Abstract
In the paper, a class of perturbed Volterra equations of convolution type with three kernel functions is considered. The kernel functions gα = tα−1/Γ(α), t > 0, α ∈ [1, 2], correspond to the class of equations interpolating heat and wave equations. The results obtained generalize our previous results from 2010.
Highlights
We study perturbed Volterra equations of the form t t u x, t u x, 0 gα t − s gα ∗ k t − s Δu x, s ds b t − s u x, s ds, 1.1 where x ∈ Rd, t > 0, gα t tα−1/Γ α, Γ is the gamma function, gα ∗ k denotes the convolution, α ∈ 1, 2, b, k ∈ L1loc R ; R, and Δ is the Laplace operator.The perturbation approach to Volterra equations of convolution type has been used by many authors, see, for example, 1
We are looking for an approximate solution to 1.1 as an element of the subspace Hnφ, spanned on nφ first basic functions {φj : j 1, 2, . . . , nφ}
In one-dimesional case, let us introduce a grid of points x1, x2, . . . , xnh, where xl − xl−1 h
Summary
We study perturbed Volterra equations of the form t t u x, t u x, 0 gα t − s gα ∗ k t − s Δu x, s ds b t − s u x, s ds, 1.1 where x ∈ Rd, t > 0, gα t tα−1/Γ α , Γ is the gamma function, gα ∗ k denotes the convolution, α ∈ 1, 2 , b, k ∈ L1loc R ; R , and Δ is the Laplace operator. The perturbation approach to Volterra equations of convolution type has been used by many authors, see, for example, 1. Such approach may be applied to more general, not necessary convolution equations, too. The authors consider the class of equations with three kernel functions which satisfy some scalar auxiliary equations. Such condition enables to construct the family of resolvent operators admitted by the Volterra equations. Two convolutions appearing in 1.1 with the kernel functions b and k, respectively, represent some perturbation acting on the Volterra equation of convolution type.
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