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ABSTRACT: The solution of a singularly perturbed type of a system of fractional integral (differential) equations is studied in this paper. The formal asymptotic solution is derived and proved to be asymptotically correct. Basic matrix algebra is used to prove the asymptotic decay in the inner layer solution.Journal of Integral Equations and Applications 06/2012; 24(2012). DOI:10.1216/JIE2012242195 · 0.40 Impact Factor 
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ABSTRACT: Several investigations have been made on singularly perturbed integral equations. This paper aims at presenting an algorithm for the construction of asymptotic solutions and then provide a proof for asymptotic correctness to singularly perturbed systems of Volterra integrodifferential equations.Quaestiones Mathematicae 11/2009; 25(2):229248. DOI:10.2989/16073600209486011 · 0.45 Impact Factor 
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ABSTRACT: It is demonstrated here that there exist initial layers to singularly perturbed Volterra equations whose thicknesses are not of the order of magnitude of O(ϵ), ϵ → 0. It is also shown that the initiallayer theory is extremely useful because it allows one to construct the approximate solution to an equation, which is almost identical to the exact solution.IMA Journal of Applied Mathematics 03/2006; 71(3). DOI:10.1093/imamat/hxh113 · 1.19 Impact Factor 
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ABSTRACT: The paper contains a discussion on so called layer type techniques with respect to the following Volterra integral and integrodifferential equations: εu(t)=ϕ(t)+∫ 0 t a(t,s) (ts) 1β f(s,u(s))ds,t≥0,0<β≤1, εu ' (t)=ϕ(t)+∫ 0 t a(t,s) (ts) 1β f(s,u(s))ds,t>0,0<β<1,u(0)=u 0 (ε), where 0<ε≤1 is sufficiently small parameter, f is assumed to be sufficiently smooth, a is sufficiently differentiable with respect to both arguments and a(0,0) is nontrivial. The function ϕ(t) is continuous for all t≥0 and ϕ ' (t) is allowed to have an integrable singularity at t=0. A special attention is drawn to the asymptotic character of the so called transcendental smallness. 
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ABSTRACT: Asymptotic solutions of nonlinear singularly perturbed Volterra integral equations with kernels possessing integrable singularity are investigated using singular perturbation methods and the Mellin transform technique. In particular, it is demonstrated that the formal approximation is asymptotically valid.Journal of Applied Mathematics 12/2004; DOI:10.1155/S1110757X04305024 · 0.72 Impact Factor 
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ABSTRACT: Nonlinear singularly perturbed Volterra integrodifferential equations with weakly singular kernels are investigated using singular perturbation methods, the Mellin transform technique, and the theory of fractional integration.International Journal of Mathematics and Mathematical Sciences 01/2003; DOI:10.1155/S0161171203301358 
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ABSTRACT: The additive decomposition singular perturbation method and the theory of fractional integration are used to study asymptotic solutions of singularly perturbed Volterra integrodifferential equations with kernels having integrable singularity. The validity of the approximation is also demonstrated.International Journal of Mathematics and Mathematical Sciences 01/2003; DOI:10.1155/S0161171203209091 
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ABSTRACT: The author studies singularly perturbed Volterra integral equations of the form εu(t)=f(t;ε)+∫ 0 t gt , s , u ( s )ds,0≤t≤T, where ε is a small parameter. The function f(t;ε) is defined for 0≤t≤T and g(t,s,u) for 0≤s≤t≤T. There are many existence and uniqueness results known that ensure that a unique continuous solution u(t;ε) exists for all small ε>0. The aim is to find asymptotic approximations to these solutions and rigorously prove the asymptotic correctness. This work is restricted to problems where there is an initial layer; various hypotheses are placed on g to exclude other behaviors.Journal of Integral Equations and Applications 06/2002; DOI:10.1216/jiea/1031328363 · 0.40 Impact Factor 
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ABSTRACT: We consider finding asymptotic solutions of the singularly perturbed linear Volterra in tegral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the MittagLeffler function.International Journal of Mathematics and Mathematical Sciences 01/2002; DOI:10.1155/S016117120201325X 
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ABSTRACT: This paper studies the behaviour of the solution u(t,ε) of the system of Volterra integral equations εu(t)=f(t)+∫ 0 t A(t,s)u(s)ds,0≤t≤T, as the positive parameter ϵ tends to zero. Both f and A are continuous, and the eigenvalues of A(t,t) are supposed to be negative. For small values of ε this leads to a fast exponential decay within a small boundary layer. The additive decomposition technique (originally developed by O’Malley and Hoppensteadt) is used to obtain an asymptotic power series expansion of u(t,ϵ) in the parameter ϵ. From this series an approximant is constructed with the property that the difference of the true solution and this approximant tends to zero uniformly on the interval [0,T] as ϵ→0. The theory is illustrated by means of two examples. In the latter of these examples the decay is not exponential within the boundary layer.01/2002; 8(2):221244. DOI:10.1515/JAA.2002.221

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