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On solution of integral equation of Abel-Volterra type. (English) Zbl 0823.45002

The authors study asymptotic expansions of the solution of the equation

φ(x)=a(x) 0 x (x-t) α-1 φ(t)dt+f(x),

as x0+, where 0<α<1. It is assumed that a and f have expansions of the form a(x) k=-1 a k x αk , f(x) k=-1 f k x αk , and it is shown that then the solution has a similar expansion φ(x) k=-1 φ k x αk when x0+. Recursive formulas for calculating the coefficients φ k are given. The case where a(x)=ax αm , m=-1,0,, and where one can find an explicit expression for the solution is studied, too.


MSC:
45E10Integral equations of the convolution type
45D05Volterra integral equations
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)