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| A BOUNDARY VALUE PROBLEM FOR A NONLINEAR ORDINARY DIFFERENTIAL EQUATION INVOLVING A SMALL PARAMETER—The Riemann problem for a Generalized Diffusion Equatiois |
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Citation： |
Wang Junyu.A BOUNDARY VALUE PROBLEM FOR A NONLINEAR ORDINARY DIFFERENTIAL EQUATION INVOLVING A SMALL PARAMETER—The Riemann problem for a Generalized Diffusion Equatiois[J].Chinese Annals of Mathematics B,1991,12(1):106~121 |
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Net amount： 797 |
Authors： |
Wang Junyu; |
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| Abstract： |
This paper studies the boundary value problem involving a small parameter
$$((k(V(t))+s)|V'(s)|^{N-1}V'(s))'+(sg(V(s))+f(V(s)))V'(s)=0 for s\in R$$,
$$V(-\infty)=A,V(+\infty)=B;A0$$,
$$U(x,0)=A for x<0,U(x,0)=B for x>0$$
under the hypotheses H1—H4 . The author's aim is not only to determine explicitly the discontinuous solution ,to the reduced problem；and the form and the number of its curves of discontinuity, but also to present, in an extremely natural way, the jump conditions which it must satisfy on each of its curves of diseontinuity. It is proved that the problem has a unique solution $U_{\varepsilon}(x,t)=V_{\varepsilon}(s),s=x/p(t),s\geq0,V_{\varepsilon}$pointwise converges to $V_{0}(s)$ as $s\downarrow0,V_{0}(s)$ has at least one jump point if and only if k(y) possesses at least one interval of degeneracy in [A-B], and there exists a one-to-one correspondence between the collection of all intervals of degeneracy of k(y) in [A-B] and the set of all jump points of $V_{0}(s)$ |
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