Y(X,E)=Y, *>»(*) + Vow'x + 0(Ear)' (1-7)
Methods and Applications of Analysis © 1995 International Press 2 (2) 1995, pp. 212-221 ISSN 1073-2772 ASYMPTOTIC SOLUTION TO A CLASS OF SINGULARLY PERTURBED VOLTERRA INTEGRAL EQUATIONS L. A. Skinner ABSTRACT. A uniformly valid asymptotic expansion for the solution to a class of singularly perturbed Volterra integral equations displaying exponential boundary layer behavior is established. Certain quasilinear ordinary differential equations are a noted special case, and a model for population growth with attrition is briefly discussed. 1. Introduction Let /(a;, e) E C^QO, 1] x [0,1]) and k(x, t, e, y) 6 C^QO, 1] x [0, x] x [0,1] x [a, 6]), where (a, b) contains the derivative /e(0,0), and assume /(0,0) = 0. We are interested in the asymptotic behavior as e —> 0+ of the solution to the Volterra integral equation €y(x,e)+ / k(x,t,e,y(t,e))dt = f(x,e). (1.1) Jo The assumption /(0, 0) = 0, which implies y(0, e) = /e(0, 0) + O(e), is nontrivial unless (1.1) is linear. If kyy(x,t,e,y) = 0, we can get /(0,0) = 0 by changing the unknown (if necessary) to ey(x^e). In the special case k(x, t, 6, y) = p(t, e)y + (x - t)[q(t, e, y) - pt(t, e)]y, (1.2) f(x, e) = ae+ tfe + ap(0, e)]x, (1.3) equation (1.1) is equivalent to the singularly perturbed initial-value problem ey" + p(x, e)yl + q{x, c, y) = 0, (1.4) 2/(0) = a, y,(0) = p.
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