Documentation

# functionalDerivative

Functional derivative

## Syntax

``D = functionalDerivative(f,y)``

## Description

example

````D = functionalDerivative(f,y)` returns the Functional Derivative of the functional $F=\int f\left(x,y\left(x\right),y\text{'}\left(x\right)...\right)dx$ with respect to the function y = y(x), where x represents one or more independent variables. If `y` is a vector of symbolic functions, `functionalDerivative` returns a vector of functional derivatives with respect to the functions in `y`, where all functions in `y` must depend on the same independent variables.```

## Examples

### Find Functional Derivative

Find the functional derivative of the function given by $f\left(y\right)=y\left(x\right)\mathrm{sin}\left(y\left(x\right)\right)$ with respect to the function `y`.

```syms y(x) f = y*sin(y); D = functionalDerivative(f,y)```
```D(x) = sin(y(x)) + cos(y(x))*y(x)```

### Find Functional Derivative of Vector of Functionals

Find the functional derivative of the function given by $H\left(u,v\right)={u}^{2}\frac{dv}{dx}+v\frac{{d}^{2}u}{d{x}^{2}}$ with respect to the functions `u` and `v`.

```syms u(x) v(x) H = u^2*diff(v,x)+v*diff(u,x,x); D = functionalDerivative(H,[u v])```
```D(x) = 2*u(x)*diff(v(x), x) + diff(v(x), x, x) diff(u(x), x, x) - 2*u(x)*diff(u(x), x)```

`functionalDerivative` returns a vector of symbolic functions containing the functional derivatives of `H` with respect to `u` and `v`, respectively.

### Find Euler-Lagrange Equation for Spring

First find the Lagrangian for a spring with mass `m` and spring constant `k`, and then derive the Euler-Lagrange equation. The Lagrangian is the difference of kinetic energy `T` and potential energy `V` which are functions of the displacement `x(t)`.

```syms m k x(t) T = sym(1)/2*m*diff(x,t)^2; V = sym(1)/2*k*x^2; L = T - V```
```L(t) = (m*diff(x(t), t)^2)/2 - (k*x(t)^2)/2```

Find the Euler-Lagrange equation by finding the functional derivative of `L` with respect to `x`, and equate it to `0`.

`eqn = functionalDerivative(L,x) == 0`
```eqn(t) = - m*diff(x(t), t, t) - k*x(t) == 0```

`diff(x(t), t, t)` is the acceleration. The equation `eqn` represents the expected differential equation that describes spring motion.

Solve `eqn` using `dsolve`. Obtain the expected form of the solution by assuming mass `m` and spring constant `k` are positive.

```assume(m,'positive') assume(k,'positive') xSol = dsolve(eqn,x(0) == 0)```
```xSol = -C3*sin((k^(1/2)*t)/m^(1/2))```

Clear assumptions for further calculations.

`assume([k m],'clear')`

### Find Differential Equation for Brachistochrone Problem

The Brachistochrone problem is to find the quickest path of descent under gravity. The time for a body to move along a curve `y(x)` under gravity is given by

`$f=\sqrt{\frac{1+y{\text{'}}^{2}}{2gy}},$`

where g is the acceleration due to gravity.

Find the quickest path by minimizing `f` with respect to the path `y`. The condition for a minimum is

`$\frac{\delta f}{\delta y}=0.$`

Compute this condition to obtain the differential equation that describes the Brachistochrone problem. Use `simplify` to simplify the solution to its expected form.

```syms g y(x) assume(g,'positive') f = sqrt((1+diff(y)^2)/(2*g*y)); eqn = functionalDerivative(f,y) == 0; eqn = simplify(eqn)```
```eqn(x) = diff(y(x), x)^2 + 2*y(x)*diff(y(x), x, x) == -1```

This equation is the standard differential equation for the Brachistochrone problem.

### Find Minimal Surface in 3-D Space

If the function u(x,y) describes a surface in 3-D space, then the surface area is found by the functional

`$F\left(u\right)=\iint f\left(x,y,u,{u}_{x},{u}_{y}\right)dxdy=\iint \sqrt{1+{u}_{x}^{2}+{u}_{y}^{2}}dxdy,$`

where ux and uy are the partial derivatives of u with respect to x and y.

Find the equation that describes the minimal surface for a 3-D surface described by the function `u(x,y)` by finding the functional derivative of `f` with respect to `u`.

```syms u(x,y) f = sqrt(1 + diff(u,x)^2 + diff(u,y)^2); D = functionalDerivative(f,u)```
```D(x, y) = -(diff(u(x, y), y)^2*diff(u(x, y), x, x)... + diff(u(x, y), x)^2*diff(u(x, y), y, y)... - 2*diff(u(x, y), x)*diff(u(x, y), y)*diff(u(x, y), x, y)... + diff(u(x, y), x, x)... + diff(u(x, y), y, y))/(diff(u(x, y), x)^2... + diff(u(x, y), y)^2 + 1)^(3/2) ```

The solutions to this equation `D` describe minimal surfaces in 3-D space such as soap bubbles.

## Input Arguments

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Expression to find functional derivative of, specified as a symbolic variable, function, or expression. The argument `f` represents the density of the functional.

Differentiation function, specified as a symbolic function or a vector, matrix, or multidimensional array of symbolic functions. The argument `y` can be a function of one or more independent variables. If `y` is a vector of symbolic functions, `functionalDerivative` returns a vector of functional derivatives with respect to the functions in `y`, where all functions in `y` must depend on the same independent variables.

## Output Arguments

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Functional derivative, returned as a symbolic function or a vector of symbolic functions. If input `y` is a vector, then `D` is a vector.

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### Functional Derivative

Consider functionals

`$F\left(y\right)=\underset{\Omega }{\int }f\left(x,y\left(x\right),y\text{'}\left(x\right),y\text{'}\text{'}\left(x\right),...\right)dx,$`

where Ω is a region in x-space.

For a small change in the value of y, δy, the change in the functional F is

The expression $\frac{\delta f\left(x\right)}{\delta y}$ is the functional derivative of f with respect to y.