Ordinary Integro-differential Equation Research Articles

Complexity for a class of elliptic ordinary integro-differential equations

Consider the variational form of the ordinary integro-differential equation (OIDE)−u″+u+∫01q(⋅,y)u(y)dy=f on the unit interval I, subject to homogeneous Neumann boundary conditions. Here, f and q respectively belong to the unit ball of Hr(I) and the ball of radius M1 of Hs(I2), where M1∈[0,1). For ε>0, we want to compute ε-approximations for this problem, measuring error in the H1(I) sense in the worst case setting. Assuming that standard information is admissible, we find that the nth minimal error is Θ(n−min⁡{r,s/2}), so that the information ε-complexity is Θ(ε−1/min⁡{r,s/2}); moreover, finite element methods of degree max⁡{r,s} are minimal-error algorithms. We use a Picard method to approximate the solution of the resulting linear systems, since Gaussian elimination will be too expensive. We find that the total ε-complexity of the problem is at least Ω(ε−1/min⁡{r,s/2}) and at most O(ε−1/min⁡{r,s/2}ln⁡ε−1), the upper bound being attained by using O(ln⁡ε−1) Picard iterations.

A similarity solution for a pre-existing fluid driven fracture in a permeable rock

The problem of a two-dimensional, pre-existing, fluid-driven fracture propagating in a permeable rock is investigated. The fracturing fluid is a viscous, incompressible Newtonian fluid and the flow of fluid inside the fracture is laminar. The elasticity of the rock is modelled using the Cauchy principal value integral derived from linear elastic fracture mechanics. With the aid of lubrication theory, a nonlinear partial integro-differential equation relating the fracture half-width to the pressure and leak-off velocity is derived. Similarity solutions are derived for the fracture half-width, pressure and leak-off velocity and are used to reduce the partial integro-differential equation to an ordinary integro-differential equation. In order to close the problem, a model in which the leak-off velocity is proportional to the fracture half-width and the gradient of the fluid–rock interface was used. Numerical results are obtained for the fracture length, fracture half-width and the net fluid pressure.

Self-similar Solution of Population Balance Equation for Aggregation with Constant Kernel

Self-similar solution of population balance equation includes the number density function which remains invariant or contains a part that is invariant. This paper describes self-similar solution of aggregation population balance model with constant kernel. Using this constant kernel, moment of the population balance system achieves the form μ_i (t)∝t^(i-1) and the aggregation population balance equation with constant kernel reduces to a first order linear ordinary integro-differential equation whose solution is exponential function with negative power.

A Simple Semi-Analytic Contact Mechanical Model for Tangential and Torsional Fretting Wear of Axisymmetric Contacts

Fretting wear of axisymmetric contacts is considered within the framework of the Hertz–Mindlin approximation and the Archard law for the linear wear. If the characteristic time scale for the wear is much larger than the duration of a single fretting oscillation, the profile change due to wear during one fretting cycle can be neglected for the contact problem as a zero-order approximation. This allows to give an exact contact solution during each fretting cycle, depending on the current worn profile, and thus for the explicit statement of an ordinary integro-differential equation system for the time-evolution of the fretting profile, which can be easily solved numerically. The proposed method gives the same results as a known, contact mechanically more rigorous simulation procedure that also operates within the framework of the Hertz–Mindlin approximation, but works significantly faster than the latter one. Tangential and torsional fretting wear are considered in detail. A comparison of the numerical prediction for the evolution of the worn profile in partial slip torsional fretting of a rubber ball on abrasive paper shows good agreement with experimental results from the literature.

A Simple Semi-Analytic Contact Mechanical Model for Tangential and Torsional Fretting Wear of Axisymmetric Contacts

Fretting wear of axisymmetric contacts is considered within the framework of the Hertz–Mindlin approximation and the Archard law for the linear wear. If the characteristic time scale for the wear is much larger than the duration of a single fretting oscillation, the profile change due to wear during one fretting cycle can be neglected for the contact problem as a zero-order approximation. This allows to give an exact contact solution during each fretting cycle, depending on the current worn profile, and thus for the explicit statement of an ordinary integro-differential equation system for the time-evolution of the fretting profile, which can be easily solved numerically. The proposed method gives the same results as a known, contact mechanically more rigorous simulation procedure that also operates within the framework of the Hertz–Mindlin approximation, but works significantly faster than the latter one. Tangential and torsional fretting wear are considered in detail. A comparison of the numerical prediction for the evolution of the worn profile in partial slip torsional fretting of a rubber ball on abrasive paper shows good agreement with experimental results from the literature.

Linear stability analysis of flashing instability based on the homogeneous equilibrium model

Flashing instability is a two-phase flow instability widely reported in low-pressure natural circulation that induces undesirable flow oscillations with large magnitudes challenging the operation of facilities. The current study performs linear stability analysis on the flashing instability using a homogenous equilibrium model developed for a typical natural circulation configuration. Physical simplifications are performed, and the governing equations are analytically solved lumping the dynamics of the flow into the evolution of the inlet flow rate governed by a linear ordinary integro-differential equation. Similarity groups governing the dynamics is extracted from the non-dimensionalization. Static stability of the flow is checked based on the classic criterion of the excursive instability, which confirms negligible potential of the static instability and proves the dynamic nature of the flashing instability. The Laplace transformation is performed providing the characteristic function governing the dynamic stability, and the D-partition method is implemented producing the stability boundaries. Through reviewing the characteristic function, a simplified model is obtained around the stability boundary with the low exit flow quality, revealing three dominant feedback effects. For conditions with a short two-phase region near the top of the chimney, the flashing is found to introduce a strong delayed feedback between the gravitational driving force and the inlet flow rate, which essentially triggers the instability under favorable oscillation frequencies properly shifting the phase differences. The extracted dominant dynamics explains the parametric effect on the stability, and analytically predicts a proportional relation between oscillation period and the fluid residence time in the hot leg. The modeling and analysis in the current study supplements the analytical basis for understanding the flashing instability and provides valuable reference for further analytical studies.

A novel deterministic forecast model for COVID-19 epidemic based on a single ordinary integro-differential equation.

In this paper, we present a new approach to deterministic modelling of COVID-19 epidemic. Our model dynamics is expressed by a single prognostic variable which satisfies an integro-differential equation. All unknown parameters are described with a single, time-dependent variable R(t). We show that our model has similarities to classic compartmental models, such as SIR, and that the variable R(t) can be interpreted as a generalized effective reproduction number. The advantages of our approach are the simplicity of having only one equation, the numerical stability due to an integral formulation and the reliability since the model is formulated in terms of the most trustable statistical data variable: the number of cumulative diagnosed positive cases of COVID-19. Once this dynamic variable is calculated, other non-dynamic variables, such as the number of heavy cases (hospital beds), the number of intensive-care cases (ICUs) and the fatalities, can be derived from it using a similarly stable, integral approach. The formulation with a single equation allows us to calculate from real data the values of the sample effective reproduction number, which can then be fitted. Extrapolated values of R(t) can be used in the model to make reliable forecasts, though under the assumption that measures for reducing infections are maintained. We have applied our model to more than 15 countries and the ongoing results are available on a web-based platform [1]. In this paper, we focus on the data for two exemplary countries, Italy and Germany, and show that the model is capable of reproducing the course of the epidemic in the past and forecasting its course for a period of four to five weeks with a reasonable numerical stability.

Variable viscosity effects for the steady flow past a sphere

The effect of a spatially dependent viscosity in the unbounded flow around a rigid spherical particle that translates with constant velocity is investigated theoretically. The variable viscosity emulates the effect of a variable concentration of an additive material in a simple solvent. The analysis is performed utilizing a smooth function for the viscosity of the additive material, which can describe qualitatively both depletion and accumulation phenomena around the particle. Assuming steady state, creeping and isothermal conditions, and no external forces and torques, the momentum and mass balances are a generalization of the classical Stokes equations for which the linearity is preserved. Manipulating suitably the governing partial differential equations, a single ordinary integrodifferential equation for the radial part of the radial velocity component is derived. This equation is solved either numerically using a Chebyshev pseudospectral method or analytically using an asymptotic technique. A decrease in the total drag on the particle as the additive material increases is predicted in the depletion case. In the accumulation case, the total drag may increase or decrease in comparison to the simple Newtonian fluid. Analysis of the total drag to its individual contributions reveals that the friction drag (due to viscous forces) is affected substantially by the change of the viscosity, while the form drag (due to pressure) varies much smoother and milder. Finally, we investigate under which conditions the variable viscosity fluid can be approximated as a constant viscosity fluid with Navier type slip at the wall.

The sharp constant for the Burkholder–Davis–Gundy inequality and non-smooth pasting

We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy (Acta Math. 124 (1970) 249–304) and Davis (Israel J. Math. 8 (1970) 187–190) for continuous martingales. For the inequalities $\mathbb{E}[\tau^{\frac{p}{2}}]\leq C_{p}\mathbb{E}[(B^{*}(\tau))^{p}]$ with $0<p<2$ we propose a connection of the optimal constant $C_{p}$ with an ordinary integro-differential equation which gives rise to a numerical method of finding this constant. Based on numerical evidence, we are able to calculate, for $p=1$, the explicit value of the optimal constant $C_{1}$, namely $C_{1}=1.27267\ldots$ . In the course of our analysis, we find a remarkable appearance of “non-smooth pasting” for a solution of a related ordinary integro-differential equation.

Existence and uniqueness results for a time-fractional nonlinear diffusion equation

In this work we consider a nonlinear ordinary integro-differential equation which arises in the studies of time-fractional porous medium equation. The nonlocality of the resulting free-boundary problem is governed by the Erdélyi–Kober operator which requires using other than classical proof techniques. To prove the existence and uniqueness of a compactly supported solution we reduce the free-boundary case to the initial-value problem. Next, we use the sub- and supersolution technique to show that there exists a globally defined unique solution. As a side product, some estimates on the exact solution are found.

Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel

This article examines questions of unique solvability for an inverse boundary value problem to recover the coefficient and boundary regime of a nonlinear integro-differential equation with degenerate kernel. We propose a novel method of degenerate kernel for the case of inverse boundary value problem for the considered ordinary integro-differential equation of second order. By the aid of denotation, the integro-differential equation is reduced to a system of algebraic equations. Solving this system and using additional conditions, we obtained a system of two nonlinear equations with respect to the first two unknown quantities and a formula for determining the third unknown quantity. We proved the single-value solvability of this system using the method of successive approximations.

Inverse problem for an ordinary integro-differential equation with degenerate kernel and nonlocal integral conditions

Inverse problem for an ordinary integro-differential equation with degenerate kernel and nonlocal integral conditions

A novel approach to construct the adjoint problem for a first-order functional integro-differential equation with general nonlocal condition

In this work, with the aim of determining Green’s solution or generalized Green’s solution, we propose a novel constructive approach by which a linear or specific nonlinear problem involving general linear nonlocal condition for a first-order functional ordinary integro-differential equation with general nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness is reduced to an integral equation. A system of two integro-algebraic equations, called “the adjoint system,” is constructed for this problem. Green’s functional for the problem with trivial kernel and generalized Green’s functional for the problem with nontrivial kernel are the unique solutions to the specific cases of this adjoint system. Green’s functional and generalized Green’s functional have two components. Their first components correspond to Green’s function and generalized Green’s function for the problem, respectively. Some illustrative applications are provided with known and unknown results.

The stochastic-volatility, jump-diffusion optimal portfolio problem with jumps in returns and volatility

This paper treats the risk-averse optimal portfolio problem with consumption in continuous time for a stochastic-jump-volatility, jump-diffusion (SJVJD) model of the underlying risky asset and the volatility. The new developments are the use of the SJVJD model with logtruncated-double-exponential jump-amplitude distribution in returns and exponential jumpamplitude distribution in volatility for the optimal portfolio problem. Although unlimited borrowing and short-selling play an important role in pure diffusion models, it is shown that borrowing and short-selling are unrealistically constrained for infinite-range jump-amplitudes. Finite-range jump-amplitude models can allow constraints to be very large in contrast to infinite range models which severely restrict the optimal instantaneous stock-fraction to [0,1]. The reasonable constraints in the optimal stock-fraction due to jumps in the wealth argument for stochastic dynamic programming jump integrals remove a singularity in the stock-fraction due to vanishing volatility. Main modifications for the usual constant relative risk aversion (CRRA) power utility model are for handling the partial integro-differential equation (PIDE) resulting from the additional variance independent variable, instead of the ordinary integro-differential equation (OIDE) found for the pure jump-diffusion model of the wealth process. In addition to natural constraints due to jumps when enforcing the positivity of wealth condition, other constraints are considered for all practical purposes under finite market conditions. Computational results are presented for optimal portfolio values, stock fraction and consumption policies.

Valuation of stock loans with jump risk

A stock loan is a special loan with stocks as collateral, which offers the borrowers the right to redeem the stocks on or before the maturity (Xia and Zhou, 2007; Dai and Xu, 2011). We investigate pricing problems of both infinite- and finite-maturity stock loans under a hyper-exponential jump diffusion model. In the infinite-maturity case, we derive closed-form formulas for stock loan prices and deltas by solving the related optimal stopping problem explicitly. Moreover, we obtain a sufficient and necessary condition under which the optimal stopping time is finite with probability one. In the finite-maturity case, we provide analytical approximations to both stock loan prices and deltas by solving an ordinary integro-differential equation as well as a complicated non-linear system. Numerical experiments demonstrate that the approximation methods for both prices and deltas are accurate, fast, and easy to implement.

Central Upwind Scheme for Solving Multivariate Cell Population Balance Models

Microbial cultures are comprised of heterogeneous cells that differ according to their size and intracellular concentrations of DNA, proteins and other constituents. Because of the included level of details, multi-variable cell population balance models (PBMs) offer the most general way to describe the complicated phenomena associated with cell growth, substrate consumption and product formation. For that reason, solving and understanding of such models are essential to predict and control cell growth in the processes of biotechnological interest. Such models typically consist of a partial integro-differential equation for describing cell growth and an ordinary integro-differential equation for representing substrate consumption. However, the involved mathematical complexities make their numerical solutions challenging for the given numerical scheme. In this article, the central upwind scheme is applied to solve the single-variate and bivariate cell population balance models considering equal and unequal partitioning of cellular materials. The validity of the developed algorithms is verified through several case studies. It was found that the suggested scheme is more reliable and effective.

Nonlinear vibration of viscoelastic sandwich plates under narrow-band random excitations

The effect of the narrow-band random excitation on the non-linear response of sandwich plates with an incompressible viscoelastic core is investigated. To model the core, both the transverse shear strains and rotations are assumed to be moderate and the displacement field in the thickness direction is assumed to be linear for the in-plane components and quadratic for the out-of-plane components. In connection to the moderate shear strains considered for the core, a non-linear single-integral viscoelastic model is also used for constitutive modeling of the core. The fifth-order perturbation method is used together with the Galerkin method to transform the nine partial differential equations to a single ordinary integro-differential equation. Converting the lower-order viscoelastic integral term to the differential form, the fifth-order method of multiple scale is applied together with the method of reconstitution to obtain the stochastic phase-amplitude equations. The Fokker–Planck–Kolmogorov equation corresponding to these equations is then solved by the finite difference method, to determine the probability density of the response. The variation of root mean square and marginal probability density of the response amplitude with excitation deterministic frequency and magnitudes are investigated and the bimodal distribution is recognized in narrow ranges of excitation frequency and magnitude.

High Resolution Finite Volume Schemes for Solving Multivariable Biological Cell Population Balance Models

Multivariable cell population balance models are commonly used to explain complicated biological phenomena associated with product formation and cell growth in microbial populations. Such models typically consist of a partial integrodifferential equation for describing cell growth and an ordinary integrodifferential equation for representing substrate consumption. Due to their mathematical complexities, the numerical solutions of such models are hard tasks for numerical schemes. In this article, semidiscrete high resolution flux-limiting finite volume schemes are applied to solve single-variate and bivariate cell population balance models. The schemes have the abilities to achieve narrow peaks and resolve sharp discontinuities in the solutions on coarse meshes. These schemes are cheaper due to their short and reliable computational coding for complex problems. Several case studies are carried out. The numerical results of the schemes are compared with each other in terms of CPU time and accuracy. After consideration of different growth rate functions and incorporating equal and unequal partitioning, the suggested schemes were found to be more reliable and effective.

Well-posedness and asymptotics of some nonlinear integro- differential equations

We will prove the well-posedness of certain second order ordinary and partial integro-differential equations with an integral term of the form t ∫ 0 m(φ(t−s))φ(s) ds, wherem is given and φ is the solution. The new aspect is the dependence of the kernel on the solution. In addition, for the ordinary integro-differential equation, the asymptotic behaviour of the solution is described for some kernels.

Stochastic-Volatility, Jump-Diffusion Optimal Portfolio Problem with Jumps in Returns and Volatility

This paper treats the risk-averse optimal portfolio problem with consumption in continuous time for a stochastic-jump-volatility, jump-diffusion (SJVJD) model of the underlying risky asset and the volatility. The new developments are the use of the SJVJD model with log-truncated-double-exponential jump-amplitude distribution in returns and exponential jump-amplitude distribution in volatility for the optimal portfolio problem. Although unlimited borrowing and short-selling play an important role in pure diffusion models, it is shown that borrowing and short-selling are unrealistically constrained for infinite-range jump-amplitudes. Finite-range jump-amplitude models can allow constraints to be very large in contrast to infinite range models which severely restrict the optimal instantaneous stock-fraction to [0,1]. The reasonable constraints in the optimal stock-fraction due to jumps in the wealth argument for stochastic dynamic programming jump integrals remove a singularity in the stock-fraction due to vanishing volatility. Main modifications for the usual constant relative risk aversion (CRRA) power utility model are for handling the partial integro-differential equation (PIDE) resulting from the additional variance independent variable, instead of the ordinary integro-differential equation (OIDE) found for the pure jump-diffusion model of the wealth process. In addition to natural constraints due to jumps when enforcing the positivity of wealth condition, other constraints are considered for all practical purposes under finite market conditions. Computational results are presented for optimal portfolio values, stock fraction and consumption policies.