A BOUNDARY VALUE PROBLEM FOR A NONLINEAR ORDINARY DIFFERENTIAL EQUATION INVOLVING A SMALL PARAMETER——The Riemann problem for a Generalized Diffusion Equation

This paper studies the boundary value problem involving a small parameter((k(V(t))+ε)V'(s)~(n-1V'(s))'+(sg(V(s))+f(V(s))V'(s)=0 for s∈R,V(-∞)=A,V(+∞)=B;A0,U(x,0)=A for x<0,U(x,0)=B for x>0,under the hypotheses H_1—H_4.The author's aim is not only to determine explicitly thediscontinuous solution U_0(x,t)=V_0(s),s=x/p(t),to the reduced problem,and the formand the number of its curves of discontinuity,but also to present,in an extremely naturalway,the jump conditions which it must satisfy on each of its curves of discontinuity.Itis proved that the problem has a unique solution U_ε(x,t)=V_ε(s),s=x/p(t),ε≥0,V_ε(s)pointwise converges to V_0(s)as ε 0,V_0(s)has at least one jump point if and only if k(y)possesses at least one interval of degeneracy inA,B],and there exists a one-to-onecorrespondence between the collection of all intervals of degeneracytof k(y)inA,B]andthe set of all jump points of V_0(s).