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March 11, 2019 Compiled on March 11, 2019 at 3:58am

1 Heat PDE

1.1 Homogeneous Heat PDE in 1D. bounded domain. (No source term)

1.1.1 Pure diﬀusion. Both ends at zero temperature

1.1.2 Pure diﬀusion. Left end at non zero temperature, right end zero temperature

1.1.3 Pure diﬀusion. Both ends insulated

1.1.4 Pure diﬀusion. Left end insulated, right end at non-zero temperature

1.1.5 Diﬀusion-Reaction. Case when both ends insulated

1.1.6 Pure diﬀusion. Left end nonhomogeneous time dependent Neumann, right end at zero temperature

1.1.7 Pure diﬀusion. Left end nonhomogeneous and time dependent Dirichlet, right end at zero temperature

1.1.8 Pure diﬀusion. Both ends at ﬁxed non-zero temperature (not time dependent)

1.1.9 Pure diﬀusion. Both ends at temperature that is time dependent

1.1.10 Diﬀusion-Reaction. Both ends at zero temperature, no source. reaction term is \(xu(x,t)\)

1.1.11 Diﬀusion-Convection. Both ends at zero temperature, no source. (also called advection)

1.2 Nonhomogeneous heat PDE in 1D bounded domain. (Source term present)

1.2.1 Both ends at temperature that depends on time. Source term depends on x only

1.2.2 Both ends at temperature that depends on time. Source term depends on x and t

1.2.3 Both ends at zero temperature. Source term depends on \(x\) only

1.2.4 Both ends at zero temperature. Source term depends on \(x\) and \(t\)

1.2.5 Left end at ﬁxed temperature, right end zero temperature. Source term depends on \(x\) and \(t\).

1.2.6 Both ends at ﬁxed temperature (not time dependent). Source term depends on x only

1.2.7 Both ends at ﬁxed temperature. Source term depends on x and t

1.3 Homogeneous Heat PDE in 1D, Semi-inﬁnite domain, No source. Laplace Transform method

1.3.1 Left end at zero temperature, No source term, zero initial conditions

1.3.2 Left end at temperature that depends on time. No source term, zero initial conditions

1.4 Heat PDE in 2D

1.4.1 Heat PDE in 2D inside a Disk, zero temperature at boundary (circumference) of disk.

1.4.2 Heat PDE in 2D inside a Disk, insulated boundary conditions

2 Wave PDE

2.1 Homogeneous Wave PDE in 1D on ﬁnite domain (plugged string) series solutions

2.1.1 Both ends of string ﬁxed, initial position zero, Fourier series solution

2.1.2 Both ends of string ﬁxed, initial velocity zero, Fourier series solution

2.1.3 Both ends of string ﬁxed, initial velocity zero, d’Alembert solution

2.1.4 Left end ﬁxed, right end free, initial velocity zero, Fourier series solution

2.1.5 Left end ﬁxed, right end free. Initial velocity zero. Damping present.

2.1.6 Both ends ﬁxed. Initial velocity zero. Dispersion term present.

2.1.7 Both ends ﬁxed. Initial velocity zero. Telegraphy PDE (dispersion and decay)

2.2 Homogeneous Wave PDE in 1D, semi-inﬁnite domain, no source

2.2.1 Left end free, initial velocity zero

2.2.2 Left end ﬁxed, initial velocity zero

2.3 Wave PDE in 1D, inﬁnite domain, D’Alembert solution

2.3.1 Initial velocity zero case

2.4 Homogeneous Wave PDE in 2D. No source

2.4.1 Rectangular membrane. Fixed on all edges

2.4.2 circular membrane, no circular symmetry, example 1

2.4.3 circular membrane, no circular symmetry, speciﬁc problem from exam

2.4.4 circular membrane, with circular symmetry

2.5 Solitons wave animation (non-linear wave PDE)

2.6 Wave PDE with inhomogeneous boundary conditions, 1D (string)

3 Laplace and Poisson PDE

3.1 Laplace PDE inside disk

3.2 Laplace PDE outside disk

3.3 Laplace PDE inside square. 3 edges homogeneous boundary conditions.

3.4 Poisson PDE, Cartesian coordinates, source term depends on y

4 Burger’s PDE

4.1 Example 1

5 Misc. PDE’s

5.1 FitzHugh-Nagumo in 2D

5.1.1 Example 1

5.1.2 Example 2

6 Appendix

6.1 Summary table

6.2 Using Mathematica to obtain the eigenvalues and eigenfunctions for heat PDE in 1D

6.2.1 \(u(0)=0,u(L)=0\)

6.2.2 \(u^{\prime }(0)=0,u^{\prime }(L)=0\)

6.2.3 \(u^{\prime }(0)=0,u(L)=0\)

6.2.4 \(u(0)=0,u^{\prime }(L)=0\)

6.2.5 \(u(0)=0,u\left ( L\right ) +u^{\prime }(L)=0\)

6.2.6 \(u\left ( 0\right ) +u^{\prime }(0)=0,u^{\prime }(L)=0\)

6.2.7 \(u(-L)=0,u\left ( L\right ) =0\)

1.1 Homogeneous Heat PDE in 1D. bounded domain. (No source term)

1.1.1 Pure diﬀusion. Both ends at zero temperature

1.1.2 Pure diﬀusion. Left end at non zero temperature, right end zero temperature

1.1.3 Pure diﬀusion. Both ends insulated

1.1.4 Pure diﬀusion. Left end insulated, right end at non-zero temperature

1.1.5 Diﬀusion-Reaction. Case when both ends insulated

1.1.6 Pure diﬀusion. Left end nonhomogeneous time dependent Neumann, right end at zero temperature

1.1.7 Pure diﬀusion. Left end nonhomogeneous and time dependent Dirichlet, right end at zero temperature

1.1.8 Pure diﬀusion. Both ends at ﬁxed non-zero temperature (not time dependent)

1.1.9 Pure diﬀusion. Both ends at temperature that is time dependent

1.1.10 Diﬀusion-Reaction. Both ends at zero temperature, no source. reaction term is \(xu(x,t)\)

1.1.11 Diﬀusion-Convection. Both ends at zero temperature, no source. (also called advection)

1.2 Nonhomogeneous heat PDE in 1D bounded domain. (Source term present)

1.2.1 Both ends at temperature that depends on time. Source term depends on x only

1.2.2 Both ends at temperature that depends on time. Source term depends on x and t

1.2.3 Both ends at zero temperature. Source term depends on \(x\) only

1.2.4 Both ends at zero temperature. Source term depends on \(x\) and \(t\)

1.2.5 Left end at ﬁxed temperature, right end zero temperature. Source term depends on \(x\) and \(t\).

1.2.6 Both ends at ﬁxed temperature (not time dependent). Source term depends on x only

1.2.7 Both ends at ﬁxed temperature. Source term depends on x and t

1.3 Homogeneous Heat PDE in 1D, Semi-inﬁnite domain, No source. Laplace Transform method

1.3.1 Left end at zero temperature, No source term, zero initial conditions

1.3.2 Left end at temperature that depends on time. No source term, zero initial conditions

1.4 Heat PDE in 2D

1.4.1 Heat PDE in 2D inside a Disk, zero temperature at boundary (circumference) of disk.

1.4.2 Heat PDE in 2D inside a Disk, insulated boundary conditions

2 Wave PDE

2.1 Homogeneous Wave PDE in 1D on ﬁnite domain (plugged string) series solutions

2.1.1 Both ends of string ﬁxed, initial position zero, Fourier series solution

2.1.2 Both ends of string ﬁxed, initial velocity zero, Fourier series solution

2.1.3 Both ends of string ﬁxed, initial velocity zero, d’Alembert solution

2.1.4 Left end ﬁxed, right end free, initial velocity zero, Fourier series solution

2.1.5 Left end ﬁxed, right end free. Initial velocity zero. Damping present.

2.1.6 Both ends ﬁxed. Initial velocity zero. Dispersion term present.

2.1.7 Both ends ﬁxed. Initial velocity zero. Telegraphy PDE (dispersion and decay)

2.2 Homogeneous Wave PDE in 1D, semi-inﬁnite domain, no source

2.2.1 Left end free, initial velocity zero

2.2.2 Left end ﬁxed, initial velocity zero

2.3 Wave PDE in 1D, inﬁnite domain, D’Alembert solution

2.3.1 Initial velocity zero case

2.4 Homogeneous Wave PDE in 2D. No source

2.4.1 Rectangular membrane. Fixed on all edges

2.4.2 circular membrane, no circular symmetry, example 1

2.4.3 circular membrane, no circular symmetry, speciﬁc problem from exam

2.4.4 circular membrane, with circular symmetry

2.5 Solitons wave animation (non-linear wave PDE)

2.6 Wave PDE with inhomogeneous boundary conditions, 1D (string)

3 Laplace and Poisson PDE

3.1 Laplace PDE inside disk

3.2 Laplace PDE outside disk

3.3 Laplace PDE inside square. 3 edges homogeneous boundary conditions.

3.4 Poisson PDE, Cartesian coordinates, source term depends on y

4 Burger’s PDE

4.1 Example 1

5 Misc. PDE’s

5.1 FitzHugh-Nagumo in 2D

5.1.1 Example 1

5.1.2 Example 2

6 Appendix

6.1 Summary table

6.2 Using Mathematica to obtain the eigenvalues and eigenfunctions for heat PDE in 1D

6.2.1 \(u(0)=0,u(L)=0\)

6.2.2 \(u^{\prime }(0)=0,u^{\prime }(L)=0\)

6.2.3 \(u^{\prime }(0)=0,u(L)=0\)

6.2.4 \(u(0)=0,u^{\prime }(L)=0\)

6.2.5 \(u(0)=0,u\left ( L\right ) +u^{\prime }(L)=0\)

6.2.6 \(u\left ( 0\right ) +u^{\prime }(0)=0,u^{\prime }(L)=0\)

6.2.7 \(u(-L)=0,u\left ( L\right ) =0\)