0.3 Boundary Value Problems

0.3 Boundary Value Problems

• The order of the differential equation is equal to the number of conditions imposed.
• We are concerned with cases where the differential equation is linear and of second order.
• Problems with elasticity often involve fourth-order equations.

• The most innocent looking boundary value problem may have one solution, no solution, or an infinite number of solutions.

• The two boundary conditions are used to supply two equations that are to be satisfied by the two constants in the general solution when the differential equation in a boundary value problem has a known general solution.
• Two linear equations can be easily solved if the differential equation is linear.

• Some physical examples of natu rally are examined in the rest of the section.

• Finding the shape of a cable that is fastened at each end is a problem.
• An important example is the cables of a suspension bridge.
• We assume the axis is horizontal.

• The shape of the cable is determined by the forces acting on it.
• The forces that hold a small segment of the cable are considered in our analy sis.
• The cable is flexible, that's the key assumption.
• This means that force inside the cable is always a tension and that direction at every point is the centerline.

• The cable is not moving.
• The horizontal and vertical components of the forces on the segment are equal.

• The endpoints where the cable is attached are included.

• The cable shows forces acting on it.

• The differential equation is not always linear.
• We can find its general solution in closed form and satisfy the boundary conditions by choosing the arbitrary constants that appear.

• This is true for a suspension bridge.

• Sections 1 and 2 can be used to find the general solution of the differential equation.

• 2 are in terms of parameters.

• The cable's shape is specified by this function.

• The heat flow is shown by the section cut from the heat-conducting cylinder.

• The Fourier's law states that the heat flow rate through a unit area of material is related to the temperature difference and thickness.

• The minus sign shows that heat moves to cooler areas.

• The Eqs are combined.

• A boundary value problem is not different from an initial value problem.
• The general solution of the differential equation must contain some arbitrary constants, and the boundary conditions must be applied to determine values for the arbitrary constants.

• The next example is not the same as the others.
• We will be looking for parameters that permit the existence of solutions of special form instead of just finding the solution of a boundary value problem.

• The upper part of the column can move up or down, but not sideways.
• The lower part of the column is needed to give this force and moment.

• a.

• a.

• This means that the column is straight and that the load is transmitted to the support.

• 2 comes up with a solution.
• The column may collapse under the load if it is a sinusoidal shape.

• The constant function 0 is always a solution for differential equations and boundary conditions.
• There is a dividing line between stable and unstable behavior.
• In later chapters, we will see them frequently.

• One of the boundary value problems has no solution, one has exactly one solution, and one has an infinite number of solutions.

• The steel column has a 2 in.
• load.

• The solution of the problem should be verified.

• Integration can be used to solve the differential equation.

• Nuclear reaction creates heat in a nuclear fuel rod.

• An assembly of nuclear fuel rods is housed in a pressure vessel shaped like a cylinder.
• Design and safety are affected by the temperature in the steel wall of the vessel.