Some Examples of the Second Order Equations in One Dimension

-d/dx(adu/dx) = q for 0 < x < L

[Taken from J. N. Reddy's Book "Introduction to the Finite Element Method", J.N. Reddy, McGraw Hill Publishers, 2nd Edition, Page 71]

Field

Primary Variable u

Constant a

Source term q

Secondary Variable Qo

Transverse Deflection of a Cable

Transverse Deflection

Tension in Cable

Heat Source Q

Axial Force

Axial Deformation of a bar

Longitudiunal Dispalcement

EA (E= Young's Modulus, A = Cross Sectional Area)

Distributed Transverse Load

Axial Force

Heat Transfer

Temperature

 

Thermal Conductivity

Friction or contact force on surface of bar

Heat

Flow Through Pipes

Hydrostatic Pressure

PD4/128m (D- Diameter, m - viscosity)

Flow Source (Generally Zero)

Flow Rate

Laminar Incompressible Flow through a Channel under Constant Pressure Gradient

Velocity

Viscosity Pressure Gradient

Axial Stress

Flow Through Porous Media

Fluid Head

Coefficient of Permeability Fluid Flux

Flow (seepage)

Electrostatics

Electroststic Potential

Dielectric Constant

Charge Density

Electric Flux

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