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## my Partial diﬀerential equations cheat sheet

May 28, 2020   Compiled on May 28, 2020 at 1:45am

2.1 Elliptic
2.2 Parabolic
2.3 Hyperbolic

### 1 introduction

These are part of my study notes on PDE’s.

Trying to classify PDE’s, here is current diagram. It is very large, but it is meant to include a summary of many methods in one place. Easier to view in a browser than in the pdf.

To derive the PDE, we start by setting up the state quantities and the ﬂow quantities, and relate these to each others by the use of the constitutive law. Then substiting this into the local conservation law, lead to the PDE.

 state quantity constitutive law ﬂow quantity density tension tempreture velocity pressure $$\Longleftrightarrow$$ momentum speciﬁc internal energy heat ﬂux entropy

### 2 linear PDE’s

#### 2.1 Elliptic

Some properties

1. Solution to the PDE represents steady state of $$u$$.
2. Only boundary conditions are used to solve. No initial conditions.
3. Relation to complex analytic functions: If $$f(z)=\phi (x,y)+i\psi (x,y)$$ is analytic, then $$\phi (x,y)$$ and $$\psi (x,y)$$ are solutions to Laplace pde’s
4. Solutions to Laplace PDE are called harmonic functions.
5. constitutive law: Either consider them as stationary process, or take the time dependent pde, and set those terms in that which depend on time to zero.

Examples of elliptic PDE’s

1. Laplace $$u_{xx}=0$$ or in general $$\nabla ^2 u=0$$
2. Poisson $$u_{xx}=-f(x)$$
3. Helmholtz in 1D $$u_{xx}+\lambda u(x) = -f(x)$$
4. Helmholtz in 2D $$u_{xx}+ u_{yy} + \lambda u(x,y) = -f(x,y)$$

#### 2.2 Parabolic

Some properties

1. Diﬀusion. Material spread is one speciﬁc example of diﬀusion. Here the state variable is the concentration of the diﬀusing matrial. The ﬂow quantity is its ﬂux. The constitutive law is Fickś law.
2. Heat spread. Here the state variable is the temprature, and the ﬂow quantity is the heat ﬂux. The constitutive law is Fourierś law.
3. Stiﬀ PDE, hence requires small time step, solved using implicit methods, not explicit for stability.
4. Numerically, use Crank-Nicleson, in 2D, can use ADI.
5. Requires initial and boundary conditions to solve.

Examples of parabolic PDE’s

1. Diﬀusion. $$u_{t}-Du_{xx}=0$$ where $$D$$ is the diﬀusion constant, must be positive quantity. For heat PDE, $$D$$ is the thermal diﬀusivity $$D=\kappa /{c_p\rho }$$ where $$\kappa$$ is thermal conductivity, $$c_p$$ is speciﬁc heat capacity, $$\rho$$ is density of medium.
2. In higher spatial dimension $$u_{t}-D\nabla ^2 u=0$$
3. Foller-Plank, Black-Sholes PDEs
4. Diﬀusion-Reaction $$u_t-Du_{xx}=F(u(x,t))$$ where $$F(u(x,t))$$ is the reaction term, which can be stiﬀ or not. Examples
1. Fischer equation, nonlinear PDE for modeling population growth. $$u_t-Du_{xx}=r u(x,t) (1-\frac{u(x,t)}{K})$$ where $$K$$ is carrying capacity, and $$r$$ is growth rate.

#### 2.3 Hyperbolic

Some properties

1. Advection PDE (or Transport or convection?). $$u_t+a u_x=0$$, Transport or drift of conserved substance (pollutant) in Fluid or Gas where $$a$$ is speed of ﬂuid. Analytic solution is $$u(x,t)=f(x-at)$$ where $$f(x)=u(x,0)$$ is the initial conditions.
2. The state variable is the concentration $$u$$ of the contaminant, and the ﬂow quantity is its ﬂux $$\phi$$. The constitutive law is $$\phi =cu$$.
3. Wave equation $$u_{tt}=c^2 u_{xx}$$. Analytic solution is $$u(x,t)=\frac{1}{2} [f(x-ct)+f(x+ct)]+\frac{1}{2c} \int _{x-ct}^{x+ct} \! g(y) \, \mathrm{d}y$$ where $$f(x)=u(x,0)$$ and $$g(x)=u_t(x,0)$$.

Examples

2. non-homogenouse advection and wave: $$u_t+a u_x=f(x,t)$$ and $$u_{tt}=c^2 u_{xx}+f(x,t)$$.
3. Klein-Gordon $$u_{tt}=c^2 u_{xx}-bu$$
4. Telegraphy $$u_{tt}+ku_t=c^2 u_{xx}+bu$$

### 3 hints

Characteristics are curves in the space of the independent variables along which the governing PDE has only total diﬀerentials

### 4 references

1. Elements of partial diﬀerential Equations, Pavel Drabek and Gabriela Holubova, 2007.
2. Applied partial diﬀerental equations. 4th edition, Richard Haberman
3. http://www.phy.ornl.gov/csep/pde/node3.html
4. http://www.me.metu.edu.tr/courses/me582/files/PDE_Introduction_by_Hoffman.pdf
5. http://en.wikibooks.org/wiki/Partial_Differential_Equations/Introduction_and_Classifications
6. http://www.scholarpedia.org/article/Partial_differential_equation
7. http://how.gi.alaska.edu/ao/sim/chapters/chap3.pdf good discussion on classiﬁcation via Characteristics lines
8. http://gwu.geverstine.com/pde.pdf table on classiﬁcation, diagram for discriminant sign