ends homogeneous BC IC hat function ✓ 18.465 ✓ 1.688 Yes Yes 156 Finite domain (bar), Both ends homogeneous BC Haberman 2.3.3 (b) ✓ 1.109 ✓ 2.759 Yes 157 Finite domain (bar), Both ends homogeneous BC Haberman 2.3.3 (c) ✓but $$n=3$$ should be special case 0.886 ✓handled $$n=3$$ case correctly. 5.885 Yes 158 Finite domain (bar), Both ends homogeneous BC Haberman 2.3.3 (d) ✓ 0.817 ✓ 2.198 Yes 159 Finite domain (bar), Both ends homogeneous BC domain from -1 to +1 ✗ 8.409 ✓ 0.643 Yes 160 Finite domain (bar), Both ends homogeneous BC domain from -1 to +1 ✗ 10.344 ✓ 0.562 Yes Yes 161 Finite domain (bar), Both ends homogeneous BC with source that depends on space only (general case) ✗ 31.709 ✓ 1.676 Yes 162 Finite domain (bar), Both ends homogeneous BC with source that depends on space only (special case) ✓ 30.659 ✓ 1.392 Yes Yes 163 Finite domain (bar), Both ends homogeneous BC with source that depends on space and time (general case) ✗ 2.272 ✓ 2.266 Yes 164 Finite domain (bar), Both ends homogeneous BC with source that depends on space and time (special case) ✓ 33.479 ✓ 7.534 Yes Yes 165 Finite domain (bar), Both ends homogeneous BC Math 4567 Exam ✓ 18.088 ✓ 1.585 Yes 166 Finite domain (bar), Both ends homogeneous BC With source ✗ 1.951 ✓ 4.825 167 Finite domain (bar), Both ends homogeneous BC special initial condition ✓ 2.705 ✗ (Timed out) 600. Yes 168 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction and Euler-Cauchy Sturm-Liouville ✗ 1.185 ✓ 2.078 169 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction. Using growth form reaction term ✗ 21.792 ✓ 0.531 Yes 170 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction, using logistic form for reaction term ✗ 1.614 ✗ 3.782 171 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction, using Aleee form for reaction term ✗ 2.769 ✗ 5.574 172 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction. Haberman 2.3.8 ✗ 1.249 ✓ 1.476 Yes 173 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction ✗ 1.033 ✓ 1.533 Yes 174 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction ✓ 0.65 ✓ 1.031 175 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction ✗ 40.533 ✓ 0.819 176 Finite domain (bar), Both ends homogeneous BC Diﬀusion convection (general case) ✗ 1.257 ✗ 3.387 Yes 177 Finite domain (bar), Both ends homogeneous BC Diﬀusion convection (special case) ✗ 31.408 ✗ (Timed out) 600. Yes Yes 178 Finite domain (bar), Both ends homogeneous BC Haberman 2.4.2 (general case) ✓ 0.366 ✓ 1.772 Yes 179 Finite domain (bar), Both ends homogeneous BC Left end zero, right end insulated, no source ✓ 0.324 ✓ 1.086 Yes 180 Finite domain (bar), Both ends homogeneous BC One end insulated ✓ 30.734 ✓ 1.54 181 Finite domain (bar), Both ends homogeneous BC Haberman 2.3.7 (general case) ✓ 0.402 ✓ 1.166 Yes 182 Finite domain (bar), Both ends homogeneous BC speciﬁc case ✓ 6.115 ✓ 0.694 Yes Yes 183 Finite domain (bar), Both ends homogeneous BC Haberman 2.4.1 (a) ✓ 0.549 ✓ 3.674 184 Finite domain (bar), Both ends homogeneous BC Both ends insulated, no source ✓ 13.565 ✓ 1.255 Yes 185 Finite domain (bar), Both ends homogeneous BC Haberman 2.4.1 (b) (special case) ✓ 0.562 ✓ 3.112 Yes 186 Finite domain (bar), Both ends homogeneous BC Haberman 2.4.1 (c) (special case) ✓ 0.846 ✓ 6.312 Yes 187 Finite domain (bar), Both ends homogeneous BC Haberman 2.4.1 (d) ✓ 0.391 ✓ 2.8 188 Finite domain (bar), Both ends homogeneous BC both ends insulated ✗ 19.042 ✓ 3.996 189 Finite domain (bar), Both ends homogeneous BC convection heat loss ✗ 30.572 ✓ 0.796 190 Finite domain (bar), Both ends homogeneous BC Pinchover and Rubinstein 6.25 ✗ 0.989 ✓ 7.672 191 Finite domain (bar), Both ends homogeneous BC external source ✗ 0.527 ✓ 18.664 192 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction (general case) ✗ 28.869 ✓ 0.726 Yes Yes 193 Finite domain (bar), Both ends homogeneous BC Pinchover and Rubinstein 6.23 ✓ 1.329 ✓ 26.237 194 Finite domain (bar), Both ends homogeneous BC Pinchover and Rubinstein 6.21 ✓ 0.374 ✓ 2.572 195 Finite domain (bar), Both ends homogeneous BC Diﬀusion Reaction. Euler-Cauchy Sturm-Liouville ✗ 1.665 ✗ (Timed out) 600. Yes 196 Finite domain (bar), Both ends homogeneous BC convection heat loss ✗ 31.058 ✓ 1.952 Yes 197 Finite domain (bar), Both ends homogeneous BC Mixed BC ✗ 1.006 ✓ 3.45 198 Finite domain (bar), Both ends homogeneous BC Haberman 8.2.1 (f) ✗ 0.52 ✓ 3.463 199 Finite domain (bar), left end homogeneous, right end not left end insulated (general case) ✓ 2.295 ✓ 2.016 Yes 200 Finite domain (bar), left end homogeneous, right end not left end insulated (special case) ✓ 5.676 ✓ 1.126 Yes Yes 201 Finite domain (bar), left end homogeneous, right end not right end nonhomogeneous BC (general case) ✓ 32.608 ✓ 0.957 Yes 202 Finite domain (bar), left end homogeneous, right end not right end nonhomogeneous BC (special case) ✓ 0.009 ✓ 2.862 Yes 203 Finite domain (bar), left end homogeneous, right end not right end nonhomogeneous BC, special case ✓ 10.052 ✓ 2.355 Yes Yes 204 Finite domain (bar), left end homogeneous, right end not convection heat loss ✗ 12.542 ✗ (Timed out) 600. Yes 205 Finite domain (bar), left end homogeneous, right end not nonhomogeneous BC ✗ 0.397 ✓ 5.377 206 Finite domain (bar), left end homogeneous, right end not nonhomogeneous BC ✗ 6.798 ✓ 1.171 207 Finite domain (bar), left end homogeneous, right end not nonhomogeneous BC ✗ 31.601 ✓ 9.329 Yes 208 Finite domain (bar), right end homogeneous, left end not left end BC depends on time (general case) ✗ 0.49 ✓ 5.963 Yes 209 Finite domain (bar), right end homogeneous, left end not left end BC depends on time (special case) ✗ 14.632 ✓ 6.313 Yes Yes 210 Finite domain (bar), right end homogeneous, left end not left end BC depends on time (special case) ✗ 18.619 ✓ 4.6 Yes Yes 211 Finite domain (bar), right end homogeneous, left end not Haberman 8.3.6 (special case) ✗ 30.966 ✓ 16.03 Yes Yes 212 Finite domain (bar), right end homogeneous, left end not BC depends on time (special case) ✓ 7.153 ✓ 5.204 213 Finite domain (bar), Periodic BC Periodic BC (general case). IC given ✗ 0.492 ✓ 5.799 214 Finite domain (bar), Periodic BC Periodic BC (general case). No IC given ✗ 0.007 ✓ 2.203 Yes 215 Finite domain (bar), Periodic BC Periodic BC (general case). Damped heat PDE. No IC given ✗ 0.007 ✓ 2.229 Yes 216 Finite domain (bar), Both ends nonhomogeneous BC both ends non-homogeneous BC (general case) ✓ 31.519 ✓ 3.014 Yes 217 Finite domain (bar), Both ends nonhomogeneous BC non-homogeneous BC (special case) ✓ 19.58 ✓ 1.43 Yes 218 Finite domain (bar), Both ends nonhomogeneous BC Articolo 8.4.1 (special case) ✓ 19.845 ✓ 1.498 Yes 219 Finite domain (bar), Both ends nonhomogeneous BC With source that depends on space only (general case) ✗ 31.438 ✓ 3.576 Yes 220 Finite domain (bar), Both ends nonhomogeneous BC With source that depends on space only (special case) ✗ 30.815 ✓ 2.017 Yes Yes 221 Finite domain (bar), Both ends nonhomogeneous BC With source that depends on space and time only (general case) ✗ 1.307 ✓ 6.288 Yes 222 Finite domain (bar), Both ends nonhomogeneous BC Both ends depend on time (general case) ✗ 8.714 ✓ 17.586 Yes 223 Finite domain (bar), Both ends nonhomogeneous BC Both ends depend on time (special case) ✓ 35.85 ✓ 13.062 Yes Yes 224 Finite domain (bar), Both ends nonhomogeneous BC both ends nonhomogeneous ✗ 30.804 ✓ 11.831 Yes 225 Finite domain (bar), Both ends nonhomogeneous BC Haberman 8.2.1 (a) (general case) ✗ 0.991 ✓ 3.046 Yes 226 Finite domain (bar), Both ends nonhomogeneous BC Haberman 8.2.1 (d) (general solution) ✗ 31.075 ✓ 1.733 Yes 227 Finite domain (bar), Both ends nonhomogeneous BC Both ends depend on time with source that depends on space only (general solution) ✗ 1.194 ✗ (Timed out) 600. Yes 228 Finite domain (bar), Both ends nonhomogeneous BC Both ends depend on time with source that depends on space only (special case) ✗ 30.834 ✓ 5.136 Yes Yes 229 Finite domain (bar), Both ends nonhomogeneous BC Both ends depend on time with source that depends on time and space (general solution) ✗ 1.318 ✓ 2.434 Yes 230 Finite domain (bar), Both ends nonhomogeneous BC Both ends depend on time with source present (special case) ✗ 30.968 ✓ 7. Yes Yes 231 Finite domain (bar), Both ends nonhomogeneous BC Pinchover and Rubinstein 6.17 ✗ 1.967 ✓ 1.941 232 Finite domain (bar), Both ends nonhomogeneous BC nonhomogeneous BC ✗ 30.863 ✓ 4.252 233 Finite domain (bar), Both ends nonhomogeneous BC Haberman 8.2.2. (a) ✗ 0.607 ✓ 3.23 Yes 234 Finite domain (bar), Both ends nonhomogeneous BC Articolo 8.4.3 ✗ 31.014 ✓ 7.222 235 Semi-inﬁnite domain left end constant (general case) ✓ 390.937 ✓ 0.439 Yes 236 Semi-inﬁnite domain left end constant (special case) ✓It fail if assumption $$x>0$$ is given. A bug 13.432 ✓ 0.925 Yes Yes 237 Semi-inﬁnite domain Logan p. 76. Left end general function of time (general case) ✓ 0.56 ✓ 1.139 Yes 238 Semi-inﬁnite domain Left end function of time (special case) ✓ 133.902 ✓ 1.431 Yes Yes 239 Semi-inﬁnite domain nonhomogeneous BC ✓ 390.726 ✓ 0.51 240 Semi-inﬁnite domain I.C. not zero ✗due to IC/BC not zero 0.008 ✓ 0.599 241 Semi-inﬁnite domain nonhomogeneous BC ✓ 394.948 ✓ 0.55 242 Semi-inﬁnite domain nonhomogeneous BC ✓ 38.713 ✓ 1.905 243 Semi-inﬁnite domain nonhomogeneous B.C. ✓ 3.438 ✓ 0.99 244 Semi-inﬁnite domain Unit triangle I.C. ✓ 29.149 ✓ 0.792 245 Semi-inﬁnite domain I.C. not at $$t=0$$ ✗ 0.008 ✓ 0.596 246 Semi-inﬁnite domain Diﬀusion with advection ✗ 0.828 ✓ 1.413 247 Semi-inﬁnite domain Practice exam problem ✓ 26.449 ✓ 0.8 248 Inﬁnite domain Inverse exponential I.C. ✓ 1.542 ✓ 0.857 Yes 249 Inﬁnite domain Advection term ✓ 0.51 ✓ 0.824 250 Inﬁnite domain UnitBox I.C. ✓ 1.974 ✓ 1.225 251 Inﬁnite domain No source ✓ 0.211 ✓ 1.515 252 Inﬁnite domain constant as source ✓ 7.7 ✓ 1.086 253 Inﬁnite domain No intial conditions ✗ 0.002 ✓ 0.109 Yes 254 Inﬁnite domain piecewise I.C. ✗due to i.c. not at zero 0.011 ✓ 0.815 Yes 255 Inﬁnite domain Practice exam problem ✓ 0.008 ✓ 0.304 256 Cartesian coordinates (Rectangle, Square) No source ✗ 0.003 ✓ 1.75 257 Cartesian coordinates (Rectangle, Square) Internal source term ✗ 0.004 ✓ 1.602 258 Cartesian coordinates (Rectangle, Square) Articolo 6.6.3 ✗ 0.003 ✓ 3.749 259 Polar coordinates (disk, sector, annulus) no $$\theta$$ dependency, insulated (General solution) ✗ 2.378 ✓Cant get series solution 15.478 Yes 260 Polar coordinates (disk, sector, annulus) no $$\theta$$ dependency, insulated (Speciﬁc solution) ✗ 1.88 ✓Do not understand Maple solution 6.748 Yes Yes 261 Polar coordinates (disk, sector, annulus) no $$\theta$$ dependency ✗ 2.27 ✓ 1.276 262 Polar coordinates (disk, sector, annulus) no $$\theta$$ dependency ✓ 0.914 ✓ 0.312 263 Polar coordinates (disk, sector, annulus) Haberman 8.3.5 (General solution) ✗ 2.176 ✓ 1.565 Yes 264 Polar coordinates (disk, sector, annulus) Speciﬁc example of the above ✗ 1.734 ✓ 1.463 Yes Yes 265 Polar coordinates (disk, sector, annulus) Speciﬁc example of the above ✗ 1.74 ✗ 1.765 Yes Yes 266 Polar coordinates (disk, sector, annulus) Articolo 6.9.1 ✗ 0.004 ✗ 1.387 267 Polar coordinates (disk, sector, annulus) Articolo 6.9.2 ✗ 0.004 ✓ 6.82 268 Polar coordinates (disk, sector, annulus) Haberman 8.2.5 with $$\theta$$ dependency (General case) ✗ 0.004 ✗ 0.515 Yes 269 Polar coordinates (disk, sector, annulus) With $$\theta$$ dependency (Speciﬁc example) ✗ 0.004 ✗ 1.213 Yes Yes 270 Polar coordinates (disk, sector, annulus) With $$\theta$$ dependency (speciﬁc example) ✗ 0.004 ✗ 1.144 Yes Yes 271 Polar coordinates (disk, sector, annulus) Insulated with $$\theta$$ dependency (General solution) ✗ 0.004 ✗ 2.447 Yes 272 Polar coordinates (disk, sector, annulus) Insulated with $$\theta$$ dependency (Speciﬁc example) ✗ 0.004 ✗ 2.788 Yes Yes 273 Polar coordinates (disk, sector, annulus) Insulated with $$\theta$$ dependency (Speciﬁc example) ✗ 0.004 ✗ 2.141 Yes Yes 274 Spherical coordinates No angle dependencies ✗ 2.195 ✓Has unresolved Laplace integrals 1.151 275 Cylinderical coordinates Haberman 7.9.4 (a) ✗ 0.004 ✓ 11.908 Yes Yes 276 Cylinderical coordinates Haberman 7.9.4 (b) ✗ 0.003 ✗ 6.707 Yes Yes 277 Cylinderical coordinates Haberman 7.9.4 (c) ✗ 0.003 ✗ 6.136 278 Cylinderical coordinates Haberman 7.9.3 (a) ✗ 0.004 ✗ 1.939 279 Cylinderical coordinates Haberman 7.9.3 (b) ✗ 0.003 ✗ 2.63 280 Cylinderical coordinates Haberman 7.9.3 (c) ✗ 0.004 ✗ 1.612

Elliptic PDE’s (Laplace, Poisson, Helmholtz)

 # PDE description Mathematica Maple hand solved? Animated? result time result time 281 Cartesian coordinates Rectangle, 3 edges zero, buttom edge not ✓ 0.246 ✓ 20.454 Yes 282 Cartesian coordinates Rectangle, 3 edges zero, right edge not ✓ 0.713 ✓ 1.493 Yes 283 Cartesian coordinates Rectangle, 3 edges zero, buttom edge has impulse ✓ 2.681 ✓ 13.09 Yes 284 Cartesian coordinates Haberman 2.5.1 (a) ✓ 0.198 ✓ 15.391 285 Cartesian coordinates Haberman 2.5.1 (b) ✓ 0.192 ✓ 18.329 286 Cartesian coordinates Haberman 2.5.1 (c) ✗ 0.405 ✓ 22.601 287 Cartesian coordinates Haberman 2.5.1 (d) ✗ 0.403 ✓ 16.85 288 Cartesian coordinates Haberman 2.5.1 (e) ✗ 0.796 ✓ 22.019 Yes 289 Cartesian coordinates Unit triangle B.C. ✓ 2.267 ✓ 3.069 290 Cartesian coordinates Top edge at inﬁnity ✗ 2.308 ✓ 1.489 Yes 291 Cartesian coordinates Top edge at inﬁnity ✗ 2.32 ✓ 1.616 Yes 292 Cartesian coordinates Right edge at inﬁnity ✗ 2.313 ✓ 1.476 Yes 293 Cartesian coordinates Right edge at inﬁnity ✗ 0.783 ✓ 9.34 Yes 294 Cartesian coordinates Right edge at inﬁnity ✗ 2.258 ✓ 11.718 295 Cartesian coordinates Laplace PDE in 2D Cartessian with boundary condition as Dirac function ✓ 0.045 ✓ 1.641 296 Cartesian coordinates One side homogeneous ✓ 0.585 ✓ 18.027 297 Cartesian coordinates In right half plane ✓ 19.451 ✓ 1.513 298 Cartesian coordinates Right edge at inﬁnity ✗ 0.829 ✓ 6.872 299 Cartesian coordinates Dirichlet problem Upper half ✓ 0.94 ✓ 1.91 300 Cartesian coordinates Right half-plane ✓ 20.57 ✓ 1.968 301 Cartesian coordinates First quadrant ✓ 15.696 ✗ 18.134 302 Cartesian coordinates Neumann problem upper half-plane ✓ 5.249 ✓used convert(sol,Int). 3.617 303 Cartesian coordinates Dirichlet problem in a rectangle ✓ 1.257 ✓ 1.789 304 Cartesian coordinates Strip in upper half ✗ 0.369 ✓ 3.974 305 Cartesian coordinates in Rectangle, right edge at inﬁnity ✗ 0.83 ✓ 7.553 306 Polar coordinates Laplace PDE inside quarter disk, Neumann BC at edge ✗ 2.091 ✓ 0.992 307 Polar coordinates $$r=4$$ and $$u=x^4$$ at boundary of disk ✓ 149.331 ✓ 5.08 Yes 308 Polar coordinates $$r=1$$ and $$u_r=x$$ at boundary of disk ✗ 0.016 ✗ 0.339 Yes 309 Polar coordinates Laplace inside disk. General solution ✓ 0.134 ✓ 1.492 310 Polar coordinates Laplace inside disk. Speciﬁc boundary conditions ✗ 0.041 ✓ 1.51 Yes 311 Polar coordinates Haberman 2.5.5 (c) ✗ 2.194 ✓ 0.695 Yes 312 Polar coordinates semi-circle ✗ 2.064 ✓ 0.67 313 Polar coordinates Haberman 2.5.8 (b) ✗ 0.011 ✓But has unresolved Invfourier and Fourier calls 35.961 Yes 314 Polar coordinates Circular annulus ✓ 2.316 ✓ 3.298 315 Polar coordinates Outside a disk ✗ 0.008 ✓ 2.042 316 Polar coordinates Outside a disk ✗ 0.038 ✓ 1.556 Yes 317 Spherical coordinates In a sphere ✗ 0.021 ✓ 1.758 318 Cylinderical coordinates Haberman 7.9.1 (a) ✗ 0.003 ✗ 1.592 319 Cylinderical coordinates Haberman 7.9.1 (b) ✗ 0.003 ✗ 1.779 320 Cylinderical coordinates Haberman 7.9.1 (c) ✗ 0.003 ✗ 1.819 321 Cylinderical coordinates Haberman 7.9.1 (d) ✗ 0.003 ✗ 1.919 322 Cylinderical coordinates Haberman 7.9.1 (e) ✗ 0.003 ✗ 1.862 323 Cylinderical coordinates Haberman 7.9.2 (a) ✗ 0.003 ✗ 2.144 324 Cylinderical coordinates Haberman 7.9.2 (b) ✗ 0.003 ✗ 7.48 325 Cylinderical coordinates Haberman 7.9.2 (c) ✗ 0.003 ✗ 9.012 326 Cylinderical coordinates Haberman 7.9.2 (d) ✗ 0.003 ✗ 1.993 327 Cartesian coordinates All boundaries at zero ✗ 0.01 ✗ 23.384 Yes 328 Cartesian coordinates Dirichlet problem in a rectangle ✓ 0.211 ✓ 0.745 329 Cartesian coordinates Poisson PDE in whole 2D plane ✗ 0.02 ✓ 0.02 Yes 330 Cartesian coordinates In rectangle ✓ 2.593 ✓ 4.516 331 Cartesian coordinates On whole plane ✗why? It solved earlier with BC? 0.004 ✓ 0.039 332 Cartesian coordinates Reduced Helmholtz Inside square ✗ 1.134 ✓ 20.8 333 Spherical coordinates Chain reaction PDE ✗ 0.036 ✗trivial solution 2.122 Yes

Hyperbolic PDE’s (Wave)

 # PDE description Mathematica Maple hand solved? Animated? result time result time 334 Finite length string General solution for both ends ﬁxed. Domain is $$0\dots L$$ ✗Surprising it could not solve this standard wave PDE on string 5.788 ✓ 5.997 Yes 335 Finite length string both ends ﬁxed, inital position zero (special case) ✓ 31.082 ✓ 1.97 Yes Yes 336 Finite length string both ends ﬁxed, inital velocity zero (special case) ✓ 31.042 ✓ 1.723 Yes Yes 337 Finite length string both ends ﬁxed but domain is $$-\pi \dots \pi$$. zero intial position, non zero initial velocity ✗ 51.161 ✓ 15.36 Yes Yes 338 Finite length string both ends ﬁxed but domain is $$-1 \dots 1$$. intial position is an impulse, zero initial velocity ✗ 5.785 ✓ 1.727 Yes 339 Finite length string Logan book, page 28. Both ends ﬁxed ✗ 0.422 ✓ 0.375 340 Finite length string non-zero initial velocity. Both ends ﬁxed ✗ 27.558 ✓ 1.625 341 Finite length string Logan book page 149) ✗ 1.244 ✓ 2.446 342 Finite length string Haberman 8.5.2 (a) ✗ 1.242 ✓ 10.801 343 Finite length string Haberman 8.5.2 (b) ✗ 1.541 ✓ 7.278 Yes 344 Finite length string Both I.C. not zero ✓ 40.463 ✓ 2.398 345 Finite length string With constant source ✗ 0.965 ✓ 6.548 346 Finite length string Logan page 213 ✗ 0.915 ✓ 4.085 347 Finite length string Telegraphy PDE ✗ 31.304 ✓But $$n = 1$$ should not be included. 3.083 348 Finite length string Dispersion term present (general case) ✗Due to adding dispersion term 1.252 ✓ 4.715 Yes 349 Finite length string Dispersion term present ✗Due to adding dispersion term 31.593 ✓ 20.208 Yes 350 Finite length string Dispersion term present (speciﬁc case) ✗Due to adding dispersion term 31.468 ✓ 3.76 Yes Yes 351 Finite length string non-zero initial position ✓But sum should not include $$n=2$$ 31.137 ✓Handled $$n=2$$ case correctly 9.71 352 Finite length string With source ✗ 30.99 ✓ 10.826 353 Finite length string Right end free (general case) ✗ 0.969 ✓ 7.717 Yes 354 Finite length string Right end free, zero initial velocity (general case) ✗ 0.967 ✓ 5.08 Yes 355 Finite length string Right end free, zero initial velocity (special case) ✗ 33.775 ✓ 2.87 Yes Yes 356 Finite length string Right end free, zero initial velocity, damping present (general case) ✗ 1.28 ✓ 6.279 Yes 357 Finite length string Right end free, zero initial velocity, damping present (special case, underdamped) ✗ 49.27 ✓ 5.674 Yes Yes 358 Finite length string Right end free, zero initial velocity, damping present (special case, critical damped) ✗ 48.952 ✓ 5.414 Yes Yes 359 Finite length string Right end free, zero initial velocity, damping present (special case, over damped) ✗ 49.823 ✓ 4.132 Yes Yes 360 Finite length string I.C. at diﬀerent times, right end free, with source ✗ 1.153 ✓ 54.131 Yes 361 Finite length string Right end oscillates ✗ 30.978 ✓ 11.337 362 Finite length string Perioidic B.C. ✗ 7.439 ✗ 2.564 363 Finite length string Mixed B.C. ✗ 21.322 ✗ 3.43 364 Finite length string Left end ﬁxed, right end non-homogeneous Neumann BC. Zero initial conditions ✗ 0.963 ✓ 4.239 Yes Yes 365 Semi-inﬁnite domain Left end ﬁxed, (general case) ✓ 2.14 ✗ 3.623 Yes 366 Semi-inﬁnite domain Left end ﬁxed with speciﬁc initial position ✓ 22.917 ✓ 3.145 Yes 367 Semi-inﬁnite domain Logan page 115, left end ﬁxed with source ✓ 0.183 ✓ 2.443 368 Semi-inﬁnite domain Left moving boundary condition ✓ 7.585 ✓ 2.458 369 Semi-inﬁnite domain moving Left end ✓ 17.273 ✓ 1.045 370 Semi-inﬁnite domain I.C. at $$t=1$$ ✓ 0.01 ✓ 0.33 371 Semi-inﬁnite domain B.C. at $$x=1$$ ✗ 10.107 ✓ 0.543 372 Semi-inﬁnite domain Left end free. zero initial velocity (general solution) ✗ 2.494 ✓ 0.341 373 Semi-inﬁnite domain Left end free. zero initial velocity (Special solution) ✓ 3.478 ✓ 0.713 Yes Yes 374 Semi-inﬁnite domain Left end ﬁxed. zero initial velocity (Special solution) ✓ 2.6 ✓ 0.536 Yes Yes 375 Semi-inﬁnite domain Left end free. zero initial position (general solution) ✗ 2.75 ✓ 0.369 376 Semi-inﬁnite domain Left end free. Non zero initial position and velocity (general solution) ✗ 3.196 ✓ 0.321 377 Semi-inﬁnite domain Left end free with source ✓ 0.182 ✓ 0.246 378 Inﬁnite domain General case. $$u_{tt} = u_{xx}$$ with $$u(x,0)=f(x),u_t(x,0)=g(x)$$ ✓ 0.062 ✓ 0.056 379 Inﬁnite domain General case. No IC given. $$u_{tt} + u_{xt} = c^2 u_{xx}$$ ✓ 0.005 ✓ 0.195 380 Inﬁnite domain $$u_{tt}= c^2 u_{xx} + f(x,t)$$, IC at $$t=1$$,$$u(x,1) = g(x),u_t(x,1)=h(x)$$ ✗ 0.008 ✓ 1.505 381 Inﬁnite domain No source. $$u_{tt} = u_{xx}$$, with $$u(x,0) =e^{-x^2},u_t(x,0)=1$$ ✓ 0.048 ✓ 0.084 382 Inﬁnite domain With source term. $$u_{tt} = u_{xx} + m$$ ✓ 0.01 ✓ 0.105 383 Inﬁnite domain non-linear (Solitons) $$u_t +6 u(x,t) u_x + u_{xxx} = 0$$ ✓ 0.03 ✓ 0.166 Yes Yes 384 Inﬁnite domain Inhomogeneous PDE $$3 u_{xx}- u_{tt} + u_{xt}=1$$ ✓ 0.003 ✓ 0.105 385 Inﬁnite domain Practice exam problem Math 5587 ✓ 0.074 ✓ 0.086 386 Inﬁnite domain Practice exam problem Math 5587 ✓ 0.031 ✓ 0.056 387 Inﬁnite domain Practice exam problem Math 5587 ✓ 24.937 ✓ 0.132 388 Inﬁnite domain zero initial velocity ✓ 0.075 ✓ 0.197 Yes Yes 389 Inﬁnite domain zero initial velocity ✓ 0.078 ✓ 0.316 Yes Yes 390 Inﬁnite domain General case $$u_{tt} = u_{xx}$$ with $$u(x,0)=\sin x, u_t(x,0)=-2 x e^{-x^2}$$ ✓ 0.134 ✓ 0.087 391 Inﬁnite domain General case. $$u_{tt} =u_{xx}$$ dAlembert solution, box function as initial position ✓ 0.032 ✓ 0.118 Yes Yes 392 Inﬁnite domain $$u_{tt}=4 u_{xx}+ \cos (t)$$ dAlembert solution with $$u(x,0)=\sin x,u_t(x,0)=\cos x$$ ✓ 0.095 ✓ 0.116 Yes Yes 393 Inﬁnite domain $$u_{tt}=c^2 u_{xx}$$ dAlembert solution with $$u(x,0)=\delta (x-a),u_t(x,0)=0$$ ✗ 35.84 ✓ 0.06 Yes 394 Inﬁnite domain system of 2 inhomogeneous linear hyperbolic system with constant coeﬃcients ✓ 0.279 ✗ 0.359 395 Cartesian coordinates Rectangular membrane. Fixed on all edges, General solution ✗ 0.028 ✓ 184.836 Yes 396 Cartesian coordinates Rectangular membrane. Fixed on all edges, zero velocity. Speciﬁc example ✗ 0.023 ✓ 2.036 Yes Yes 397 Cartesian coordinates All 4 edges ﬁxed, zero initial velocity, Speciﬁc example ✓ 5.053 ✓ 12.351 Yes Yes 398 Cartesian coordinates All 4 edges ﬁxed, zero initial velocity, Speciﬁc example, delta in center ✓ 1.229 ✓ 10.169 Yes Yes 399 Cartesian coordinates All 4 edges ﬁxed ✓ 0.997 ✓ 5.191 400 Cartesian coordinates All edges ﬁxed (Haberman 8.5.5 (a) ✗ 0.003 ✗ 2.079 401 Cartesian coordinates 2 edgs ﬁxed, 2 free, zero initial velocity ✗ 0.003 ✓ 9.615 402 Cartesian coordinates All 4 edges ﬁxed, zero initial velocity, general solution ✓ 0.1 ✗ (Timed out) 600. Yes 403 Cartesian coordinates With damping ✗ 0.003 ✓ 18.699 404 Cartesian coordinates On the whole plane ✓ 0.104 ✗ 0.086 405 Polar coordinates no $$\theta$$ dependency, ﬁxed boundary, general case ✓ 0.723 ✓Has unresolved Invlaplace calls 0.785 Yes 406 Polar coordinates no $$\theta$$ dependency. Speciﬁc example. Both initial conditions not zero ✓ 31.224 ✓Has unresolved Invlaplace calls 42.668 407 Polar coordinates no $$\theta$$ dependency. Speciﬁc example. Both initial conditions not zero ✓ 31.077 ✓Has unresolved Invlaplace calls. How to get series solution? 6.716 Yes Yes 408 Polar coordinates no $$\theta$$ dependency. Using integral transforms. Source present. Speciﬁc example ✗ 1.772 ✓ 10.237 409 Polar coordinates no $$\theta$$ dependency. Using integral transforms. Source present. Speciﬁc example ✗ 1.954 ✓ 3.56 410 Polar coordinates $$\theta$$ dependency, ﬁxed on edges, general solution ✗ 0.004 ✗ 1.901 Yes 411 Polar coordinates $$\theta$$ dependency, ﬁxed on edges, zero initial velocity, general solution ✗ 0.004 ✗ 4.776 Yes 412 Polar coordinates $$\theta$$ dependency, ﬁxed on edges, zero initial velocity, speciﬁc example ✗ 0.004 ✗ 1.127 Yes Yes 413 Polar coordinates $$\theta$$ dependency, ﬁxed on edges, zero initial position, speciﬁc example ✗ 0.005 ✗ 1.689 Yes Yes 414 Polar coordinates $$\theta$$ dependency, ﬁxed on edges, zero initial position with internal source (Haberman 8.5.5. (b) ✗ 0.027 ✗ 0.398 Yes 415 Spherical coordinates No I.C. no B.C. ✗ 0.031 ✓ 3.537 416 Cylindrical coordinates No I.C. no B.C. ✗ 0.003 ✓ 0.268

Handbook of ﬁrst order partial diﬀerential equations

 # PDE description Mathematica Maple hand solved? Animated? result time result time 417 problem number 1 ✓ 0.005 ✓ 0.046 418 problem number 2 ✓ 0.003 ✓ 0.004 419 problem number 3 ✓ 0.008 ✓ 0.004 Yes 420 problem number 4 ✓ 0.006 ✓ 0.003 421 problem number 5 ✓ 0.022 ✓ 0.017 Yes 422 problem number 6 ✓ 0.021 ✓ 0.012 423 2.1 problem number 1 ✓ 0.004 ✓ 0.024 Yes 424 2.1 problem number 2 ✓ 0.005 ✓ 0.027 Yes 425 2.1 problem number 3 ✓ 0.037 ✓ 0.033 Yes 426 2.1 problem number 4 ✓ 0.009 ✓ 0.029 Yes 427 2.1 problem number 5 ✓ 0.012 ✓ 0.128 Yes 428 2.1 problem number 6 ✓ 0.03 ✓ 0.145 Yes 429 2.1 problem number 7 ✓ 0.026 ✓ 1.447 Yes 430 2.1 problem number 8 ✗ 1.783 ✓ 2.075 431 2.2 problem number 1 ✓ 0.005 ✓ 0.008 432 2.2 problem number 2 ✓ 0.171 ✓ 0.185 433 2.2 problem number 3 ✓ 0.084 ✓ 0.019 434 2.2 problem number 4 ✓ 0.249 ✓ 0.135 435 2.2 problem number 5 ✓ 0.152 ✓ 0.181 436 2.2 problem number 6 ✓ 0.139 ✓ 0.102 437 2.2 problem number 7 ✓ 0.144 ✓ 0.206 438 2.2 problem number 8 ✓ 0.162 ✓ 0.191 439 2.2 problem number 9 ✓ 0.238 ✓ 0.405 440 2.2 problem number 10 ✓ 0.145 ✓ 0.182 441 2.2 problem number 11 ✓ 0.264 ✓ 0.106 442 2.2 problem number 12 ✓ 0.578 ✓ 0.439 443 2.2 problem number 13 ✓ 0.122 ✓ 0.036 444 2.2 problem number 14 ✓ 0.242 ✓ 0.274 445 2.2 problem number 15 ✓ 0.537 ✓ 0.302 446 2.2 problem number 16 ✓ 1.094 ✓ 0.225 447 2.2 problem number 17 ✓ 0.868 ✓ 0.408 448 2.2 problem number 18 ✓ 0.245 ✓ 0.07 449 2.2 problem number 19 ✓ 0.261 ✓ 0.194 450 2.2 problem number 20 ✗ 0.176 ✓ 0.37 451 2.2 problem number 21 ✓ 0.158 ✓ 0.198 452 2.2 problem number 22 ✓ 0.356 ✓ 0.638 453 2.2 problem number 23 ✓ 0.416 ✓ 0.313 454 2.2 problem number 24 ✗ 0.278 ✓ 0.244 455 2.2 problem number 25 ✗ (Timed out) 600. ✗ (Timed out) 600. 456 2.2 problem number 26 ✗ (Timed out) 600. ✗ 33.256 457 2.2 problem number 27 ✗ 101.682 ✗ 1.896 458 2.2 problem number 28 ✗ 0.495 ✓solution contains RootOf 0.147 459 2.2 problem number 29 ✗ 101.287 ✗ 1.356 460 2.2 problem number 30 ✗ 0.237 ✓ 0.24 461 2.2 problem number 31, Hesse’s equation ✗ 235.141 ✓ 10.764 462 2.3 problem number 1 ✗But it can’t solve it when assuming $$b>0$$ which is strange. 40.89 ✓ 1.633 463 2.3 problem number 2 ✓ 0.055 ✓ 0.178 464 2.3 problem number 3 ✓ 0.147 ✓ 0.906 465 2.3 problem number 4 ✗ 0.177 ✓ 1.41 466 2.3 problem number 5 ✗ 0.56 ✓Answer contains RootOf 0.155 467 2.3 problem number 6 ✗ 0.209 ✓Answer contains RootOf 0.067 468 2.3 problem number 7 ✓ 0.178 ✓ 0.174 469 2.3 problem number 8 ✓ 0.216 ✓ 0.066 470 2.3 problem number 9 ✗ (Timed out) 600. ✓ 0.049 471 2.4 problem number 1 ✓ 0.021 ✓ 0.019 472 2.4 problem number 2 ✓ 0.2 ✓ 0.292 473 2.4 problem number 3 ✓ 0.159 ✓ 0.27 474 2.4 problem number 4 ✓ 0.404 ✓ 1.785 475 2.4 problem number 5 ✗ 0.378 ✗ (Timed out) 600. 476 2.4 problem number 6 ✓ 1.831 ✓ 0.602 477 2.4 problem number 7 ✗ 1.475 ✓ 2.763