Volterra-type Integral Equations Research Articles

LOCALMINIMUM PRINCIPLE FOR OPTIMIZATION PROBLEMS WITH DIFFERENT TYPES OF CONTROL SYSTEMS SUBJECT TO MIXED STATE-CONTROL CONSTRAINTS

This paper discusses the first-order optimality conditions for optimal control problems with two different types of control systems, considered on a fixed time interval: systems of ordinary differential equations and systems of Volterra-type integral equations

Fixed point theorems for multivalued and single-valued contractive mappings on Menger PM spaces with applications

In this paper, by introducing multivalued $$(\alpha ,\eta )$$ – $$\psi $$ -contractive mappings, we obtain new fixed point theorems for multivalued and single-valued mappings and also coupled fixed point theorems in complete Menger PM and partially ordered Menger PM spaces. We have improved, extended and generalized probabilistic version of the very important generalization of the Banach contraction principle. Some examples and also application of our results in metric spaces and an application to existence of solution of Volterra-type integral equation are given to support the obtained results.

ON COMMON FIXED POINTS THEOREMS FOR ORDERED $F$-CONTRACTIONS WITH APPLICATION

We study the conditions for existence of a unique common fixed point of ordered $F$-contractions defined on an ordered partial metric space; in particular, we present a common fixed point result for a pair of ordered $F$-contractions satisfying a generalized rational type contractive condition and discuss its consequences. It is remarked that the notion of an $F$-contraction in partial metric spaces is more general than that in metric spaces. As application of our findings, we demonstrate the existence of common solution of the system of Volterra type integral equations.

Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives

The generalized Caputo fractional derivative is a name attributed to the Caputo version of the generalized fractional derivative introduced in Jarad et al. (J. Nonlinear Sci. Appl. 10:2607–2619, 2017). Depending on the value of ρ in the limiting case, the generality of the derivative is that it gives birth to two different fractional derivatives. However, the existence and uniqueness of solutions to fractional differential equations with generalized Caputo fractional derivatives have not been proven. In this paper, Cauchy problems for differential equations with the above derivative in the space of continuously differentiable functions are studied. Nonlinear Volterra type integral equations of the second kind corresponding to the Cauchy problem are presented. Using Banach fixed point theorem, the existence and uniqueness of solution to the considered Cauchy problem is proven based on the results obtained.

Transient solution for the queue-size distribution in a finite-buffer model with general independent input stream and single working vacation policy

• Transient queue-size distribution in GI/M/1/N-type model with working vacation policy is studied. • The idea of embedded Markov chain, total probability law and linear algebra are applied. • Final result is given for the Laplace transform in a compact form. • Illustrative numerical examples are attached. A single-channel finite-buffer queueing model with a general independent input stream of customers, exponential processing times and a working vacation policy is considered. Every time , when the server becomes idle, an exponentially distributed single working vacation period is being initialized, during which the processing is provided with another (slower) rate. After the completion of the vacation period, the service is being continued normally, with the original speed. Using the idea of an embedded Markov chain, the systems of Volterra-type integral equations for the time-dependent queue-size distributions, conditioned by the initial buffer state and related to each other, are built for models beginning the operation in normal and working vacation modes, separately. The solutions of the corresponding systems written for the Laplace transforms are obtained in compact forms using the linear algebraic approach. The numerical illustrative examples are attached as well.

Study of the Solvability of Some Volterra-Type Integral and Integro-Differential Equations of Third Kind

For Volterra integral equations of the third kind and for Volterra-type integrodifferential equations of the third kind, theorems on the existence of solutions in Sobolev spaces (i.e., regular solutions) are proved. The proofs are based on the theory of boundary value problems for degenerate ordinary differential equations and on the theory of boundary value problems for parabolic equations with a changing evolution direction.

Semi-Analytical Solution of Equation for Phonon Radiative Transport Pertinent to Thin Films

Thermal energy transfer in thin films is governed by the phonon transport, and the Fourier heating law fails to predict correct temperature rise because of the assumption of infinite speed of the heat wave. The equation of phonon radiative transport predicts accurately thermal characteristics of the thin film when subjected to a thermal disturbance. Because the numerical solution of the transient equation phonon transport is very expensive in terms of computational efforts and run time, the analytical solution of the equation becomes fruitful. In the present study, a semi-analytical solution of transient equation for phonon radiative transport across the thin film is presented. In the analysis, the governing transport equation is transformed into two-dimensional linear mixed Fredholm–Volterra-type integral equation, and the solution is obtained through Liouville–Neumann series. The results obtained from the semi-analytical solution are compared to that of the numerical predictions, and findings revealed that both results are in good agreement. The semi-analytical solution reduces the computational run time by 30 fold shorter. Reducing the film thickness increases the temperature jump at the high-temperature edge of the film, which is more pronounced during the early heating period.

An improved PC scheme for nonlinear fractional differential equations: Error and stability analysis

A novel computationally effective fractional predictor–corrector (PC) scheme is proposed to solve fractional differential equations involving Caputo derivative. The properties of the Caputo derivative are used to reduce the fractional differential equation into a Volterra integral equation. To design high order numerical solution of FDEs, the Simpson’s 3/8 rule is applied to the Volterra type integral equation. The scheme is capable of handling both linear and nonlinear fractional differential equations. A detailed error analysis and stability analysis of the numerical scheme are rigorously established. The proposed scheme is compared with the PC schemes of literature for illustrating the effectiveness of the algorithm.

Mathematical analysis of a power-law form time dependent vector-borne disease transmission model

In the last few years, fractional order derivatives have been used in epidemiology to capture the memory phenomena. However, these models do not have proper biological justification in most of the cases and lack a derivation from a stochastic process. In this present manuscript, using theory of a stochastic process, we derived a general time dependent single strain vector borne disease model. It is shown that under certain choice of time dependent transmission kernel this model can be converted into the classical integer order system. When the time-dependent transmission follows a power law form, we showed that the model converted into a vector borne disease model with fractional order transmission. We explicitly derived the disease-free and endemic equilibrium of this new fractional order vector borne disease model. Using mathematical properties of nonlinear Volterra type integral equation it is shown that the unique disease-free state is globally asymptotically stable under certain condition. We define a threshold quantity which is epidemiologically known as the basic reproduction number (R0). It is shown that if R0 > 1, then the derived fractional order model has a unique endemic equilibrium. We analytically derived the condition for the local stability of the endemic equilibrium. To test the model capability to capture real epidemic, we calibrated our newly proposed model to weekly dengue incidence data of San Juan, Puerto Rico for the time period 30th April 1994 to 23rd April 1995. We estimated several parameters, including the order of the fractional derivative of the proposed model using aforesaid data. It is shown that our proposed fractional order model can nicely capture real epidemic.

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

We study an optimal control problem with a nonlinear Volterra-type integral equation considered on a nonfixed 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 linear–positive independence of the gradients of active mixed constraints with respect to the control. We obtain first-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the corresponding ones for problems with ordinary differential equations.

Particle in a box with a time-dependentδ-functionpotential

In quantum information processing, one often considers inserting a barrier into a box containing a particle to generate one bit of Shannon entropy. We formulate this problem as a one-dimensional Schr\"{o}dinger equation with a time-dependent $\delta$-function potential. It is a natural generalization of the particle in a box, a canonical example of quantum mechanics, and we present analytic and numerical investigations on this problem. After deriving an exact Volterra-type integral equation, composed of an infinite sum of modes, we show that approximate formulas with the lowest-frequency modes correctly capture the qualitative behavior of the wave function. If we take into account hundreds of modes, our numerical calculation shows that the quantum adiabatic theorem actually gives a very good approximation even if the barrier height diverges within finite time, as long as it is sufficiently longer than the characteristic time scale of the particle. In particular, if the barrier is slowly inserted at an asymmetric position, the particle is localized by the insertion itself, in accordance with a prediction of the adiabatic theorem. On the other hand, when the barrier is inserted quickly, the wave function becomes rugged after the insertion because of the energy transfer to the particle. Regardless of the position of the barrier, the fast insertion leaves the particle unlocalized so that we can obtain meaningful information by a which-side measurement. Our numerical procedure provides a precise way to calculate the wave function throughout the process, from which one can estimate the amount of this information for an arbitrary insertion protocol.

Orthogonal sets: The axiom of choice and proof of a fixed point theorem

In this paper, we prove some fixed point theorem on orthogonal spaces. Our result improve the main result of the paper by Eshaghi Gordji et al. [On orthogonal sets and Banach fixed point theorem, to appear in Fixed Point Theory]. Also we prove a statement which is equivalent to the axiom of choice. In the last section, as an application, we consider the existence and uniqueness of a solution for a Volterra-type integral equation in L p space.

Semi-parametric survival function estimators deduced from an identifying Volterra type integral equation

Based on an identifying Volterra type integral equation for randomly right censored observations from a lifetime distribution function F , we solve the corresponding estimating equation by an explicit and implicit Euler scheme. While the first approach results in some known estimators, the second one produces new semi-parametric and pre-smoothed Kaplan–Meier estimators which are real distribution functions rather than sub-distribution functions as the former ones are. This property of the new estimators is particular useful if one wants to estimate the expected lifetime restricted to the support of the observation time. Specifically, we focus on estimation under the semi-parametric random censorship model (SRCM), that is, a random censorship model where the conditional expectation of the censoring indicator given the observation belongs to a parametric family. We show that some estimated linear functionals which are based on the new semi-parametric estimator are strong consistent, asymptotically normal, and efficient under SRCM. In a small simulation study, the performance of the new estimator is illustrated under moderate sample sizes. Finally, we apply the new estimator to a well-known real dataset.

Integral Representation of the Manifold Solution for New Class of the Volterra Type Integral Equation with a Boundary Singularity in the Case, When Kernel Contain Logarithmic Singularity and its Power

In this work we suggestion new methods investigation the model Volterra type integral equation with logarithmic singularity, kernel which consisting from composition polynomial function with logarithmic singularity and function with singular point. The problem investigation this type integral equation at 2 reduce to problem investigate the Volterra type integral equation (1) for 2 the theory for which was constructed in [2]. In this work, we investigation integral equation (1) at 2 1. In this case, we investigate integral equation (1) reduction it’s to integral equation type [2] (I) and one the following integral equation [1] . (II)

On solving an n×n system of nonlinear Volterra integral equations by the Newton-Kantorovich method

We consider an nxn system of nonlinear integral equations of Volterra type (nonlinear VIEs) arising from an economic model. By applying the Newton-Kantorovich method to the nonlinear VIEs we linearize them into linear Volterra type integral equations (linear VIEs). Uniqueness of the solution of the system is shown. An idea has been proposed to find the approximate solution by transforming the system of linear VIEs into a system of linear Fredholm integral equations by using sub-collocation points. Then the backward Newton interpolation formula is used to find the approximate solution at the collocation points. Each iteration is solved by the Nystrom type Gauss-Legendre quadrature formula (QF). It is found that by increasing the number of collocation points of QF with fewer iterations, a high accurate approximate solution can be obtained. Finally, an illustrative example is demonstrated to validate the accuracy of the method.

Some new fixed point theorems for single and set-valued admissible mappings in Menger PM-spaces

In this paper, the new concepts of \(\alpha \)-admissible mappings with respect to \(\eta \) in single-valued case and \(\alpha _{*}\)-admissible mappings with respect to \(\eta _{*}\) in set-valued case in Menger PM-spaces are introduced, and some new fixed point results for both single-valued and set-valued mappings under certain contractive conditions are studied, which extend and generalize some results in corresponding literatures. Some examples are provided to illustrate the validity of our main results, and an application is also given to discuss the existence of solution for Volterra type integral equations.

Analytical Solution for Time-Dependent Queue-Size Behavior in the Manufacturing Line with Finite Buffer Capacity and Machine Setup and Closedown Times

A single reliable-machine manufacturing line with finite buffer capacity is considered, in which each idle period is preceded by a random closedown time, and each busy period is preceded by a random setup time. The stream of arriving jobs is described by a simple Poisson process, while processing (service), closedown and setup times are generally distributed random variables. A system of Volterra-type integral equations for the transient queue-size distribution conditioned by the initial level of buffer saturation is found by using the Markov property of successive service completion epochs and the continuous version of total probability law. A parallel system written for Laplace transforms is obtained and written in a specific form. The solution of the latter system is derived in a compact-form applying the linear algebraic approach. The final representations are found in terms of “input” system parameters (transforms of processing, closedown and setup times’ distributions) and certain sequence defined recursively.

Fixed point theorems and the Krein-S̆mulian property in locally convex spaces

AbstractThe aim of this paper is to establish some new fixed point theorems for the superposition operators in locally convex spaces which satisfy the Krein-S̆mulian property. We employ a family of measures of noncompactness in conjunction with the Schauder-Tychonoff fixed point theorem. As an application, the existence of solutions to a quite general nonlinear Volterra type integral equation is considered in locally integrable spaces.

Some fixed point theorems in Menger PbM-spaces with an application

In this paper, we establish the structure of Menger PbM-spaces as a generalization of Menger PM-spaces. We present some fixed point theorems for a new class of contractive mappings in the framework of Menger PbM-spaces. We also provide examples to illustrate the results presented herein. Then we utilize our main result to obtain the existence and uniqueness of a solution for a Volterra type integral equation.

On computing first and second order derivative spectra

Enhancing resolution in spectral response and an ability to differentiate spectral mixing in delineating the endmembers from the spectral response are central to the spectral data analysis. First and higher order derivatives analysis of absorbance and reflectance spectral data is commonly used techniques in differentiating the spectral mixing. But high sensitivity of derivative to the noise in data is a major problem in the robust estimation of derivative of spectral data. An algorithm of robust estimation of first and second order derivative spectra from evenly spaced noisy normal spectral data is proposed. The algorithm is formalized in the framework of an inverse problem, where based on the fundamental theorem of calculus a matrix equation is formed using a Volterra type integral equation of first kind. A regularization technique, where the balancing principle is used in selecting a posteriori optimal regularization parameter is designed to solve the inverse problem for robust estimation of first order derivative spectra. The higher order derivative spectra are obtained while using the algorithm in sequel. The algorithm is tested successfully with synthetically generated spectral data contaminated with additive white Gaussian noise, and also with real absorbance and reflectance spectral data for fresh and sea water respectively.