Differential Equations In Space Research Articles

Differential equations with maximal monotone operators

The paper deals with multivalued differential equations in abstract spaces. Nonlocal conditions are assumed. The model includes an m-dissipative multioperator which generates an equicontinuous, not necessarily compact, semigroup. The regularity of the nonlinear term also depends on the Hausdorff measure of noncompactness. The existence of integral solutions is discussed, with a topological index argument. A transversality condition is required. The results are applied to a partial differential inclusion in a bounded domain in Rn with nonlocal integral conditions. The model also includes an m-dissipative but not necessarily compact semigroup generated by a suitable subdifferential operator.

Energy based methods applied in mechanics by using the extended Noether's formalism

AbstractPhysical systems are modeled by field equations; these are coupled, partial differential equations in space and time. Field equations are often given by balance equations and constitutive equations, where the former are axiomatically given and the latter are thermodynamically derived. This approach is useful in thermomechanics and electromagnetism, yet challenges arise once we apply it in damage mechanics for generalized continua. For deriving governing equations, an alternative method is based on a variational framework known as the extended Noether's formalism. Its formal introduction relies on mathematical concepts limiting its use in applied mechanics as a field theory. In this work, we demonstrate the power of extended Noether's formalism by using tensor algebra and usual continuum mechanics nomenclature. We demonstrate derivation of field equations in damage mechanics for generalized continua, specifically in the case of strain gradient elasticity.

Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals.We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.

Stochastic analysis of small-scale beams with internal and external damping

Random flexural vibrations of small-scale Bernoulli–Euler beams with internal and external damping are investigated in the paper. Such a kind of problem occurs in design and optimization of structural components of smart miniaturized electromechanical systems. Internal damping is related to the viscoelastic stress–strain law describing the beam time-dependent behavior while the external damping represents the interaction between the structure and the surrounding viscous environment. In the proposed formulation, Boltzmann superposition integral and fractional-order viscoelasticity are exploited to capture the above mentioned non-conventional phenomena. Size effects caused by long range interactions arising at small-scale are also taken into account and modeled by means of an integral nonlocal formulation based on the stress-driven approach. Moreover, in order to reproduce stochastic vibrations of small-size devices, such as resonators and sensors, the external loading are modeled by taking into account their random nature. The relevant dynamic problem is thus described by a fractional-order stochastic partial differential equation in space and time equipped with standard and non-classical boundary conditions. A proper differential eigenanalysis for the nonlocal viscoelastic bending problem is performed to find the modal shapes and closed-form expressions of power spectral density and stationary variance of displacements are provided. The proposed methodology, accounting for size-effects, hereditariness behaviors, viscous properties of surrounding environment and randomness of external loadings, is suitable to model and capture the effective behavior of miniaturized devices. The presented theoretical findings and numerical outcomes show the influence of size-effects and viscoelastic parameters on the structural response.

Well-Posedness of the Complex Cauchy Problem for Systems of Partial Differential Equations in Spaces of Entire Functions with Integral Metrics

Well-Posedness of the Complex Cauchy Problem for Systems of Partial Differential Equations in Spaces of Entire Functions with Integral Metrics

Field analogue of the Ruijsenaars-Schneider model

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of L-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical elliptic spin chain. In this way, one obtains a lattice field analogue of the Ruijsenaars-Schneider model with continuous time. The second method is based on investigation of general elliptic families of solutions to the 2D Toda equation. We derive equations of motion for their poles, which turn out to be difference equations in space with a lattice spacing η, together with a zero curvature representation for them. We also show that the equations of motion are Hamiltonian. The obtained system of equations can be naturally regarded as a field generalization of the Ruijsenaars-Schneider system. Its lattice version coincides with the model introduced via the first method. The limit η → 0 is shown to give the field extension of the Calogero-Moser model known in the literature. The fully discrete version of this construction is also discussed.

Solvability of infinite systems of fractional differential equations in the space of tempered sequence space $m^beta(phi)$

The purpose of this article, is to establish the existence of solution of infinite systems of fractional differential equations in space of tempered sequence $m^beta(phi)$ by using techniques associated with Hausdorff measures of noncompactness. Finally, we provide an example to highlight and establish the importance of our main result.

Uniformly convergent computational method for singularly perturbed unsteady burger-huxley equation

This paper deals with the numerical treatment of a singularly perturbed unsteady non-linear Burger-Huxley problem. Due to the simultaneous presence of a singular perturbation parameter and non-linearity in the problem applying classical numerical methods to solve this problem on a uniform mesh are unable to provide oscillation-free results unless they are applied with very fine meshes inside the region. Thus, to resolve this issue, a uniformly convergent computational scheme is proposed. The scheme is formulated:•First, the non-linear singularly perturbed problem is linearized using the Newton-Raphson-Kantorovich quasilinearization technique.•The resulting linear singularly perturbed problem is semi-discretized in time using the implicit Euler method to yield a system of singularly perturbed ordinary differential equations in space.•Finally, the system of singularly perturbed ordinary differential equations are solved using fitted exponential cubic spline method.The stability and uniform convergence of the proposed scheme are investigated. The scheme is stable and ε−uniformly convergent with first order in time and second order in space directions. To validate the applicability of the proposed scheme several test examples are considered. The obtained numerical results depict that the proposed scheme provides more accurate results than some methods available in the literature.

Solution of linear and non-linear partial differential equation of fractional order

We know that the solution of partial differential equations by analytical method is better than the solution by approximate or series solution method. In this paper, we discuss the solution of linear and non-linear fractional partial differential equations involving derivatives with respect to time or space variables by converting them into the partial differential equations of integer order. Also we develop an analytical formulation to solve such fractional partial differential equations. Moreover, we discuss the method to solve the fractional partial differential equations in space as well as time variables simultaneously with the help of some examples.

Numerical investigation for the effect of deformation and dynamic pressure on the fast drainage of porous materials

In the case of a fast drainage process, water movements result in a rapid change in capillary pressure and shrinkage of a porous material. Until now the models used to predict the deformation of most porous media are based on the quasi-static behaviour laws, and dynamic effects are completely neglected. In this paper, a fully coupled model with dynamic capillarity effect considered is proposed for unsaturated water-air flow within deforming porous media. Then, this model is validated with respect to documented experiment on investigation of dynamic capillarity effect in unsaturated sand. The Finite Element Method (FEM) and backward Euler method are used for the discrete approximation of the partial differential equations in space and time, respectively. Numerical results provided by dynamic model have high agreement with recorded experiment data. Moreover, numerical results also reveal that dynamic effects on capillary pressure could cause unexpected deformation (shrinkage) for porous media. All investigations suggest that dynamic effects on capillary pressure should be taken into account to predict unsaturated water-air flow and shrinkage for porous media.

THE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS

For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Dirichlet problem. This work dedicated to systems of first-order partial differential equations of elliptic and hyperbolic types consisting of four equations with three unknown variables. An explicit solution of the CauchyDirichlet problem is constructed using the method of an exponential – differential operator. Giving a very simple example of the co-solution of the Cauchy problem for a second-order differential equation and the Cauchy problem for systems of first-order hyperbolic differential equations.

Free vibration response of internally-thickness-tapered laminated composite square plates based on an energy method

In the present work, the free vibration response of symmetric linearly-thickness-tapered laminated composite square plates is considered for a variety of taper configurations. Since exact and closed-form solutions for the natural frequencies and mode shapes of the plates could not be obtained from the corresponding complex partial differential equations in space and time coordinates, the Ritz method in conjunction with the Classical Laminated Plate Theory (CLPT) and then the First-order Shear Deformation Theory (FSDT) is used to obtain the system’s mass and stiffness matrices for out-of-plane bending vibrations. The free vibration analysis of the plates is conducted using these system matrices and the natural frequencies and mode shapes are determined. Different boundary conditions are considered for the free vibration response of the plates and the computational work is performed in the MATLAB® environment. The convergence of the series expansions used for the out-of-plane displacement functions and the Ritz solutions for plates with various boundary conditions are established. The demonstration of solution accuracy is performed by conducting a so-called “layer reduction test” and also by comparing the results obtained by using the Ritz method with the solution obtained based on the Finite Element Method (FEM) using ANSYS®. A parametric study is then conducted to investigate the influences of taper configurations and taper parameters on the free vibration response of the composite plates.

Numerical Solution of European and American Option with Dividends using Finite Difference Methods

Numerical methods form an important part of the pricing of financial derivatives where there is no closed form analytical formula. Black-Scholes equation is a well known partial differential equation in financial mathematics. In this paper, we have studied the numerical solutions of the Black-Scholes equation for European options (Call and Put) as well as American options with dividends. We have used different approximate to discretize the partial differential equation in space and explicit (Forward Euler’s), fully implicit with projected Successive Over-Relaxation (SOR) algorithm and Crank-Nicolson scheme for time stepping. We have implemented and tested the methods in MATLAB. Finally, some numerical results have been presented and the effects of dividend payments on option pricing have also been considered.

Random vibrations of stress-driven nonlocal beams with external damping

Stochastic flexural vibrations of small-scale Bernoulli-Euler beams with external damping are investigated by stress-driven nonlocal mechanics. Damping effects are simulated considering viscous interactions between beam and surrounding environment. Loadings are modeled by accounting for their random nature. Such a dynamic problem is characterized by a stochastic partial differential equation in space and time governing time-evolution of the relevant displacement field. Differential eigenanalyses are performed to evaluate modal time coordinates and mode shapes, providing a complete stochastic description of response solutions. Closed-form expressions of power spectral density, correlation function, stationary and non-stationary variances of displacement fields are analytically detected. Size-dependent dynamic behaviour is assessed in terms of stiffness, variance and power spectral density of displacements. The outcomes can be useful for design and optimization of structural components of modern small-scale devices, such as Micro- and Nano-Electro-Mechanical-Systems (MEMS and NEMS).

NONLOCAL CAUCHY PROBLEM FOR SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS AND FRACTIONAL DIFFERENTIAL EQUATIONS

The aim of this paper is to prove the existence and uniqueness of mild and classical solutions of a second order evolution equation with functional dependence on the solutions and on derivatives of the solutions. The theory of strongly continuous cosine families of linear operators in a Banach space is applied. Further we discuss the existence of solutions of nonlinear fractional differential equations in abstract spaces.

Angular Momentum in Bipolar Outflows: Dynamical Evolutionary Model

Abstract We model molecular outflows produced by the time-dependent interaction between a stellar wind and a rotating cloud envelope in gravitational collapse, studied by Ulrich. We consider spherical and anisotropic stellar winds. We assume that the bipolar outflow is a thin shocked shell, with axial symmetry around the cloud rotation axis, and obtain the mass and momentum fluxes into the shell. We solve numerically a set of partial differential equations in space and time and obtain the shape of the shell, the mass surface density, the velocity field, and the angular momentum of the material in the shell. We find that there is a critical value of the ratio between the wind and the accretion flow momentum rates β that allows the shell to expand. As expected, the elongation of the shells increases with the stellar wind anisotropy. In our models, the rotation velocity of the shell is in the range of 0.1–0.2 km s−1, a factor of 5–10 lower than the values measured in several sources. We compare our models with those of Wilkin & Stahler for early evolutionary times and find that our shells have the same sizes at the pole, although we use different boundary conditions at the equator.

Continuum Gating Current Models Computed with Consistent Interactions

The action potential of nerve and muscle is produced by voltage-sensitive channels that include a specialized device to sense voltage. The voltage sensor depends on the movement of charges in the changing electric field as suggested by Hodgkin and Huxley. Gating currents of the voltage sensor are now known to depend on the movements of positively charged arginines through the hydrophobic plug of a voltage sensor domain. Transient movements of these permanently charged arginines, caused by the change of transmembrane potential V, further drag the S4 segment and induce opening/closing of the ion conduction pore by moving the S4-S5 linker. This moving permanent charge induces capacitive current flow everywhere. Everything interacts with everything else in the voltage sensor and protein, and so it must also happen in its mathematical model. A Poisson-Nernst-Planck (PNP)-steric model of arginines and a mechanical model for the S4 segment are combined using energy variational methods in which all densities and movements of charge satisfy conservation laws, which are expressed as partial differential equations in space and time. The model computes gating current flowing in the baths produced by arginines moving in the voltage sensor. The model also captures the capacitive pile up of ions in the vestibules that link the bulk solution to the hydrophobic plug. Our model reproduces the signature properties of gating current: 1) equality of ON and OFF charge Q in integrals of gating current, 2) saturating voltage dependence in the Q(charge)-voltage curve, and 3) many (but not all) details of the shape of gating current as a function of voltage. Our results agree qualitatively with experiments and can be improved by adding more details of the structure and its correlated movements. The proposed continuum model is a promising tool to explore the dynamics and mechanism of the voltage sensor.

Energy estimates and model order reduction for stochastic bilinear systems

ABSTRACT In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretised systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied, the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an -error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.

Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation

We develop two linear, second order energy stable schemes for solving the governing system of partial differential equations of a hydrodynamic phase field model of binary fluid mixtures. We first apply the Fourier pseudo-spectral approximation to the partial differential equations in space to obtain a semi-discrete, time-dependent, ordinary differential and algebraic equation (DAE) system, which preserves the energy dissipation law at the semi-discrete level. Then, we discretize the DAE system by the Crank-Nicolson (CN) and the second-order backward differentiation/extrapolation (BDF/EP) method in time, respectively, to obtain two fully discrete systems. We show that the CN method preserves the energy dissipation law while the BDF/EP method does not preserve it exactly but respects the energy dissipation property of the hydrodynamic model. The two new fully discrete schemes are linear, unconditional stable, second order accurate in time and high order in space, and uniquely solvable as linear systems. Numerical examples are presented to show the convergence property as well as the efficiency and accuracy of the new schemes in simulating mixing dynamics of binary polymeric solutions.

A simplified model for high-rate actuation of shape memory alloy torque tubes using induction heating

Shape memory alloy actuators deliver high forces while being compact and reliable, making them ideal for consideration in aerospace applications. Induction heating of shape memory alloy actuators, specifically tubes that twist about the longitudinal axis, has recently been studied experimentally and computationally using finite element analysis. Reduced-order models for the torsional behavior of shape memory alloy tubes and induction heating of general metallic tubes exist, and thus, it is possible for these thermal and mechanical models to be combined and numerically solved. This work develops and implements an engineering model for inductively heated shape memory alloy tubes based on the reduction of the governing partial differential equations in space and time to an ordinary differential equation in time. An example solution is compared to finite element analysis results and agrees well. Finally, the ordinary differential equation is linearized and solved analytically. The linearized model agrees well with the nonlinear ordinary differential equation and finite element model.