4.1.

Solve

where the initial condition is given along the x ₁ axis as w ₀( x ₁). The final result does contain a difficult integral, which you do not have to evaluate.

4.2.

Solve

with the boundary condition that w is given as some function w ₀ in ( x ₁, x ₃, x ₄) space for x ₂ = 0. That is,

The final result does contain a difficult integral, which you do not have to evaluate.

4.3.

Solve the wave equation

for a wave reflection at x = 0 using d'Alembert's method. For simplicity consider only waves on the positive x axis and assume the initial conditions below are valid.

where clearly one must demand that w ₀(0) = 0 and w ₀( x) are defined only for positive values of x.

This problem is not as straightforward as it might seem. The initial manifold is the positive x axis, but the solution obtained from this manifold is not valid for all positive arguments of x and t because w ₀, which appears in the solution, is defined only for positive arguments. Figuring out how to solve this puzzle is the main motive for this assignment.

4.4.

Solve

This model can be interpreted as the cell mass population balance equation of a population of cells that have lost the ability to divide, but still grow, in...