Understanding Partial Differential Equations: Key Concepts and Applications

Posted on Jan 28, 2025 in Mathematics

Key Partial Differential Equations (PDEs)

Wave Equation:                       u_tt-c²u_xx=0

Laplace’s Equation:                 u_xx+u_yy=0

Heat Equation:                         u_t=ku_xx

Diffusion Equation:                  u_t-Du_xx=0

Transport/Advection Equation: u_t+cu_x=0

Advection-Diffusion Equation: u_t+cu_x-Du_xx=0

Advection-Diffusion-Decay Equation:

                                                    u_t+cu_x-Du_xx=-ʎu

Completing the Square

ax²+bx+c=0 to a(x+d)²+e=0

d=b/2a

e=c-b²/2a

Section 1.1: PDE Models

PDE integration yields arbitrary functions, whereas ODE integration yields constants.

Cauchy Problems: ϵ x, no boundary conditions, initial condition(s) are needed.

Operator Functions

Linear when: L(u+w)=L(u)+L(w) & L(cu)=cL(u)

Section 1.2: Conservation Laws

Conservation law: The rate at which a quantity changes is equal to the rate of flow (flux) across the boundary plus the rate at which it is created or destroyed.

• u=density (amount/volume)
• AФ=flux, amount crossing cross section at location x at time t (positive if to the right, negative if to the left)
• f(x,t)=sink (negative) or source (positive)

u_t+Ф_x-f=0 conservation law states this must equal zero

Change of Coordinates->ξ=x-ct, τ=t   ;   u(x,t)->U(ξ, τ)

 u_t=U_ξ ξ_t+U_ττ_t, u_x=U_ξ if ξ=x-ct, τ=t

For example:

u_t+cu_x+au=f -> U_τ+aU=F(ξ, τ)

 if u_t+u_x-3u=t -> U_τ-U=τ -> integrate and then return to normal coordinates

When c=f(x,t), dx/dt=c, separate and integrate and solve for constant.

For example:

if u_t+xtu_x=0, c=xt

dx/dt=xt

dx/x=tdt

lnx=t²/2+c

c=ξ=lnx-t²/2                            

Section 1.3: Diffusion

Fick’s Law: Ф=-Du_x

Fourier’s Law: Ф=-KƟ_x (K=thermal conductivity, Ɵ(x,t)=temperature assuming energy is proportional to temperature)

If u_xx>0, or concentration profile is concave up, u_t>0

Dirichlet Condition: u(0,t)=g(t), t>0

Neumann Condition: =Ku_x(0,t)=h(t), t>0

            If h(t)=0 it is insulated and homogeneous.

Newton’s Law of Cooling: -Ku_x(0,t)=-ɳ(u(0,t)-Ѱ(t)), t>0

            Ѱ(t)=environment temperature

            ɳ=constant of proportionality (heat loss factor)

Steady state-> time derivatives vanish, integrations are constants not functions.

Section 1.4: Diffusion and Randomness

Fischer’s Equation: u_t=Du_xx+ru(1-u/K)

• Advection graph– shifts
• Diffusion graph- “smears”
• Advection diffusion– shifts and “smears”

Point source=fundamental solutions-> those that satisfy boundary conditions=Green’s functions

Section 1.5: Vibrations and Acoustics

Often only assume movement in one direction (string being plucked, assume no horizontal, just vertical movement).

General Solution to the Wave Equation (u_tt=c²u_xx): u(x,t)=F(x-ct)+G(x+ct)

Section 1.7: Heat Conduction in Higher Dimensions

The Laplacian: Δu= ∇∇u= ∇²u=u_xx+u_yy+u_zz

Gradient acts on a scalar to make a vector:

grad(u)= ∇u=(u_x,u_y,u_z) ; u is scalar, ∇u is vector.

Divergence acts on a vector to make it scalar:

div(Ф)= ∇Ф

div(grad(u))= ∇·(u_x,u_y,u_z)= u_xx+u_yy+u_zz

            dot product

Divergence of a scalar and a vector yields a scalar:

div(u*ṽ)= ṽ·grad(u) + udiv(ṽ)

Can’t take divergence of a scalar, take gradient.

dV=dxdydz

Divergence Theorem: , ɳ=normal vector

Section 1.8: Laplace’s Equation

u(x,y)=(1/4)*[u(x-h,y)+u(x+h,y)+u(x,y-h)+u(x,y+h)] -temperature>

The Maximum Principle: If u is not a constant function, then the maximum and minimum values of u are attained on the boundary of D.

u(x,y)->u(r)

r=

u_x=(d/dx) u(r)=u_r (dr/dx)

u_y=(d/dy) u(r)=u_r(dr/dy)

Solve for all u derivatives and plug in to original equation; solve for u(r) using boundary conditions.

Section 1.9: Classification of PDEs

Au_xx+Bu_xt+Cu_tt+F(x,t,u,u_x,u_t)=0

D=B²-4AC, the discriminant

ξ=ax+bt, τ=cdt

D>0, hyperbolic (wave)

If a=c=1

b=(-B+sqrt(D))/2C

d=(-B-sqrt(D))/2C

if C=0, b=d=1

a=(-B+sqrt(D))/2A

c=(-B-sqrt(D))/2A

D<0, elliptic (Laplace’s)

D=0, parabolic (diffusion)

If a=c=1

d=-B/2C

b=0                                                  

u_x= U_ξξ_x+U_ττ_x =aU_ξ+cU_τ

u_t = U_ξξ_t+U_ττ_t =bU_ξ+dU_τ

u_xx= a²U_ξξ+2acU_ξτ+c²U_ττ

u_tt= b²U_ξξ+2bdU_ξτ+d²U_ττ

u_xt= abU_ξξ+(ad+cb)U_ξτ+cdU_ττ

Au_xx+Bu_xt+Cu_tt= U_ξξ(Aa²+Bab+Cb²)+U_ξτ(2Aac+B(ad+cb)+C(2bd))+U_ττ(Ac²+Bcd+Cd²)

Section 2.1: Cauchy Problem for the Heat Equation

Cauchy problem=pure initial value problem

    Heat Kernel/Fundamental Solution

G(x,t)=      Temperature surface from x=0 and t=0

u(x,t)=

erf(z)=

erfc= =1-erf

*Don’t forget to change limits and derivatives when replacing r value.

Section 2.2: Cauchy Problem for the Wave Equation

u(x,t)=F(x-ct)+G(x+ct) general solution

d’Alembert’s Formula

u(x,t)=

u(x,0)=f(x) initial displacement

u_t(x,0)=g(x) initial velocity

Section 2.3: Well-Posed Problems

A problem is well-posed if:

1. (existence) it has a solution

2. (uniqueness) the solution is unique

3. (stability) the solution depends continuously on the initial and/or boundary data

If a problem does not meet these requirements, it is ill-posed.

Steps

1.Plug in 0 for t in u(x,0)

2.Set f(x) equal to 1 or another arbitrary number to solve for x.

3.Change n to a large number, like 100, and plug in to f(x) (where t=0).

4.Compare value calculated to value expected (should be close).

5.Now pick arbitrary numbers for t in u(x,t) and set n at the same large number—should be a large number, and incorrect, thus ill-posed.

Section 2.4: Semi-Infinite Domains

Even Functions: f(-x)=f(x)

Odd Functions: -f(x)=f(-x)

Break apart integrals in sections of different initial conditions, use dummy variables for -y when necessary.

Section 2.5: Duhamel’s Principle

PDE contains a source term, f(x,t)

Duhamel’s Principle

The solution to the problem,

y’(t)+ay=F(t), t>0, y(0)=0

is given by

y(t)=

where w(t, τ) solves the homogeneous problem,

w’(t;τ)+aw(t;τ)=0, t>0; w(0;τ)=F(τ)

u(x,t)=

where w(x,t;τ) solves

w_t=kw_xx

w(x,0;τ)=f(x, τ)

u(x,t)=

Section 2.6: Laplace Transforms

Linear

L(u+v)=L(u)+L(v)

L(cu)=cL(u)

L^-1(U)=u

Convolution theorem

L(u*v)(s)=U(s)V(s)

Where,

(u*v)(t)=

Replace given equations with Laplace transforms, solve, and revert back.

Must be bounded with exponentials (set function to 0 in front of exponential).

Solve for remaining function with boundary condition(s).

t changes, x does not.

Section 2.7: Fourier Transforms

x changes, t does not.

Schwartz Class of Functions

S=the set of rapidly decreasing functions of all real numbers that have continuous derivatives of all orders.

Definition of S: Rapidly decreasing functions are functions that along with all their derivatives decay to zero as x->infinity faster than any power function.

Integration by Parts

u=f(x) given -> take derivative for du

du=f’(x)

v=g(x)

dv=g’(x) given -> integrate for v