Form Of Integro-differential Equation Research Articles

An Integro-Differential Equation Formulation for the Ideal Veselago Lens

An Integro-Differential Equation Formulation for the Ideal Veselago Lens

Evolution of the rotational motion of a viscoelastic planet with a core on an elliptical orbit

The work is devoted to the study of the evolution of the rotational motion of a planet in the central Newtonian field of forces. The planet is modeled by a body consisting of a solid core and a viscoelastic shell rigidly attached to it. A limited formulation of the problem is considered, when the center of mass of the planet moves along a given Keplerian elliptical orbit. The equations of motion are derived in the form of a system of Routh equations using the canonical Andoyer variables, which are “action-angle” variables in the unperturbed problem and have the form of integro-differential equations with partial derivatives. The technique developed by V.G. Vilke is used for mechanical systems with an infinite number of degrees of freedom. A system of ordinary differential equations is obtained by the method of separation of motions. The system describes the rotational motion of the planet taking into account the perturbations caused by elasticity and dissipation. An evolutionary system of equations for the “action” variables and slow angular variables is obtained by the averaging method. A phase portrait is constructed that describes the mutual change in the modulus of the angular momentum vector G of the rotational motion and the cosine of the angle between this vector and the normal to the orbital plane of the planet’s center of mass. A stationary solution of the evolutionary system of equations is found, which is asymptotically stable. It is shown that in stationary motion, the angular momentum vector G is orthogonal to the orbital plane, and the limiting value of the modulus of this vector depends on the eccentricity of the elliptical orbit. The constructed mathematical model can be used to study the tidal evolution of the rotational motion of planets and satellites. The results obtained in this work are consistent with the results of previous studies in this area.

ASYMPTOTIC EXPANSION OF SOLUTION OF BVP WITH INITIAL JUMPS FOR SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATION

We consider the undivided boundary value problem for singularly perturbed linear integro-differential equations of th order with Fredholm’s integral terms have an initial jump of the phenomenon th order on the left end of the segment. Defined regular and boundary layer members of the asymptotic expansions of the solutions. Regular members of the asymptotics constructed in the form of integro-differential equations, which different from the usual unperturbed equations by the presence of additional term called initial jump of integral terms. The values of the initial jumps are defined. Boundary conditions for the regular members of the asymptotics also contain additional term, called an initial jump of the derivatives th order. Thus, for the determination of the regular members of the asymptotics obtained boundary value problems for linear integro-differential with additional parameter. To determine the boundary layer of the asymptotic expansion of solution we obtained initial problems for homogeneous an inhomogeneous ordinary differential equations with constant coefficients. Exponential estimates for boundary layer terms of the asymptotics are obtained. A theorem of existence and uniqueness and the asymptotic representations of solution with an estimate of the remainder term of the asymptotics. It is found that the constructed asymptotics approximation to the solution of the original singularly perturbed integro-differential boundary value problem is uniformly throughout the considered interval.

Direct and inverse electromagnetic scattering problems for bi-anisotropic media**This paper is dedicated to the memory of Professor Armin Lechleiter.

This paper is concerned with the direct and inverse scattering of time-harmonic electromagnetic waves from bi-anisotropic media. For the direct problem, we establish an integro-differential equation formulation, its Fredholm property, and the uniqueness of a weak solution. Using this integro-differential formulation we study a fast spectral Galerkin method for the numerical solution to the direct problem. Numerical examples are presented and convergence of the spectral method is proved via Gårding estimates for (strongly singular) integro-differential equations. We solve the inverse problem of recovering bi-anisotropic scatterers from far-field data using orthogonality sampling methods. These methods aim to construct imaging functionals which are robust to noise, computationally cheap, and require data for only one or a few incident fields. We provide some theoretical analysis as well as numerical simulations for the proposed imaging functionals.

A thermodynamically consistent peridynamics model for visco-plasticity and damage

This article presents a unified visco-plastic-damage model in the peridynamics set-up which may be applied across different regime of strain rates and temperatures. In the model, we introduce two internal variables, one describing plastic flow and other the damage in the material. Exploiting the idea of master balance, in addition to the conventional momentum balances, we postulate micro-force balances for both plastic flow and damage evolution in terms of additional peridynamic force states. The equations of motion are in the form of integro-differential equations and do not require continuity of field variables. Using the idea of energy equivalence and entropy equivalence, constitutive relations for the peridynamic force states are determined. The proposed peridynamic visco-plastic-damage model may be thought as a non-trivial extension of the recently developed peridynamic visco-plasticity model (Rahaman et al.,2017). The current scheme couples the visco-plasticity and damage in a thermo-dynamically consistent manner and provides temperature evolution which reflects contribution from both plasticity and damage. The efficacy of the model is demonstrated through simulations of the adiabatic shear band propagation as observed in Kalthoff–Winkler experiment and the shear plugging failure of Weldox 460 E steel plates along with the determination of the ballistic limit.

Asymptotic behavior of a neural field lattice model with a Heaviside operator

Motivated by the importance of discrete structures of neuron networks, a neural field lattice system arising from the discretization of neural field models in the form of integro-differential equations is studied. The neural field lattice system is first formulated as a differential inclusion on a weighted space of infinite sequences, due to the switching effects. Then the existence of solutions of the resulting differential inclusion is proved by a series of sequential finite-dimensional approximations. The solutions are shown to generate a nonautonomous set-valued dynamical system which possesses a pullback attractor. Forward omega limit sets for the set-valued dynamical system are also discussed.

Trapped modes due to narrow cracks in thin simply-supported elastic plates

We consider an elastic plate of infinite length and constant width supported simply along its two parallel edges and having a finite length crack along its centreline. In particular, we look for and find trapped modes (localised oscillations) in the presence of the crack. An explicit wide-spacing approximation based on the Wiener–Hopf technique applied to incident wave scattering by semi-infinite cracks is complemented by an exact formulation of the problem in the form of integro-differential equations. An application of a Galerkin method for the numerical calculation of results from the latter method leads to a novel explicit ‘small-spacing’ approximation. In combination with the wide-spacing results this is shown to provide accurate results for all lengths of crack.

Pricing Exotic Variance Swaps Under 3/2-Stochastic Volatility Models

We consider pricing of various types of exotic discrete variance swaps, like the gamma swaps and corridor swaps, under the 3/2-stochastic volatility models with jumps. The class of stochastic volatility models (SVM) that use a constant-elasticity-of-variance (CEV) process for the instantaneous variance exhibit nice analytical tractability when the CEV parameter takes just a few special values (namely, 0, 1/2, 1 and 3/2). The popular Heston model corresponds to the choice of the CEV parameter to be 1/2. However, the stochastic volatility dynamics derived from the Heston model fails to agree with empirical findings from actual market data. The choice of 3/2 for the CEV parameter in the SVM shows better agreement with empirical studies while it maintains a good level of analytical tractability. By using the partial integro-differential equation formulation, we manage to derive quasi-closed form pricing formulas for the fair strike values of various types of discrete variance swaps. Pricing properties of these exotic discrete variance swaps under different market conditions are explored.

Scattering From Isotropic Connected Wire Medium Metamaterials: Three-, Two-, and One-Dimensional Cases

Scattering problems involving wire media are computationally intensive due to the spatially dispersive nature of homogenized wire media. In this work, an integro-differential equation based on a transport formulation is proposed instead of the convolution-form integral equation that directly arises from spatial dispersion. The integro-differential equation is much faster to solve than the convolution equation form, and its effectiveness is confirmed by solving several examples in one-, two-, and three-dimensions. As experimental confirmation of both the integro-differential equation formulation and the homogenized wire medium parameters, several isotropic connected wire medium spheres have been fabricated on a rapid-prototyping machine, and their measured extinction cross sections compared with simulation results. Wire parameters (period and diameter) are varied to the point where homogenization theory breaks down, which is reflected in the measurements.

Analysis of Variance of Integro-Differential Equations with Application to Population Dynamics of Cotton Aphids

The population dynamics of cotton aphids is usually described by mechanistic models, in the form of integro-differential equations (IDEs), with the IDE parameters representing some key properties of the dynamics. Investigation of treatment effects on the population dynamics of cotton aphids is a central issue in developing successful chemical and biological controls for cotton aphids. Motivated by this important agricultural problem, we propose a framework of analysis of variance (ANOVA) of IDEs. The main challenge in estimating the IDE-based ANOVA model is that IDEs usually have no analytic solution, and repeatedly solving IDEs numerically leads to a high computational cost. We propose a penalized spline method in which spline functions are used to estimate the IDE solutions and the penalty function is defined by the IDEs. The estimated IDE solutions, as implicit functions of the parameters, are inputs in a nonlinear least squares criterion, which in turn is minimized by a Gauss–Newton algorithm. The proposed method is illustrated using simulation and an observed cotton aphid data set.

A dynamic non-isothermal model for a hydrocracking reactor: Model development by the method of continuous lumping and application to an industrial unit

Hydrocracking is an important refinery process which is carried out in catalytic reactors to convert heavy petroleum fractions into valuable products. Because of the large number of species and complex reactions involved, modeling of hydrocracking is a challenging task. In this paper a dynamic, non-isothermal reactor model has been constructed using the method of continuous lumping which treats the complex reactive mixture as a continuum. In doing so concentrations are characterized in terms of reactivity which is a monotonic function of the true boiling point of the mixture. The material and energy balances are developed in the form of integro-differential equations. The significant modeling parameters are identified and estimated using data from an industrial reactor. Steady-state and dynamic predictions of the model outputs such as reactor temperature, product yields and hydrogen consumption are shown to be in good agreement with plant data.

The maximum surplus before ruin and related problems in a jump-diffusion renewal risk process

In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.

Traveling wave solutions in an integro-differential competition model

We study the existence of traveling wave solutions for the two-species Lotka-Volterra competition model in the form of integro-differential equations. The model incorporates asymmetric dispersal kernels that describe long distance dispersal processes of competing species in space. Using lower and upper traveling wave solutions, we show that the model has traveling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.

An integro-differential equation for surface ocean waves with finite depth

An explicit integro-differential equation formulation is derived for surface ocean waves with finite depth. The equation involves only 2D surface variables. For this equation, we establish the stability and existence of solutions, and explain the effect of depth on surface wave properties.

Waves, bumps, and patterns in neural field theories

Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axo-dendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.

Static Model Describing the Inhomogeneous Deformation of Unidirectional Fibrous Composites under Plane Strain

A mathematical model is proposed to describe the static deformation of a unidirectional fiber-reinforced composite under plane-strain conditions in a plane perpendicular to the fiber direction. The model is based on the stochastic static equations for an elastic homogeneous two-component medium with nonzero body forces. Applying the method of conditional moments and double Fourier transform yields a nonlocal model in the form of integro-differential equations. Expanding the integral kernels into series in expansion coefficients yields differential equations whose order depends on the number of terms in the series. The zero-order approximation leads to the theory of effective moduli, the first-order approximation to the refined theory of effective moduli, and the second-order approximation to fourth-order equilibrium equations for mean displacements (mathematical expectations) and formulas for displacements, strains, and stresses averaged over the composite and its components. All the coefficients in theses formulas are expressed in terms of the elastic constants of the components and the geometric parameters of the structure. The model is valid for heavy stress gradients. In a particular case, the static theory of two-component elastic mixtures follows from the model

Odor interactions and learning in a model of the insect antennal lobe

We present a new model of insect antennal lobe in the form of integro-differential equation with short-range inhibition. The learning in the model modifies the odor-dependent input by adding a term that is proportional to the firing rates of the network in the pre-learning steady state. We study the modification of odor-induced spatial patterns (steady states) by combination of odors (binary mixtures) and learning. We show that this type of learning applied to the inhibitory network “increases the contrast” of the network's spatial activity patterns. We identify pattern modifications that could underlie insect behavioral phenomena.

3-D analysis of temperature distribution in the material during pulsed laser and material interaction

Recently, lasers are being increasingly used in the industry owing to their precision and low cost. Material is heated and evaporated during laser and material interaction due to the absorption of laser beams by the material. In this study, a 3-D Laser heating model including evaporation has been solved using the electron- kinetic theory approach. The basis in examining the problem using the kinetic theory approach is to describe the heat conduction through electron-phonon and molecule-phonon collisions. The problem is solved by using the electron-kinetic theory approach in such a way that heat conduction is taken into account until the material is heated to its melting temperature and non-conduction limited heat transfer is considered after the melting temperature is reached. Non-conduction limited heat transfer through the phase change process is resulted from vacancy-molecule collisions. A numerical scheme is introduced to solve the governing equation, owing to the fact that the energy equation resulted is in the form of integro-differential equation. Four different materials, namely iron, nickel, tantalum and titanium are chosen in this study determine the material response to laser pulse heating. For each material, time dependent temperature distribution through the depth of the material and on the surface of the material is computed and analyzed for four different materials.

Coupled channel formulation of the perturbed finite-temperature atomic random phase approximation: single channel approximation

We present the spin and angular momentum analysis of the inhomogeneous coupled partial integro-differential equation formulation of the finite-temperature random phase approximation of an atom in a plasma. We demonstrate the cancellation of the direct and exchange self-interaction terms in the three-dimensional form of the equations. The direct self-interaction terms are also shown to cancel for each partial wave in the radial equations obtained from the spin and angular momentum analysis. We discuss how this inhomogeneous formulation avoids normalization difficulties encountered in our previous homogeneous eigenvalue formulation. Finally, the photoabsorption of the atom is given in terms of its polarizability.

3-dimensional modeling of laser repetitive pulse heating: a phase change and a moving heat source considerations

Laser repetitive pulse heating may offer thermal integration at the workpiece surface during laser material processing. However, the condition for the thermal integration highly depends on the material response to successive laser pulses. In the present study, 3-dimensional laser heating of semi-infinite steel subjected to repetitive pulsation is considered. In modeling the process, an electron kinetic theory approach is employed and governing equations are derived. Since the governing equations derived are in the form of integro-differential equations, they do not yield analytical solutions. Therefore, the numerical method employing an explicit scheme is introduced to solve the governing equations. In the analysis, conduction and phase change processes are taken into account and the laser beam is considered as scanning the surface with a constant velocity. The repetitive pulses are introduced such that the ratio of successive pulse intensity to the first pulse intensity is kept constant and the cooling period between two consecutive pulses is also kept constant. To investigate the power intensity ratio on the resulting temperature profiles, three ratios are considered separately. Moreover, the condition for the thermal integration and isothermal heating are introduced. It is found that the temperature profiles do not follow the pulse profile and that thermal integration is possible for the low power intensity ratio considered in the present study.