Partial Differential Equation

Author

Eric W. Weisstein
May 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/PartialDifferentialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PartialDifferentialEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Comments

This notebook contains some of the named PDEs and systems of PDEs from Zwillinger's Handbook of Differential Equations, together with a few others.

Please send any additional named PDEs not in this list to eww@wolfram.com for inclusion in this master document and in MathWorld.

Benjamin-Bona-Mahony Equation

(Subscript[u, t])+(Subscript[u, x])-(Subscript[u, xxt])+(Subscript[uu, x])==0

DSolve[D[#,t]+D[#,x]-D[#,{x,2},{t,1}]+# D[#,x]==0,#,{x,t}]&@u[x,t]

Biharmonic Equation

DSolve[D[u[x],{x,4}]==0,u[x],x]

DSolve[Plus@@(D[u[x,y],{#,4}]&/@{x,y})==0,u[x,y],{x,y}]

DSolve[Plus@@(D[u[x,y,z],{#,4}]&/@{x,y,z})==0,u[x,y,z],{x,y,z}]

Boussinesq Equation

DSolve[D[u[x,t],{t,2}]-(α^2)D[u[x,t],{x,2}]==(β^2)D[u[x,t],{x,2},{t,2}],u[x,t],{x,t}]

Cauchy-Riemann Equations

DSolve[{
D[u[x,y],x]==D[v[x,y],y],
D[u[x,y],y]==-D[v[x,y],x]},
{u[x,y],v[x,y]},{x,y}]

Chaplygin's Equation

DSolve[D[#,{x,2}]+y^2/(1-y^2/c^2)D[#,{y,2}]+y D[#,y]&@u[x,y]==0,u[x,y],{x,y}]

Euler-Darboux Equation

DSolve[D[#,x,y]+((α D[#,x]-β D[#,y])/(x-y))&@u[x,y]==0,u[x,y],{x,y}]

Heat Conduction Equation

DSolve[D[T[t,x],t]==κ D[T[t,x],{x,2}],T[t,x],{t,x}]

DSolve[D[T[t,x,y],t]==κ(D[T[t,x,y],{x,2}]+ D[T[t,x,y],{y,2}]),T[t,x,y],{t,x,y}]

Helmholtz Differential Equation

DSolve[D[u[x],{x,2}]==k^2 u[x],u[x],{x}]

DSolve[D[u[x,y],{x,2}]+D[u[x,y],{y,2}]==k^2 u[x,y],u[x,y],{x,y}]

DSolve[Plus@@(D[u[x,y,z],{#,2}]&/@{x,y,z})==k^2 u[x,y,z],u[x,y,z],{x,y,z}]

Klein-Gordon Equation

DSolve[D[#,{x,2}]-D[#,{t,2}]/c^2==k^2 #,#,{x,t}]&@u[x,t]

DSolve[D[#,{x,2}]+D[#,{y,2}]-D[#,{t,2}]/c^2==k^2#,#,{x,y,t}]&@u[x,y,t]

Korteweg-de Vries-Burger Equation

TraditionalForm[D[#,t]+D[#,{x,3}]-6#D[#,x]==0]&@u[x,t]

DSolve[D[#,t]+D[#,{x,3}]-6#D[#,x]==0,#,{x,t}]&@u[x,t]

Korteweg-de Vries Equation

u_t+u_{xxx}-6uu_x=0.

DSolve[D[u[x,t],t]+D[u[x,t],{x,3}]-6u[x,t] D[u[x,t],x]==0,u[x,t],{x,t}]

Krichever-Novikov Equation

DSolve[D[u[t,x],t]/D[u[t,x],x]==1/4 D[u[t,x],{x,3}]/D[u[t,x],x]-3/8 D[u[t,x],{x,2}]^2/D[u[t,x],x]^2+3/2/4 (4u[t,x]^3-(Subscript[g, 2])u[t,x]-(Subscript[g, 3]))/D[u[t,x],x]^2,u[t,x],{t,x}]

Laplace's Equation

DSolve[D[u[x],{x,2}]==0,u[x],{x}]

DSolve[D[u[x,y],{x,2}]+D[u[x,y],{y,2}]==0,u[x,y],{x,y}]

DSolve[Plus@@(D[u[x,y,z],{#,2}]&/@{x,y,z})==0,u[x,y,z],{x,y,z}]

Lin-Tsien Equation

DSolve[2D[#,t,x]+D[#,x]D[#,{x,2}]-D[#,{y,2}]==0,#,{x,y,t}]&@u[x,y,t]

Sine-Gordon Equation

Spherical Harmonic Differential Equation

DSolve[1/Sin[θ] D[Sin[θ] D[#,θ],θ]+1/Sin[θ]^2 D[#,{ϕ,2}]+l(l+1)#==0,#,{θ,ϕ}]&@u[θ,ϕ]

Tricomi Equation

DSolve[D[#,{y,2}]==y D[#,{x,2}],#,{x,y}]&@ u[x,y]

Wave Equation

DSolve[v^2 D[ψ[x,t],{x,2}]==D[ψ[x,t],{t,2}],ψ[x,t],{x,t}]

DSolve[v^2 (D[ψ[x,y,t],{x,2}]+D[ψ[x,y,t],{y,2}])==D[ψ[x,y,t],{t,2}],ψ[x,y,t],{x,y,t}]