ODE Boundary Value Problems and Finite Differences

Steady State Heat and Diffusion:

If we regard as the movement of heat in a long thin object (like a metal bar) it is known that the temperature u(x, t) at a location x and time t convince the partial differential equation:

u_t− u_xx = g(x, t),

Where g(x, t) is the consequence of any external heat source. The similar equation as well describes the diffusion of a chemical in a one-dimensional environment. For illustration the environment might be a canal and then g(x, t) would stand for how a chemical is introduced.

Sometimes we are interested merely in the steady state of the system, supposing g(x, t) = g(x) and u(x, t) = u(x). In this case:

u_xx= −g(x).

This is a linear second-order ordinary differential equation that we could discover its solution exactly if g(x) isn’t too complicated. If the environment or else object we consider has length L then typically one would have conditions on each end of the object such as u(0) = 0, u(L) = 0. Therefore instead of an initial value problem we have a boundary value problem or BVP.

Beam With Tension:

Consider a simply supported beam with modulus of elasticity E moment af inertia I, a uniform load w and end tension T. If y(x) denotes the deflection at every point x in the beam then y(x) satisfies the differential equation:

With boundary conditions y(0) = y(L) = 0. This equation is nonlinear as well as there is no hope to solve it exactly. If the deflection is little then (y′)2 is negligible compared to 1 and the equation approximately simplifies to:

This is a linear equation as well as we can find the exact solution. We can rewrite the equation as:

y′′ − αy = βx(L − x),

A merely supported beam with a uniform load w and end tension T

Where

α =T/EI    and β =w/2EI

and then the precise solution is:

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