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Partial differential equation
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Words: 2k
Articles: 114
A partial differential equation relates a multivariable function to its partial derivatives.
Table of contents
2k
114
Modified Helmholtz equation
Partial differential equation
14
Poisson equation
Partial differential equation
32
2
Poisson kernel for the upper half-space
Poisson equation
32
1
Poisson integral
Poisson kernel for the upper half-space
21
Diffusion equation
Partial differential equation
720
29
Fick's first law
Diffusion equation
24
Heat equation
Diffusion equation
213
8
Sinusoidally forced heat equation on a half-line
Heat equation
27
Potential Burgers equation
Heat equation
22
Heat kernel
Heat equation
97
3
Gaussian approximate identity
Heat kernel
33
Heat-kernel solution
Heat kernel
34
1
Heat equation with interval-indicator initial data
Heat-kernel solution
13
Duhamel principle
Heat equation
54
1
Cancellation of two heat-kernel impulses
Duhamel principle
28
Advection-diffusion equation
Diffusion equation
31
1
Advection-diffusion heat-kernel solution
Advection-diffusion equation
9
Nonlinear diffusion equation
Diffusion equation
52
2
Two-dimensional Barenblatt solution with diffusivity proportional to concentration
Nonlinear diffusion equation
34
1
Linear-reaction time change for quadratic nonlinear diffusion
Two-dimensional Barenblatt solution with diffusivity proportional to concentration
16
Reaction-diffusion system
Diffusion equation
388
13
Morphogen reaction-diffusion equation
Reaction-diffusion system
54
2
Mixed Dirichlet-Neumann modes on an interval
Morphogen reaction-diffusion equation
28
1
Critical length for a linearly growing morphogen
Mixed Dirichlet-Neumann modes on an interval
17
Fast-inhibitor elimination in a reaction-diffusion system
Reaction-diffusion system
52
1
Dispersion relation after fast-inhibitor elimination
Fast-inhibitor elimination in a reaction-diffusion system
29
Turing instability
Reaction-diffusion system
127
3
Two-species diffusion-driven instability criterion
Turing instability
98
2
Impossibility of a two-species Turing instability at equal diffusivities
Two-species diffusion-driven instability criterion
31
Near-unity diffusivity ratio for a Turing instability
Two-species diffusion-driven instability criterion
24
Brusselator
Reaction-diffusion system
143
3
Brusselator trapping region
Brusselator
53
Brusselator periodic-orbit criterion
Brusselator
39
Turing threshold of the Brusselator
Brusselator
29
Similarity solution
Partial differential equation
16
Separation of variables
Partial differential equation
31
1
Dirichlet problem on an annulus for one Fourier mode
Separation of variables
15
Polynomial ansatz
Partial differential equation
16
Harmonic function
Partial differential equation
377
14
Harmonic function as the real part of a holomorphic function
Harmonic function
28
Subharmonic function
Harmonic function
66
1
Maximum principle for subharmonic functions
Subharmonic function
38
Maximum principle for harmonic functions
Harmonic function
35
Harmonic Liouville theorem
Harmonic function
55
1
Surjectivity of a nonconstant entire harmonic function
Harmonic Liouville theorem
22
Scale-periodic harmonic function from an elliptic function
Harmonic function
48
Harmonic polynomial
Harmonic function
10
Harmonic conjugate
Harmonic function
85
2
Local harmonic conjugate
Harmonic conjugate
22
Log modulus has no global harmonic conjugate on the punctured plane
Harmonic conjugate
49
Harmonic functions are smooth
Harmonic function
18
Harmonic functions are real analytic
Harmonic function
23
1
Unique continuation for harmonic functions
Harmonic functions are real analytic
16
Smooth bump function
Partial differential equation
41
1
Disjoint-support zero-product construction
Smooth bump function
20
Laplace operator
Partial differential equation
122
8
Laplace equation
Laplace operator
49
2
Green function of the Laplacian
Laplace equation
16
Laplace equation in polar coordinates
Laplace equation
24
Radial Laplacian
Laplace operator
9
Laplacian in spherical coordinates
Laplace operator
27
1
Axisymmetric harmonic function
Laplacian in spherical coordinates
10
Laplacian eigenfunction
Laplace operator
26
1
Dirichlet Laplacian eigenfunction
Laplacian eigenfunction
14
Wave equation
Partial differential equation
681
36
Elastic wave
Wave equation
259
13
Helmholtz separation of elastic waves
Elastic wave
22
P-wave
Elastic wave
30
1
Longitudinal polarization
P-wave
15
S-wave
Elastic wave
98
4
Transverse polarization
S-wave
15
SV-wave
S-wave
16
SH-wave
S-wave
54
1
Guided SH modes between rigid and free planes
SH-wave
42
Elastic-interface boundary conditions
Elastic wave
46
2
P-SV mode conversion at a solid interface
Elastic-interface boundary conditions
17
Snell law for elastic waves
Elastic-interface boundary conditions
17
Acoustic reflection and transmission at an interface
Elastic wave
43
1
Equal displacement-amplitude condition at an acoustic interface
Acoustic reflection and transmission at an interface
25
Wave equation on a string
Wave equation
91
4
Linearly damped string
Wave equation on a string
72
3
Critical damping
Linearly damped string
27
Separated solution of the damped string equation
Linearly damped string
18
Energy dissipation identity for a linearly damped string
Linearly damped string
15
Normal mode
Wave equation
38
2
Modal energy distribution
Normal mode
13
Node of a standing wave
Normal mode
13
Fourier transform method for the wave equation
Wave equation
14
D'Alembert formula
Wave equation
49
2
D'Alembert formula with initial velocity
D'Alembert formula
43
1
Central zero region for odd compactly supported initial velocity
D'Alembert formula with initial velocity
35
Wavenumber
Wave equation
18
Dispersion relation
Wave equation
130
5
Growth rate
Dispersion relation
25
Phase velocity and group velocity
Dispersion relation
83
3
Wave crest
Phase velocity and group velocity
20
Wave packet
Phase velocity and group velocity
20
Ninth-order dispersive advection equation
Phase velocity and group velocity
13
Klein-Gordon equation
Wave equation
73
2
Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray
Klein-Gordon equation
62
1
Upward zero crossings of an oscillatory stationary-phase tail
Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray
21
Green second identity
Partial differential equation
38
1
Green's third identity
Green second identity
22
Green's first identity
Partial differential equation
17
Lie point symmetry
Partial differential equation
233
9
Infinitesimal generator of a Lie point symmetry
Lie point symmetry
14
Prolongation of a Lie point symmetry
Lie point symmetry
58
1
Second prolongation of a Lie point symmetry
Prolongation of a Lie point symmetry
19
Scaling symmetry of a partial differential equation
Lie point symmetry
42
1
Simultaneous spacetime scaling symmetry of the wave equation
Scaling symmetry of a partial differential equation
26
Symmetry reduction of a partial differential equation
Lie point symmetry
97
3
Group-invariant solution
Symmetry reduction of a partial differential equation
31
Projective Lie symmetry of the potential Burgers equation
Symmetry reduction of a partial differential equation
36
Similarity variable
Symmetry reduction of a partial differential equation
12
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Analysis
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