On the solutions of linear Volterra equations of the second kind with sum kernels

Solution to the Volterra integral equations of the first kind with piecewise continuous kernels in class of Sobolev-Schwartz distributions

Sufficient conditions for existence and uniqueness of the solution of the Volterra integral equations of the first kind with piecewise continuous kernels are derived in framework of Sobolev-Schwartz distribution theory. The asymptotic approximation of the parametric family of generalized solutions is constructed. The method for the solution's regular part refinement is proposed using the successive approximations method.

Volterra Series Based Time-domain Macro-modeling of Nonlinear Circuits

Volterra series representation is a powerful mathematical model for nonlinear circuits. However, the difficulties in determining higher-order Volterra kernels limited its broader applications. In this work, a systematic approach that enables a convenient extraction of Volterra kernels from X-parameters is presented. A concise and general representation of the output response due to arbitrary number of input tones is given. The relationship between Volterra kernels and X-parameters is explicitly formulated. An efficient frequency sweep scheme and an output frequency indexing scheme are provide.…

Correct solvability of hyperbolic Volterra equations with kernels depending on the parameter

We study the correct solvability of an abstract functional differential equations in Hilbert space, which includes integro-differential equations describing evolution of thermal phenomena, heat transfer in materials with memory or sound propagation in viscoelastic media.

Volterra积分方程组解的稳定性

Stability of solutions of systems of Volterra integral equations

针对Volterra线性积分方程与积分微分方程组，本文利用系数及核导数所成矩阵的对数范数，给出右端及初始条件扰动下解稳定性的新充分条件。

On Fractional Eulerian Numbers and Equivalence of Maps with Long Term Power-Law Memory (Integral Volterra Equations of the Second Kind) to Gr$\ddot{u}$nvald-Letnikov Fractional Difference (Differential) Equations

In this paper we consider a simple general form of a deterministic system with power-law memory whose state can be described by one variable and evolution by a generating function. A new value of the system's variable is a total (a convolution) of the generating functions of all previous values of the variable with weights, which are powers of the time passed. In discrete cases these systems can be described by difference equations in which a fractional difference on the left hand side is equal to a total (also a convolution) of the generating functions of all previous values of the system's v…

The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions

For kernels $ν$ which are positive and integrable we show that the operator $g\mapsto J_νg=\int_0^x ν(x-s)g(s)ds$ on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a "contractive" effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor $N(x)=\int_0^x ν(s)ds$. For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of…