# Contents of the Exact Analytical Conduction Toolbox

### Heat Equation, Cartesian, 1D Infinite Composite Body

X0C10 Infinite Composite Body of Two Dissimilar Materials Joined at x=0 with an Impusive Heat Source (Green's Function)

X0C30 Infinite Composite Body Separated at x=0 by a Thermal Resistance With an Impulse Heat Source(Green's Function)

### Heat Equation, Cartesian, 1D Semi-infinite Body

X10B(3,J)T0 Power variation of surface temperature on semi-infinite body

X10B0T0G5 Semi-infinite Cartesian Body with a Zero Surface Temperature and Volumetric Heating

X10B1T0 Semi infinite Cartesian body with step change in boundary temperature

X20 Semi-Infinite Slab Insulated on the Boundary x=0 with an Impulsive Heat Source (Green's Function)

X20B(3,J)T0 Semi-infinite body with power law heat flux at x = 0

X20B-T0 Semi-infinite body with piecewise-in-time exponential surface heat flux for a reciprocating plasma probe

X20B0T0G5 Semi-infinite Cartesian Body with an Insulated Surface and Volumetric Heating

X20B1T0 Semi infinite Cartesian body with jump in boundary heat flux

X20B4T0 Semi-infinite slab with exponential heat flux at x = 0

X30B(3,J)T0 Semi-infinite body heated by surface convection with power-law varying fluid temperature

X30B1T0 Semi infinite Cartesian body with jump in boundary ambient temperature

X40B(3,J)T0 Semi-infinite body with high-conductivity surface layer with power-law varying heat flux

X40B0T01 Semi-infinite Cartesian body in perfect contact with a thermally insulated high-conductivity surface layer having zero initial temperature

X40B0T10 Semi-infinite Cartesian body in perfect contact with a thermally insulated high-conductivity surface layer having different uniform initial temperatures

X40B1T00 Semi infinite Cartesian body with high-conductivity surface layer with jump in boundary heat flux

X60B1T00 Semi-infinite Cartesian body initially at zero temperature and in imperfect contact with a high-conductivity surface layer with zero initial temperature and subject to a jump in boundary heat flux

### Heat Equation, Cartesian, 1D Layered Semi-infinite Body

### Heat Equation, Cartesian, 1D Slab Body

X11B(3,J)0T0 Power variation at x = 0 for the X11 case

X11B10T0 Slab with jump in temperature at one boundary, zero temperature at other boundary

X11B60T0 Slab body with cosine-periodic temperature variation at x=0, and zero temperature at x=L

X12B(3,J)0T0 Power variation at the surface for the X12 case

X12B10T0 Slab with jump in temperature at one boundary, zero heat flux at other boundary

X12B60T0 Slab body with cosine-periodic temperature variation at x=0, and insulated boundary at x=L.

X13B10T0 Slab with jump in temperature at one boundary, dissipating heat by convection into a zero temperature environment at x=L

X13B60T0 Slab body with cosine-periodic temperature variation at x=0 and convection at x=L.

X21B(3,J)0T0 Slab body with power law heat flux at x = 0 and fixed temperature at x = L

X21B10T0 Slab with jump in heat flux at one boundary, zero temperature at other boundary

X21B60T0 Slab with cosine-periodic heating at one boundary, zero temperature at other boundary.

X22B00T0Gx5 Slab with partial volumetric heat source and zero heat flux at both boundaries

X22B10T0 Slab with jump in heat flux at one boundary, zero heat flux at other boundary

X22B60T0 Slab with cosine-periodic heating at one boundary, insulated at other boundary

X23B10T0 Slab with jump in heat flux at one boundary, dissipating heat by convection into a zero temperature environment at x=L.

X23B40T0 Slab with exponential heat flux at one boundary, convection at other boundary

X23B60T0 Slab body with cosine-periodic heat flux at x=0 and convection at x=L

X31B10T0 Slab heated by convection with jump in environment temperature at one boundary and zero temperature at other boundary

X31B60T0 Slab body with cosine-periodic fluid convection at x=0 and zero temperature at x=L.

X32B10T0 Slab heated by convection with environment at constant temperature on one boundary and thermally insulated on the other boundary

X32B60T0 Slab body with cosine-periodic fluid convection at x=0 that is thermally insulated at x=L.

X33B00Tx4 Slab body with exponential initial condition at x=0 from laser flash and convection into ambient environment at x=0 and x=L

X33B11T1G1 Slab body with prescribed convection, surface heat flux, volumetric source, and initial temperatures

X33B50G(x5)T5 Slab body with piecewise-in-time convection at x = 0 and homogeneous convection at x = L, volumetric heating that is piecewise-in-space and constant-in-time, and piecewise initial condition.

X33B60T0 Slab body with cosine-periodic fluid convection at x=0 and convection cooling at x=L.

X33B70T0 Slab body initially at ambient temperature with flash heating at x=0 and convection into ambient environment at x=0 and x=L

X42B10T00 Slab in perfect contact with a high-conductivity surface layer with zero initial temperature and subject to a jump in boundary heat flux, initially at zero temperature and zero heat flux on the back side

X62B10T00 Slab in imperfect contact with a high-conductivity surface layer with zero initial temperature and subject to a jump in boundary heat flux, initially at zero temperature and zero heat flux on the back side

### Heat Equation, Cartesian, 1D Two-layer Slab

X1C11B00G11 Two-layer slab with volumetric heat source, zero temperature at both boundaries

X1C11B00T11 Two-layer slab with perfect contact between layers; zero temperature at first boundary and second boundary

X1C11B10T0 Two-layer slab with perfect contact between layers; jump in temperature at first boundary and zero temperature at second boundary

X1C12B00G11 Two-layer slab with volumetric heat source, zero temperature at one boundary, zero heat flux at other boundary

X1C12B00T11 Two-layer slab with perfect contact between layers; zero temperature in one boundary, zero heat flux at other boundary

X1C12B10T0 Two-layer slab with perfect contact between layers; jump in temperature in one boundary, zero heat flux at other boundary

X1C13B00G11 Two-layer slab with volumetric heat source, zero temperature at one boundary, dissipating heat by convection to environment at x=L

X1C13B00T11 Two-layer slab with perfect contact between layers; with zero temperature at one boundary, dissipating heat by convection into a zero temperature environment at x=L

X1C13B10T0 Two-layer slab with perfect contact between layers; with jump in temperature at one boundary, dissipating heat by convection into a zero temperature environment at x=L

X2C11B00G11 Two-layer slab with volumetric heat source, heat flux at one boundary, zero temperature at other boundary

X2C11B00T11 Two-layer slab with perfect contact between layers; with zero heat flux at one boundary, zero temperature at other boundary

X2C11B10T0 Two-layer slab with perfect contact between layers; with heat flux at one boundary, zero temperature at other boundary

X2C12B00G11 Two-layer slab with volumetric heat source, zero heat flux at both boundaries

X2C12B00T11 Two-layer slab with perfect contact between layers; with zero in heat flux at one boundary, zero heat flux at other boundary

X2C12B10T0 Two-layer slab with perfect contact between layers; with jump in heat flux at one boundary, zero heat flux at other boundary

X2C13B00G11 Two-layer slab with volumetric heat source, heat flux at one boundary, dissipating heat by convection into a zero temperature environment at x=L

X2C13B00T11 Two-layer slab with perfect contact between layers; with zero heat flux at one boundary, dissipating heat by convection into a zero temperature environment at x=L

X2C13B10T0 Two-layer slab with perfect contact between layers; with jump in heat flux at one boundary, dissipating heat by convection into a zero temperature environment at x=L

X3C11B00G11 Two-layer slab with volumetric heat source, heated by convection with jump in environment temperature at one boundary, zero temperature at other boundary

X3C11B00T11 Two layer slab with non-zero initial condition, cooled by convection at one boundary and zero temperature at the other boundary

X3C11B10T0 Two-layer slab with perfect contact between layers; heated by convection with jump in environment temperature at one boundary and zero temperature at other boundary

X3C12B00G11 Two-layer slab with volumetric heat source, heated by convection with jump in environment temperature at one boundary, zero flux at other boundary

X3C12B00T11 Two layer slab with non-zero initial condition, cooled by convection at one boundary and zero heat flux at the other boundary

X3C12B10T0 Two-layer slab with perfect contact between layers; heated by convection with jump in environment temperature at one boundary and zero heat flux at other boundary

X3C13B00G11 Two-layer slab with volumetric heat source, heated by convection with jump in environment temperature at one boundary, dissipating heat by convection into a zero temperature environment at x=L

X3C13B00T11 Two layer slab with non-zero initial condition, cooled by convection at both boundaries

X3C13B10T0 Two-layer slab with perfect contact between layers; heated by convection with jump in environment temperature at one boundary and dissipating heat by convection into a zero temperature environment at x=L

X3C13B70T0 Two-layer slab with flash heating on one surface and convective cooling on both surfaces

### Heat Equation, Cartesian, 1D Fin

X20B1F0T0 Straight semi-infinite fin with jump in heat flux at base

X21B10F0T0 Straight fin with jump in heat flux at base and constant-temperature tip

X22B10F0T0 Straight fin with jump in flux at base of the fin and insulated tip

X23B10F0T0 Straight fin with jump in heat flux at base and convection at tip

X33B10F0 Straight fin, steady, with contact conductance at base and convection losses at tip

XIJB11F1G1T1 Straight fin with prescribed convection, surface heat flux, volumetric source and initial condition

### Heat Equation, 1D Two Layer Cylinder

R0C11B0T00G11 Two-layer concentric cylinders with prescribed surface temperature and volumetric heat sources

R0C11B0T11 Two-layer concentric cylinders with fixed boundary temperature and initial temperatures

R0C12B0T11 Two-layer concentric cylinder with adiabatic outer surface and prescribed initial temperatures

R0C12B1T00G11 Two-layer concentric cylinders with volumetric heat sources and specified heat flux on the outer surface

R0C13B0T00G11 Two-layer concentric solid cylinder with a convective surface and prescribed volumetric heat sources

R0C13B0T11 Two-layer concentric solid cylinder with a convective surface and prescribed initial temperatures

R1C11B00G11T00 Two layer cylinder with zero boundary temperatures and prescribed volumetric sources

R1C11B11T11 Two layer hollow cylinder with prescribed surface temperatures and prescribed initial temperatures

R1C12B00T11 Two-layer hollow cylinder with inner surface at zero temperature, outer surface insulated, and prescribed initial temperatures.

R1C12B11G11T00 Two layer hollow cylinder with prescribed surface temperature at inner surface prescribed flux at outer surface, and prescribed volumetric heating

R1C13B00T00G11 Two layer concentric hollow cylinders with prescribed volumetric heat sources

R1C13B11T11 Two layer concentric hollow cylinders with prescribed surface conditions and initial temperatures

R2C11B10T00G11 Two layer concentric cylinders with prescribed volumetric heat sources

R2C11B11T11 Two-layer concentric hollow cylinders with prescribed surface conditions and initial temperatures

R2C12B00T11 Two layer concentric hollow cylinders with insulated boundaries and prescribed initial temperatures

R2C12B11T00G11 Two layer concentric cylinders with prescribed boundary conditions and volumetric heat sources

R2C13B10T00G11 Two layer concentric cylinders with prescribed volumetric heat sources.

R2C13B11T11 Two layer concentric hollow cylinders with prescribed surface conditions and initial temperatures.

R3C11B00T00G11 Two-layer concentric cylinders with prescribed volumetric heat sources

R3C11B11T11 Two-layer concentric hollow cylinders with prescribed surface conditions and initial temperatures.

R3C12B00T11 Two-layer concentric hollow cylinders with prescribed initial temperatures and surface conditions

R3C12B11T00G11 Two layer concentric cylinders with prescribed boundary conditions and volumetric heat sources.

R3C13B00T00G11 Two layer concentric hollow cylinders with prescribed surface conditions and volumetric heat sources

R3C13B11T11 Two-layer concentric hollow cylinders with prescribed initial temperatures and surface conditions.

### Heat Equation, 1D Solid Cylinder

R01B1T0 Solid cylinder with jump in temperature at outer boundary and zero initial temperature

R02B1T0 Solid cylinder with jump in heat flux at outer boundary and zero initial temperature

### Heat Equation, 1D Hollow Cylinder

R11B01T0 Hollow cylinder with temperature jump at outer boundary and zero temperature at inner boundary

R11B10T0 Hollow Cylinder, temperature jump at inner boundary and zero temperature at outer boundary

R12B01T0 Hollow cylinder with zero temperature at inner radius and constant heat flux at outer radius

R12B10T0 Hollow Cylinder, temperature jump at inner boundary and zero heat flux at outer boundary

R13B01T0 Hollow cylinder with zero temperature at inner radius and heating through convection at outer radius

R13B10T0 Hollow cylinder with jump in temperature at inner radius and convection at outer radius

R21B01T0 Hollow cylinder insulated at inner radius with a step change in temperature at outer radius

R21B10T0 Hollow cylinder with jump in heat flux at inner radius and zero temperature at outer radius

R22B01T0 Hollow Cylinder, heat flux at outer boundary and insulated at inner boundary

R22B10T0 Hollow Cylinder, zero heat flux at outer boundary and jump in heat flux at inner boundary

R23B01T0 Hollow cylinder with heating through convection at outer surface and insulated at inner surface

R23B10T0 Hollow cylinder with jump in heat flux at inner boundary and convection at outer boundary

R31B10T0 Hollow cylinder with heating through convection at inner radius and constant zero temperature at outer radius

R32B01T0 Hollow cylinder with convection at inner radius and heating through constant heat flux at outer radius

R32B10T0 Hollow cylinder with heating through convection at inner radius and insulated at outer radius

R33B10T0 Hollow cylinder with convection heating at inner radius and convection cooling at outer radius

### Heat Equation, 1D Solid Sphere

RS01B1T0 Solid sphere with suddenly applied temperature at outer surface

RS03B0Gr5T0 Solid sphere with piecewise-constant internal heating and convection boundary

### Heat Equation, 1D Hollow Sphere

RS22B11T1G1 hollow sphere with surface heat fluxes and a prescribed volumetric heat source

RSIJB11T1G1 hollow sphere with convective surfaces and a prescribed volumetric heat source

### Heat Equation, Cartesian, 1D steady

### Heat Equation, Cartesian, Two-dimensional

X00Y11B(x5)0 Steady 2D conduction in the infinite slab with zero temperature on one surface and piecewise temperature on the other surface

X00Y21B(x5)0 Steady 2D conduction in the infinite slab with zero temperature on one surface and piecewise heat flux on the other surface

X11B00Y23B10 Steady 2D Conduction in Rectangle with homogeneous temperature on opposite boundaries, constant heat flux on one boundary, heat dissipation to zero temperature on other boundary

X22Y22Tx5y5 Rectangle with insulated boundaries and piecewise initial condition.

X33B00Y33B00T5 Rectangular plate with homogeneous convection boundary conditions and piecewise initial condition.

X33Y33Gx5y5F0T0 Rectangular plate with piecewise internal heating, out-of-plane heat loss, and homogeneous convection boundary conditions at the edges of the plate.

### Heat Equation, Cylinder, 2D Steady

R02B0Z23B50 Finite cylinder with piecewise concentric heating at z = 0, convection at z = L, and insulated on curved boundary

R22B0z5Z22B00 Finite-length hollow cylinder with Neumann boundaries and heat added/removed circumferentially

### Lumped body

### Thermal Wave Equation, Cartesian 1D Semi-infinite Body

X10B1T00 Semi-infinite body with step in surface temperature, initially zero temperature and zero time derivative

X20B1T00 Semi-infinite body with step in surface heat flux, initially zero temperature and zero time derivative

### Thermal Wave Equation, 1D Slab Body

X12B10T00 Thermal wave response of slab body with jump in temperature at x=0 and insulated boundary at x=L

X22B10T00 Thermal wave response of slab body with sudden heat flux at x=0 and insulated boundary at x=L

### Two diffusion equations weakly coupled, Cartesian

### Heat Equation, Cartesian, Three Dimensional

X00Y00Z33B00G5T0 Infinite region between parallel planes with piecewise internal heating and convection cooling at the planar boundaries.

X22Y22Z22Tx5y5z5 Parallelepiped with insulated boundaries and piecewise initial condition.

X33Y33Z33G5T0 Parallelepiped with convection boundaries and piecewise internal heating.

X33Y33Z33Tx5y5z5 Parallelepiped with homogeneous convection boundary conditions and piecewise initial condition