2 In the continuity equation (3.4), external factors that break the conservation of mass are accounted for by adding terms to the right-hand side....

A question on Partial Differential Equations.

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3.2 In the continuity equation (3.4), external factors that break the conservation ofmass are accounted for by adding terms to the right-hand side. (a) A forcing term f (t, x) is independent of the existing concentration. (In thebloodstream model of Sect. 3.1, this could represent intravenous injection, forexample.) Assume that c is constant, f E C1(]R2), and g E C1(R). Solve theequation _+c_= , u(0,x)=g(x), to find an explicit formula for u(t, x) in terms of f and g. (b) A reaction term depends on the concentration u. The simplest case is a linearterm yu where the coefficient is some function y(t, x). (This could representabsorption of oxygen into the walls of the artery, for example.) Assume that c isconstant, y E C1(R2), and g E C1(R). Solve the equation 3 3,7”: Mg = yu, u<0,x) = g(x), to find an explicit formula for u(t, x) in terms of y and g.