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There is a lot to explore with VisualPDE.

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Explore how heat diffuses over time

\frac{\partial T}{\partial t} = D_{T} \nabla^{2} T

Play with waves and vibrations

\frac{\partial^{2} u}{\partial t^{2}} = D \nabla^{2} u

Turing on Turing

Turing patterns in Turing's image

\frac{\partial u}{\partial t} = \nabla^{2} u + (1 - T (x, y)) - u + u^{2} v,

\frac{\partial v}{\partial t} = D \nabla^{2} v + 1 - u^{2} v

2D Navier-Stokes

\frac{\partial u}{\partial t} + u \cdot \nabla u = ν \nabla^{2} u - \nabla p,

\nabla \cdot u = 0

Shallow water equations

Water waves and ripples

\frac{\partial h}{\partial t} = - \nabla \cdot (u (h + H_{e})),

\frac{\partial u}{\partial t} = ν \nabla^{2} u - g \nabla h - k u - (u \cdot \nabla) u - f \times u

Thermal convection

Instability and mixing

\frac{\partial ω}{\partial t} = ν \nabla^{2} ω - \frac{\partial ψ}{\partial y} \frac{\partial ω}{\partial x} + \frac{\partial ψ}{\partial x} \frac{\partial ω}{\partial y} + \frac{\partial b}{\partial x}

0 = \nabla^{2} ψ + ω

\frac{\partial b}{\partial t} = κ \nabla^{2} b - (\frac{\partial ψ}{\partial y} \frac{\partial b}{\partial x} - \frac{\partial ψ}{\partial x} \frac{\partial b}{\partial y})

Fluids via the vorticity equation

Spinning fluids

\frac{\partial ω}{\partial t} = ν \nabla^{2} ω - \frac{\partial ψ}{\partial y} \frac{\partial ω}{\partial x} + \frac{\partial ψ}{\partial x} \frac{\partial ω}{\partial y}

0 = \nabla^{2} ψ + ω

Swift–Hohenberg equation

Criticality and localisation in pattern formation

\frac{\partial u}{\partial t} = r u - (1 + D \nabla^{2})^{2} u + a u^{2} + b u^{3} + c u^{5}

with periodic boundary conditions

Logistic travelling waves

Classic models of invasion

\frac{\partial u}{\partial t} = \nabla^{2} u + r u (1 - \frac{u}{K})

with periodic boundary conditions

Validating VisualPDE

VisualPDE versus exact solutions

Ripples on a pond

Making waves in shallow water

Method of images

Boundaries in potential flow

\nabla^{2} ϕ = 0,

(u, v) = \nabla ϕ

Survival in a harsh environment

Impacts of boundary conditions on survival

\frac{\partial u}{\partial t} = \nabla^{2} u + u (1 - \frac{u}{K})

with no-flux boundary conditions

Complex Ginzburg–Landau

A world of complexity

\frac{\partial ψ}{\partial t} = (D_{r} + i D_{i}) \nabla^{2} ψ + (a_{r} + i a_{i}) ψ + (b_{r} + i b_{i}) ψ | ψ |^{2}

Virus transmission

Visualising airborne infections

Chemical Basis of Morphogenesis

A transformative text

\frac{\partial u}{\partial t} = \nabla^{2} u + a - u + u^{2} v,

\frac{\partial v}{\partial t} = D \nabla^{2} v + b - u^{2} v

Inhomogeneous heat equation

Sources and sinks of heat

\frac{\partial T}{\partial t} = \nabla \cdot (g (x, y) \nabla T) + f (x, y)

Dipoles in potential flow

Playing with dipoles

\nabla^{2} ϕ = f,

(u, v) = \nabla ϕ

Bistability: invasion and persistence

Spatial Allee effects and changing wavespeeds

\frac{\partial u}{\partial t} = D \nabla^{2} u + u (u - a) (1 - u)

Cahn–Hilliard equation

Phase separation

\frac{\partial u}{\partial t} = \nabla^{2} (F (u) - g \nabla^{2} u) + f (u)

Cleaning up contaminants

Chemical decontamination in porous media

Beating hearts and slime moulds

\frac{\partial u}{\partial t} = \nabla^{2} u +

You spin me right ’round, baby

\frac{\partial u}{\partial t} = D_{u} \nabla^{2} u + a u - (u + c v) (u^{2} + v^{2}),

\frac{\partial v}{\partial t} = D_{v} \nabla^{2} v + a v + (c u - v) (u^{2} + v^{2})

Inhomogeneous wave equation

Waves through a complex medium

\frac{\partial^{2} u}{\partial t^{2}} = \nabla \cdot (f (x, y) \nabla u)

Kuramoto–Sivashinsky equation

Spatiotemporal chaos

\frac{\partial u}{\partial t} = - \nabla^{2} u - \nabla^{4} u - | \nabla u |^{2}

Defending against surging rivers

A-maze-ing PDEs

Searching for love in all the wrong places

\frac{\partial 🐀}{\partial t} = D_{🐀} \nabla \cdot (\nabla 🐀 - g (🐀) \nabla 🧀) + f (🐀, 🧀),

\frac{\partial 🧀}{\partial t} = D_{🧀} \nabla^{2} 🧀 + g (🐀, 🧀)

Schrödinger equation

Interactive quantum mechanics

i ℏ \frac{\partial ψ}{\partial t} = - \frac{ℏ}{2 m} \nabla^{2} ψ + V (x, t) ψ

Schnakenberg pattern formation

Spots and stripes

\frac{\partial u}{\partial t} = \nabla^{2} u + a - u + u^{2} v,

\frac{\partial v}{\partial t} = D \nabla^{2} v + b - u^{2} v

Gray–Scott model

Intricate reaction–diffusion patterning

\frac{\partial u}{\partial t} = \nabla^{2} u + u^{2} v - (a + b) u,

\frac{\partial v}{\partial t} = D \nabla^{2} v - u^{2} v + a (1 - v)

Ducks, advection and ocean flows

Linearly elastic models

Bending beams and deforming plates

\frac{\partial^{2} u}{\partial t^{2}} = - D^{2} \nabla^{4} u - Q

Brusselator pattern formation

Turing instability regions

\frac{\partial u}{\partial t} = \nabla^{2} u + a - (b + 1) u + u^{2} v,

\frac{\partial v}{\partial t} = D \nabla^{2} v + b u - u^{2} v

Viscous Burgers' equation

Nonlinear waves

\frac{\partial u}{\partial t} = - u \frac{\partial u}{\partial x} + ε \frac{\partial^{2} u}{\partial x^{2}}

Convection–diffusion

Movement along streamlines

\frac{\partial u}{\partial t} = D \nabla^{2} u - v \cdot \nabla u

Gierer–Meinhardt pattern formation

Spots and stripes

\frac{\partial u}{\partial t} = \nabla^{2} u + a + \frac{u^{2}}{v} - b u,

\frac{\partial v}{\partial t} = D \nabla^{2} v + u^{2} - c v

Pattern formation & advection

Going with the flow?

FitzHugh–Nagumo and excitability

Patterns, spiral waves, and chaos

\frac{\partial u}{\partial t} = \nabla^{2} u + u - u^{3} - v,

\frac{\partial v}{\partial t} = D \nabla^{2} v + ε_{v} (u - a_{v} v - a_{z})

Korteweg–De Vries & Zakharov–Kuznetsov equations

\frac{\partial ϕ}{\partial t} = - \frac{\partial^{3} ϕ}{\partial x^{3}} - 6 ϕ \frac{\partial ϕ}{\partial x}

Keller–Segel chemotaxis

Slime moulds and cell movement

\frac{\partial u}{\partial t} = \nabla^{2} u - \nabla \cdot (χ (u) \nabla v) + f_{u} (u),

\frac{\partial v}{\partial t} = D \nabla^{2} v + f_{v} (u, v)

Perona–Malik equation

Image denoising via nonlinear anisotropic diffusion

\frac{\partial u}{\partial t} = \nabla \cdot (e^{- D | \nabla u |^{2}} \nabla u)

Reaction–cross-diffusion systems

Dark solitons and extended Turing spaces

\frac{\partial u}{\partial t} = \nabla \cdot (D_{u u} \nabla u + D_{u v} \nabla v) + a - u + u^{2} v,

\frac{\partial v}{\partial t} = \nabla \cdot (D_{v u} \nabla u + D_{v v} \nabla v) + b - u^{2} v

Hyperbolic reaction–diffusion systems

Turing wave instabilities

τ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} = D_{u} \nabla^{2} u + f (u, v),

τ \frac{\partial^{2} v}{\partial t^{2}} + \frac{\partial v}{\partial t} = D_{v} \nabla^{2} v + g (u, v)

Cyclic competition models

Rock, paper, scissors – spiral waves!

\frac{\partial u}{\partial t} = D_{u} \nabla^{2} u + u (1 - u - a v - b w)

\frac{\partial v}{\partial t} = D_{v} \nabla^{2} v + v (1 - b u - v - a w)

\frac{\partial w}{\partial t} = D_{w} \nabla^{2} w + w (1 - a u - b v - w)

Nonlinearly elastic beams

Exploring overdamped state-dependent stiffness

\frac{\partial y}{\partial t} = - \frac{\partial^{2}}{\partial x^{2}} [E (y) \frac{\partial^{2} y}{\partial x^{2}}]

Heterogeneous reaction-diffusion systems

Isolated patterns and moving spikes

\frac{\partial u}{\partial t} = \nabla^{2} u + a + G (x) + \frac{u^{2}}{v} - [b + H (x)] u,

\frac{\partial v}{\partial t} = D \nabla^{2} v + u^{2} - c v

Stochastic partial differential equations

Randomness in time and space

\frac{\partial u}{\partial t} = D \nabla^{2} u + f (u) + \frac{d W_{t}}{d t}

Vegetation patterns

Vegetation moving towards water

\frac{\partial w}{\partial t} = a - w - w n^{2} + v \frac{\partial w}{\partial x} + \nabla^{2} w,

\frac{\partial n}{\partial t} = w n^{2} - m n + \nabla^{2} n

Turing instabilities are not enough

Beyond the limits of linear theory

Spatial resilience in cancer immunotherapy

Turing patterning affects cancer treatment

\frac{\partial u}{\partial t} = δ_{u} \nabla^{2} u + α v - μ_{u} u + \frac{ρ_{u} u w}{1 + w} + σ_{u} + K_{u} t

\frac{\partial v}{\partial t} = \nabla^{2} v + v (1 - v) - u v / (γ_{v} + v)

\frac{\partial w}{\partial t} = δ_{w} \nabla^{2} w + \frac{ρ_{w} u v}{γ_{w} + v} - μ_{w} w + σ_{w} + K_{w} t

Diffusively coupled Lorenz model

Butterfly effects in space

\frac{\partial X}{\partial t} = D \nabla^{2} X + σ (Y - X)

\frac{\partial Y}{\partial t} = D \nabla^{2} Y + X (ρ - Z) - Y

\frac{\partial Z}{\partial t} = D \nabla^{2} Z + X Y - β Z

Diffusively coupled oscillators

Nonlinearity and oscillation in space

\frac{\partial^{2} X}{\partial t^{2}} = D \nabla^{2} X + μ (1 - X^{2}) \frac{\partial X}{\partial t} - X

Kovalevskaya on chaos

Playing in time and space

\frac{\partial ψ}{\partial t} = (D_{r} + i D_{i}) \nabla^{2} ψ + (a_{r} + i a_{i}) (1 - I_{S} (x, y)) ψ + (b_{r} + i b_{i}) ψ | ψ |^{2}

Hills and valleys

Effects of topography on models of water and vegetation

Flowing bacteria

A toy model of decay and advection in rivers