01 - ODEs and PDEs

ODE

Equation for an unknown function $y(x)$ of one variable $x$ in form

$$G(x,y(x),y^{\prime }(x),y^{\prime \prime }(x),\cdots ,y^{(n)}(x))=0$$

where $y^{(n)}(x)=\frac{d^{n}y}{d{x}^n}(x)$

This can be solved for the highest derivative of $t$ in the form

$$y^{(n)}(x)=F(x,y,y^{\prime },\cdots ,y^{(n-1})$$

Order of an ODE

Order $n$ of an ODE is order of the highest derivative that appears

Non Uniqueness of solutions to ODEs with initial values

Consider IVP

$$y^{\prime }(x)=3y(x{)}^{2/3},y(0)=0$$

By separation of variables you get

$$\int \frac{dy}{3y^{2/3}}=\int dx$$

Solving that you get

$$y(x)=(x+A{)}^3\text{ where }A\text{ is a constant}$$

Using initial condition $y(0)=0$ we get

1. Solution $y(x)=x^3$
2. Trivial Solution $y(x)=0$
3. Infinite Solutions for $c,d$ where $c\le 0\le d$ then

$$y_{c,d}(x):=\{\begin{array}{ll}(x-c{)}^3& \text{ for }x<c\\ 0& \text{ for }c\le x\le d\\ (x-d{)}^3& \text{ for }x>d\end{array}$$

All these functions satisfy the initial conditions and are differentiable everywhere

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General form of the IVP

$$y^{\prime }(x)=f(x,y(x))\text{ and }y(a)=b$$

Plane Autonomous System of ODEs

Pair of ODEs in the form

$$\begin{array}{r}\frac{dx}{dt}=X(x,y)\\ \frac{dy}{dt}=Y(x,y)\end{array}$$

Autonomous meaning there is no $t-$dependence in $X$ or $Y$
Plane meaning just two equations i.e. $(x,y)$

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First Order Semi-Linear PDE

$$P(x,y)\frac{\partial z}{\partial x}+Q(x,y)\frac{\partial z}{\partial y}=R(x,y,z(x,y))$$

for some unknown function $z=z(x,y)$

Generally assumed

1. $P(x,y)$ and $Q(x,y)$ are Lipschitz continuous in $x,y$
2. $R(x,y,z)$ is continuous and Lipschitz Continuous in $z$

Note that PDE is quasi-linear if $P,Q$ depend on $z$

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Second Order Semi-Linear PDEs

$$\underset{\text{principal part}}{\underbrace{a(x,y)u_{xx}+2b(x,y)u_{xy}+c(x,y)u_{yy}}}=f(x,y,u,u_x,u_y)$$

PDE is linear if $f$ is linear in $u,u_x,u_y$ otherwise is semi-linear
PDE is quasi-linear if $a,b,c$ also depend on $u,u_x,u_y$

Normal form for Second Order Elliptic PDE

$$u_{xx}+u_{yy}=f(x,y,u,u_x,u_y)$$

Normal Form for Second Order Parabolic PD

$$u_{xx}=F(x,t,u,u_t,u_x)$$