Differential equations

This page examines basic forms of differential equations as these are often used to build up more sophisticated numerical solutions to complex non-linear partial differential equations (PDEs).

Homogeneous vs heterogeneous forms

A homogeneous differential equation only includes terms that have factors of the solution variable and its derivatives.

Homogeneous:

$\displaystyle\sum_{n=0}^{N}a_n(x)\frac{\partial^n u}{\partial x^n}=0$

Heterogeneous:

$\displaystyle\sum_{n=0}^{N}a_n(x)\frac{\partial^n u}{\partial x^n}=g(x)$

Linear vs non-linear forms

A linear differential equation is one that can be written as linear operator.

$\mathcal{L}u=\displaystyle\sum_{n=0}^{N}a_n(x)\frac{\partial^n u}{\partial x^n}$

Where

$\mathcal{L}=\displaystyle\sum_{n=0}^{N}a_n(x)\frac{\partial^n}{\partial x^n}$

Equation order

The order of a differential equation refers to the maximum level of differentiation involved. $\frac{\partial^n}{\partial x^n}$ is the $n$-th order differential operator.

Specific examples

First order, linear, homogeneous

$\frac{\partial u}{\partial t}=au$

In this case we have the family of trivial solutions, of which a linear combination is also a solution:

$u=ce^{at}$

$u=\displaystyle\sum_{n=0}^{N}c_n e^{a_n t}$

Where $c$ is solved for using boundary conditions.

First order, linear operator, constant heterogeneous term

$\frac{\partial u}{\partial t}=b+au$

$u=c_1e^{at}+c_0$

We then substitute this into the ODE:

$c_1ae^{at}=b+a(c_1e^{at}+c_0)$

And solve for $c_0$:

$0=b+ac_0$

$c_0=-\frac{b}{a}$

So we have a solution of the form:

$u=c_1e^{at}-\frac{b}{a}$

First order, linear operator, linear heterogeneous term

$\frac{\partial u}{\partial t}=b+au+wt$

$u=c_1e^{at}+c_0+c_2t$

$c_1ae^{at}+c_2=b+ac_1e^{at}+ac_0+ac_2t+wt$

$c_2=ac_0+b$

$ac_2t+wt=0$

$c_2=-\frac{w}{a}$

$c_0=-\frac{w}{a^2}-\frac{b}{a}$

Multivariate Linear ODE

We can extend this to a system of linear ODEs as follows:

$\frac{\partial\mathbf{u}}{\partial t}=\mathbf{A}\mathbf{u}+\mathbf{b}$

For the solution we use the same form as that for a scalar equation/function, exploiting the concepts used in matrix exponentiation (assuming certain conditions on $\mathbf{A}$).

$\mathbf{u}=c_1e^{\mathbf{A}t}+\mathbf{c_0}$

$\mathbf{A}=\mathbf{U}\Sigma\mathbf{U}^*$

Substituting this into the ODE we get.

$c_1\mathbf{A}e^{\mathbf{A}t}=\mathbf{A}c_1e^{\mathbf{A}t}+\mathbf{A}\mathbf{c}_0+\mathbf{b}$

Then solving for $\mathbf{c}_0$ we get

$\mathbf{c}_0=-\mathbf{A}^{-1}\mathbf{b}$

Which is analogous to what we have above. So we end up with the following solution family:

$\mathbf{u}=c_1e^{\mathbf{A}t}-\mathbf{A}^{-1}\mathbf{b}$

Linear time dependent PDEs

We are solving for a function that is both dependent upon time and space, $u(x,t)$.

$\frac{\partial u}{\partial t}=au+bx$

Directional derivatives

The Gateaux derivate is a form of directional differentiation. Under reasonable assumptions it is equal the inner product of the gradient and direction, like other definitions of directional differentiation.

https://en.wikipedia.org/wiki/Gateaux_derivative