Homogeneous Volterra Research Articles

Stability Properties for the Delay Integro-Differential Equation

In this paper stability inequalities for the linear nonhomogeneous Volterra delay integro-differential equation (VDIDE) is being established. The particular problems are encountered to show the applicability of the method and to confirm the predicted theoretical analysis.

Definition of the solution of the homogeneous Volterra integral equation

In this paper, we study the solvability of a second-kind Volterra integral equation. By replacing the right-hand side and the unknown function, the integral equation is reduced to an integral equation, the kernel of which is not «compressiblek. Using the Laplace transform, the obtained equation is reduced to an ordinary first-order differential equation (linear). Its solution is found. The solution of the homogeneous integral equation corresponding to the original nonhomogeneous integral equation found in explicit form. Special cases of a homogeneous integral equation and its solutions are written for different values of the parameter k. Classes are indicated in which the integral equation has a solution. Singular integral equations were considered in works [1–3]. Their kernels were also «incompressiblek, but kernels had an another form. In this connection, the weight classes of the solution existence differ from the class of the solution existence for the equation considered in this work.

Differential and integral equations for Legendre-Laguerre based hybrid polynomials

UDC 517.9 In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial differential equations are established. The analogous results for the three-variable Hermite – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, –E uler and – Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by deriving homogeneous Volterra integral equations for these polynomials and for their relatives.

Specific Asymptotic Stability of Solutions to a Linear Homogeneous Volterra Integro-Differential Equation of the Fourth Order

Specific Asymptotic Stability of Solutions to a Linear Homogeneous Volterra Integro-Differential Equation of the Fourth Order

Numerical analysis of Volterra integro-differential equations for viscoelastic rods and membranes

We consider the initial boundary value problems for a homogeneous Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain Ω. The memory kernel of the equation is made up in a complicated way from the (distinct) moduli of stress relaxation for compression and shear, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will study the second order continuous time Galerkin approximation for its space discretization and propose a fully discrete scheme employing the Crank–Nicolson method for the time discretization. Then we derive the uniform long time error estimates in the norm Lt1(0,∞;Lx2) for the finite element solutions. Some numerical results are presented to illustrate our theoretical error bounds.

Boundedness and Asymptotic Stability for the Solution of Homogeneous Volterra Discrete Equations

We consider homogeneous linear Volterra Discrete Equations and we study the asymptotic behaviour of their solutions under hypothesis on the sign of the coefficients and of the first- and second-order differences. The results are then used to analyse the numerical stability of some classes of Volterra integrodifferential equations.

Differential and integral equations associated with some hybrid families of Legendre polynomials

The article aims to explore some new classes of differential and integral equations for some hybrid families of Legendre polynomials. Beginning with the recurrence relations and shift operators, the authors derived the differential, integro-differential and partial differential equations for the hybrid Legendre-Appell polynomials. Certain examples are framed for the hybrid Legendre-Bernoulli, Legendre-Euler and Legendre-Genocchi polynomials to show the applications of main results. Further, the homogeneous Volterra integral equations for the hybrid Legendre-Appell and other hybrid families of special polynomials are derived. The inclusion of integral equations is a bonus to this article.

Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra–Hammerstein integral equations. To do this, we introduce the concept of “blow-up collocation solution” and analyze numerically some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finally, we discuss the relationships between necessary conditions for blow-up of collocation solutions and exact solutions.

Transverse instability of a plane front of fast impact ionization waves

The transverse instability of a plane front of fast impact ionization waves in p+-n-n+ semiconductor structures with a finite concentration of donors N in the n layer has been theoretically analyzed. It is assumed that the high velocity u of impact ionization waves is ensured owing to the avalanche multiplication of the uniform background of electrons and holes whose concentration σb ahead of the front is high enough for the continuum approximation to be applicable. The problem of the calculation of the growth rate s of a small harmonic perturbation with wavenumber k is reduced to the eigenvalue problem for a specific homogeneous Volterra equation of the second kind containing the sum of double and triple integrals of an unknown eigenfunction. This problem has been solved by the method of successive approximations. It has been shown that the function s(k) for small k values increases monotonically in agreement with the analytical theory reported in Thermal Engineering 58 (13), 1119 (2011), reaches a maximum sM at k = kM, then decreases, and becomes negative at k > k01. This behavior of the function s(k) for short-wavelength perturbations is due to a decrease in the distortion of the field owing to a finite thickness of the space charge region of the front and “smearing” of perturbation of concentrations owing to the transverse transport of charge carriers. The similarity laws for perturbations with k ≳ kM have been established: at fixed σb values and the maximum field strength on the front E0M, the growth rate s depends only on the ratio k/N and the boundary wavenumber k01 ∝ N. The parameters sM, kM, and k01, which determine the perturbation growth dynamics and the upper boundary of the instability region for impact ionization waves, have been presented as functions of E0M. These dependences indicate that the model of a plane impact ionization wave is insufficient for describing the operation of avalanche voltage sharpers and that fronts of fast streamers in the continuum approximation should be stable with respect to transverse perturbations in agreement with the previously reported numerical simulation results. The results have been confirmed by the numerical simulation of the evolution of small harmonic perturbations of the steady-state plane impact ionization wave.

Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations

We analyse collocation methods for nonlinear homogeneous Volterra—Hammerstein integral equations with non-Lipschitz nonlinearity. We present different kinds of existence and uniqueness of nontrivial collocation solutions and give conditions for such existence and uniqueness in some cases. Finally, we illustrate these methods with an example of a collocation problem and give some examples of collocation problems that do not fit in the cases studied previously.

Runge–Kutta convolution quadrature methods for well-posed equations with memory

Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity, of the exact solution. Numerical illustrations are provided.

Composite Surface Integral Transforms in Neutron Therapy of Cancer

Composite surface finite integral transforms are applied to formulate the optimal ballistic property for a temporally tuned multibeam neutron cancer 3D therapy as a single-valued dynamical system. By invoking Pontryagin's maximum principle, with the operation functions of the beams constituting the control vector, it is proved, in an inverse problem formulation, that for every spatial configuration of the neutron beams, there exists an optimal temporal control vector satisfying an a priori system of linear homogeneous Volterra integral equations of the first kind and convolution type. A version of this newly advanced, temporally optimalized, multibeam 3D irradiation therapy, with a linearized ballistic property, is shown to result from a shooting-type solution to a related, semihomogeneous dual system of linear integral equations of the first kind. A criterion for the controllability of this optimization problem has also been established.

Bifurcation theory of the time-dependent von Kármán equations

In this paper the author studies existence and bifurcation of a nonlinear homogeneous Volterra integral equation, which is derived as the first approximation for the solution of the time dependent generalization of the von Kármán equations. The last system serves as a model for stability (instability) of a thin rectangular visco-elastic plate whose two opposite edges are subjected to a constant loading which depends on the parameters of proportionality of this boundary loading.

The problem of eigenvalues in some singular homogeneous Volterra integral equations

It is shown that when the kernel of a homogeneous Volterra integral equation is singular, it is possible for there to be a continuous spectrum of eigenvalues.

The Problem of Eigenvalues in Some Singular Homogeneous Volterra Integral Equations

It is shown that when the kernel of a homogeneous Volterra integral equation is singular, it is possible for there to be a continuous spectrum of eigenvalues.