Volterra-type Integral Equations Research Articles

Analytical solutions to the axisymmetric elasticity and thermoelasticity problems for an arbitrarily inhomogeneous layer

In this paper, we present a technique for constructing an analytical solution to the axisymmetric elasticity and thermoelasticity problems in terms of stresses for an inhomogeneous layer, whose elastic and thermophysical properties vary arbitrarily within the thickness-coordinate. By making use of the direct integration method, the equilibrium and compatibility equations are reduced to the governing Volterra-type integral equations accompanied with both integral and local boundary conditions for the key functions. To solve the obtained governing equations, we employed the resolvent-kernel technique which results in closed-form analytical expressions for the key functions. Having determined the key functions, the stress-tensor components are found through the relationship established by the integration of equilibrium equations. The same solution procedure is employed for solving the steady-state heat-conduction problem in an inhomogeneous layer. Typical numerical examples are discussed.

Uniaxial load analysis under stretch-dependent fiber remodeling applicable to collagenous tissue

This article investigates the relation between stress and deformation of a fibrous material that is subjected to a constant fiber creation rate and deformation-dependent fiber dissolution rate. Experimental studies show such behavior in certain highly collagenous tissue. The investigation focuses on two different constitutive hypotheses: (a) fibers that reassemble in an undeformed state and (b) fibers that reassemble in a pre-strained state that is equal to the strain state of the surrounding matrix. The matrix component of the material is treated as isotropic and hyperelastic, and its mechanical properties remain constant in time. Separate processes of step displacement and step load are considered. In each case, an initial state of homeostasis is disrupted by the step input. The system then evolves to a new state of homeostasis that can be predicted in advance. The approach to this new homeostasis requires the consideration of a Volterra-type integral equation.

On an integral equation under Henstock–Kurzweil–Pettis integrability

In this paper, we investigate the set of solutions for nonlinear Volterra type integral equations in Banach spaces in the weak sense and under Henstock–Kurzweil–Pettis integrability. Moreover, a fixed point result is presented for weakly sequentially continuous mappings defined on the function space C(K, X), where K is compact Hausdorff and X is a Banach space. The main condition is expressed in terms of axiomatic measure of weak noncompactness.

Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval

We study an optimal control problem with Volterra-type integral equation,considered on a nonfixed time interval, subject to endpoint constraintsof equality and inequality type. We obtain first-order necessary optimalityconditions for an extended weak minimum, the notion of which is a naturalgeneralization of the notion of weak minimum with account of variationsof the time. The conditions obtained generalize the Euler--Lagrange equationand transversality conditions for the Lagrange problem in the classicalcalculus of variations with ordinary differential equations.

Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval

We reduce the initial value problem for the generalized Schroedinger equation with piecewise-constant leading coefficient to the system of Volterra type integral equations and construct new useful integral representations for the fundamental solutions of the Schroedinger equation. We also investigate some significant properties of the kernels of these integral representations. The integral representations of fundamental solutions enable to obtain the basic integral equations, which are a powerful tool for solving inverse spectral problems.

Обратная задача для нелинейного интегро-дифференциального уравнения Фредгольма четвертого порядка с вырожденным ядром

Рассмотрены вопросы об однозначной разрешимости обратной задачи для одного нелинейного интегро-дифференциального уравнения типа Фредгольма в частных производных четвертого порядка с вырожденным ядром. Развит метод вырожденного ядра для случая обратной задачи для рассматриваемого интегро-дифференциального уравнения типа Фредгольма в частных производных четвертого порядка. С помощью обозначения интегро-дифференциальное уравнение типа Фредгольма сведено к системе интегральных уравнений. Система интегральных уравнений путем дифференцирования сведена к системе дифференциальных уравнений. При выполнении определенного условия система дифференциальных уравнений заменена системой алгебраических уравнений. При регулярных значениях спектрального параметра решена система алгебраических уравнений методом Крамера. Используя дополнительное условие, относительно основной неизвестной функции получено нелинейное интегральное уравнение типа Вольтерра второго рода, и относительно функции восстановления получено специальное интегральное уравнение типа Вольтерра первого рода. Применен метод последовательных приближений в сочетании с методом сжимающих отображений. Далее определяется функция восстановления. Данная работа является дальнейшим развитием теории интегро-дифференциальных уравнений типа Фредгольма с вырожденным ядром.

On the theory of a class of Volterra-type integral equations with a fixed boundary singularity in the kernel

On the theory of a class of Volterra-type integral equations with a fixed boundary singularity in the kernel

On Volterra-type integral equations in noncompact metric space

In the present work, we consider one of the possible generalizations of linear and nonlinear Volterra integral equations of the second kind in the case when the independent variable belongs to an arbitrary noncompact metric space. Sufficient conditions are obtained for the existence of solutions of Volterra-type integral equations in the nonhomogeneous case. Some applications of the obtained results to the integral inequalities are given. MSC:35Q99, 35R35, 65M12, 65M70.

Thermal properties of thin films studied by ultrafast laser spectroscopy: Theory and experiment

The one-dimensional non-homogeneous hyperbolic energy equation within a semi-infinite domain, applied to pump–probe transient reflectance measurements for subpicosecond and picosecond laser pulses was studied. The result of this study is an analytical model in the form of a Volterra-type integral equation that describes the surface temperature response of the thin film induced by an ultra-short laser pulse and takes into account contributions of the transient heat flux and the volumetric heat source. The criterion based on dimensionless time ξp=t/2τ (where τ is the relaxation time of phonons due to collisions) is introduced, in order to predict the temperature response caused by the surface heat flux and/or volumetric source. This analytical solution is validated by using experimental pump–probe transient reflectance measurements of thin gold with a time resolution of 50fs. A good agreement between the experimental results and the theoretical model is found.

Hybrid coincidence and common fixed point theorems in Menger probabilistic metric spaces under a strict contractive condition with an application

We prove some coincidence and common fixed point theorems for two hybrid pairs of mappings in Menger spaces satisfying a strict contractive condition. An illustrative example is given to support the genuineness of our extension besides deriving some related results. Then, we establish the corresponding common fixed point theorems in metric spaces. Finally, we utilize our main result to obtain the existence of a common solution for a system of Volterra type integral equations.

On the uniform convergence of solutions to a controlled system of integral equations with weakly convergent controls

A controlled system of Volterra-type integral equations is considered which is linear with respect to the control and has integrand measurable with respect to the integration variable. It is proved that if a sequence of controls weakly converges in the space L1, then, for its members with large numbers, the system has solutions uniformly converging to the solution corresponding to the limit control.

Some new fixed point theorems in Menger PM-spaces with application to Volterra type integral equation

We establish some fixed point theorems by introducing two new classes of contractive mappings in Menger PM-spaces. First, we prove our results for an α-ψ-type contractive mapping and then for a generalized β-type contractive mapping. Some examples and an application to Volterra type integral equation are given to support the obtained results.

Series solutions to linear integral equations

Linear Volterra-type integral equations with kernels having a series expansion in the first variable have series solutions with coefficients given iteratively. Their resolvents may be expanded likewise. The associated homogeneous equation Kf=f generally has Frobenius series solutions when the kernel is singular, whereas Kf=0 generally has such solutions regardless of singularity: the proviso in each case is that associated “indicial equation” has solutions.

Necessary Conditions for a Weak Minimum in Optimal Control Problems with Integral Equations Subject to State and Mixed Constraints

The first order necessary optimality conditions for a weak minimum are derived for optimal control problems with Volterra-type integral equations, considered on a fixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the uniform linear-positive independence of the gradients of active mixed constraints with respect to the control. The conditions obtained generalize the Euler--Lagrange equation (as a stationarity condition) for the Lagrange problem in the classical calculus of variations with ordinary differential equations.

An Application of Random Fixed Point Theorem in Integral Equation

In this paper random fixed point theorem has been applied seeking solution of Volterra type integral equation involving more generous kernel.

Application of fixed point theorem of convex-power operators to nonlinear Volterra type integral equations

Application of fixed point theorem of convex-power operators to nonlinear Volterra type integral equations

Multi-valued $$F$$ F -contractions in 0-complete partial metric spaces with application to Volterra type integral equation

We study the existence of fixed points for multi-valued mappings that satisfy certain generalized contractive conditions in the setting of 0-complete partial metric spaces. We apply our results to the solution of a Volterra type integral equation in ordered 0-complete partial metric spaces.

Three Fixed Point Theorems: Periodic Solutions of a Volterra Type Integral Equation with Infinite Heredity

AbstractIn this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's and Schaefer's fixed point theorems are employed in the analysis. The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov's direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes.

To Theory One Class Linear Model Noclassical Volterra Type Integral Equation with Left Boundary Singular Point

In this work, we investigate one class of Volterra type integral equation, in model case, when kernels have first order fixed singularity and logarithmic singularity. In detail study the case, when n = 3. In depend of the signs parameters solution to this integral equation can contain three arbitrary constants, two arbitrary constants, one constant and may have unique solution. In the case when general solution of integral equation contains arbitrary constants, we stand and investigate different boundary value problems, when conditions are given in singular point. Besides for considered integral equation, the solution found cane represented in generalized power series. Some results obtained in the general model case.

A Novel Analytical Implementation of Nonlinear Volterra Integral Equations

This article aims at preferring a new and viable algorithm, specifically a two-step homotopy perturbation transform algorithm (TSHPTA). This novel technique is a feasible way in finding exact solutions with a small amount of calculations. As a simple but typical example, it demonstrates the strength and the great potential of the two-step homotopy perturbation transform method to solve nonlinear Volterra-type integral equations efficiently. The results reveal that the proposed scheme is suitable for the nonlinear Volterra equations.