Partial Differential Equation

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3.2 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) A forcing term f (t, x) is independent of the existing concentration. (In the bloodstream model of Sect. 3.1, this could represent intravenous injection, for example.) Assume that c is constant, f E C1(]R2), and g E C1(R). Solve the equation _+c_= , u(0,x)=g(x), to nd 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 linear term yu where the coefcient is some function y(t, x). (This could represent absorption of oxygen into the walls of the artery, for example.) Assume that c is constant, y E C1(R2), and g E C1(R). Solve the equation 3 3 ,7\": Mg = yu, u