Linear Inhomogeneous Equation Research Articles

Interval riccati equation-based and non-probabilistic dynamic reliability-constrained multi-objective optimal vibration control with multi-source uncertainties

Structural vibration control is used to ensure the safety of structures in service and has become an active research area in the past few years. In this study, uncertainties in structural dynamics are considered to propose an interval linear quadratic regulator (LQR) method based on an interval Riccati equation. Multi-objective optimization with non-probabilistic dynamic reliability as a constraint is performed to design an optimal uncertain vibration control system. To alleviate the difficulty of quantifying the probabilistic uncertainty and accurately characterize uncertainty information for small samples, multi-source uncertainties are regarded as interval parameters. The bounds on these uncertainties are determined, and the deterministic and uncertain parts of an interval state-space equation for vibration control are then formulated using an expanded-order matrix and the interval analysis method. Next, the interval perturbation method with a matrix series is used to develop a novel solution method for the linear inhomogeneous time-invariant equation in the interval state-space equation for the vibration control system and is used to accurately predict the interval bounds of the state responses. The conventional LQR method from modern control theory is used to propose a novel optimal control method based on a novel interval Riccati equation using the Lyapunov equation and interval uncertainty propagation. The proposed interval control method can be used to estimate the uncertain bounds of the controlled gain and cost function. A non-probabilistic dynamic reliability model is established to evaluate the reliability index of the system, which can overcome the limitations of the large computational cost of determining the probabilistic reliability. The deterministic controlled cost and uncertain states are considered as two optimization objectives with the proposed reliability constraint. The resulting constrained multi-objective optimal vibration control problem is designed and solved using c-DPEA. A flowchart is presented as a clear overview of the uncertain vibration control method, and the effectiveness of the proposed method is validated using two numerical examples. Different optimal control designs are obtained using different reliability constraints for quantitatively analyzing the safety of the control system.

Finite Element Mode-ling of Damping of a Squeeze Gas Film in High-Frequency Microelectromechanical Switches with Capacitive Contact

The damping effects of a squeeze gas film are one of the main components of damping and naturally arise in the designs of dynamic micromechanical devices, in particular, in high-frequency micromechanical switches with an electrostatic activation mechanism, in which the electrostatic drive consists of a fixed electrode in the form of a plate and a movable electrode in the form of a movable plate suspended on elastic suspensions for the case of capacitive contact. In the static mode of the device, a thin film of gas consisting of air molecules or other inert gas surrounding the device is located and held between the electrodes of the electrostatic drive. This article presents a mathematical model based on the obtained linear inhomogeneous Reynolds differential equation and performs finite element modeling of the pressure distribution in the plate of a suspended movable electrode during compression of a gas film during electrostatic activation of micromechanical switches with a capacitive contact and parallel arrangement of the electrodes of an electrostatic drive depending on the magnitude of the coefficient of this component of damping in a three-dimensional coordinate system, the dependences between the damping force and the elastic force of the compressible gas film on the value of the damping coefficient are also presented. The presented results are applicable to assess the effect of isothermal damping of a squeeze gas film on the dynamics of micromechanical devices, in particular, on the dynamic characteristics of high-frequency micromechanical switches with an electrostatic activation mechanism and a capacitive contact.

COMPOSITIONS OF FRACTIONAL DERIVATIVES AS DZHRBASHYAN - NERSESYAN DERIVATIVE

The issues of unique solvability for a special initial value problem to two classes of a linear inhomogeneous equation with a derivative of Dzhrbashyan - Nersesyan are investigated. In one of the classes, the operator at the unknown function is bounded, in the other it is sectorial. The representation of the composition of any number of Gerasimov - Caputo and/or Riemann - Liouville differential operators in the form of a Dzhrbashyan - Nersesyan derivative is also proved.

LINEAR AND QUASILINEAR EQUATIONS WITH SEVERAL GERASIMOV - CAPUTO DERIVATIVES

A representation of a solution of the Cauchy problem for a linear inhomogeneous equation solved with respect to the oldest derivative with several fractional Gerasimov - Caputo derivatives and with a sectorial pencil of linear closed operators at them in the case of the Holder function in the right-hand side of the equation is obtained; the uniqueness of the solution is proved. This result is used to reduce the Cauchy problem for the corresponding quasilinear equation to an integro-differential equation. The existence of a unique local solution is proved by the method of contraction operators in the case of local Lipschitz nonlinear operator depending on several Gerasimov - Caputo derivatives in the equation and a single global solution under the Lipschitz condition for this operator.

Об интегральном подходе при использовании метода коллокации

The article describes a matrix method of polynomial Chebyshev approximation using an integral approach to construct a solution to a nonhomogeneous fourth-order differential equation with mixed boundary conditions of the first kind. The proposed method is based on the expansion of the fourth-order derivative of the desired function into a series in terms of Chebyshev polynomials of the first kind and the representation of the partial sum of this series as a product of matrices whose elements are, respectively, the Chebyshev polynomials and the coefficients in this expansion. In this paper, using analytical formulas for calculating integrals of Chebyshev polynomials, we obtain a representation of the desired function in terms of the product of the matrices defined above. The use of points of extrema and zeros of Chebyshev polynomials of the first kind as nodes, as well as the properties of the sums of products of Chebyshev polynomials at these points, made it possible to reduce the boundary value problem by the collocation method to a system of inhomogeneous linear algebraic equations with a sparse matrix of this system. It is shown that the solution constructed in this way satisfies the differential equation at all nodes, including the boundary ones, in contrast to the approximate solution obtained by approximating the exact solution in the form of a finite sum of the Chebyshev series. The effectiveness of the proposed method is demonstrated by considering a boundary value problem with a known analytical solution. The convergence analysis of the constructed solution is carried out.

Tiresia: A code for molecular electronic continuum states and photoionization

The Tiresia program [1] provides access to numerically accurate solutions of the one-particle Schrödinger equation for highly excited states of complex polyatomic molecules, both bound and continuum, that cannot be described by conventional Quantum Chemistry approaches. It is based on an expansion of the required solution in a local multicentric basis set, with primitive functions built as products of a radial B-spline times a real spherical harmonic. In conjunction with Density Functional Theory (DFT), it has been extensively employed in a large variety of photoionization studies, also for rather large systems. Highly excited bound states as well as wavepacket propagation can also be accurately described. In fact, the flexibility of the basis essentially allows accurate solutions of linear operator equations, like Poisson or inhomogeneous perturbative equations, which are employed in the code. The program is parallelized with standard MPI-I instructions and makes extensive use of the Scalapack linear algebra library. Ancillary programs are available for the evaluation of photoionization cross sections and angular distributions from randomly to fully oriented molecules. Program summaryProgram Title: TiresiaCPC Library link to program files:https://doi.org/10.17632/fcrjxwgjxh.1Licensing provisions: GPLv3Programming language: Fortran77, Fortran90, MPISupplementary material: Program manual documentNature of problem: Accurate solutions for highly excited and continuum electronic states in complex polyatomic molecules. Molecular photoionization cross sections and angular distributions under high energy, high-intensity radiation pulses from Synchrotron radiation and laser sources, photoelectron imaging in pump-probe experiments, basis for electronic wavepackets under ultrafast or nonperturbative excitation.Solution method: Solution of the Schrödinger and similar linear operator equations in a finite domain is obtained via basis set expansion. Flexible basis set, that may approach practical completeness within the domain, is obtained as a multicenter set of B-spline radial functions times spherical harmonics. Accurate numerical integration is employed for the evaluation of matrix elements, and conventional diagonalization for bound states, or Galerkin approach for the full multichannel solution in the continuum. Full hamiltonian and dipole matrices in the spectral basis are available for time propagation. DFT many-body description is available, and strong correlations in the bound states may be incorporated via Dyson orbitals.Additional comments including restrictions and unusual features: The code is noted for computational efficiency, which allows fast yet reasonably accurate photoionization calculations for medium-sized molecules, allowing, e.g., calculations at many molecular geometries as required to describe time-resolved photoelectron spectra in pump-probe experiments.

Bounded on the semi-axis multiperiodic solution of a linear finite-hereditarity integro-differential equation of parabolic type

The question of the existence of a solution of linear integro-differential systems of parabolic type limited on the semiaxis in a spatial variable and multiperiodic in time variables was considered. Sufficient conditions of multiperiodic oscillations in time variables in a linear homogeneous equation with a boundary condition and in a linear inhomogeneous equation were established. A linear homogeneous and inhomogeneous finitehereditarity integro-differential equation of convective-diffusion type were investigated.

Integrated Resolving Functions for Equations with Gerasimov–Caputo Derivatives

The concept of a β-integrated resolving function for a linear equation with a Gerasimov–Caputo fractional derivative is introduced into consideration. A number of properties of such functions are proved, and conditions for the solvability of the Cauchy problem to linear homogeneous and inhomogeneous equations are found in the case of the existence of a β-integrated resolving function. The necessary and sufficient conditions for the existence of such a function in terms of estimates on the resolvent of its generator are obtained. The example of a β-integrated resolving function for the Schrödinger equation is given. Thus, the paper discusses some aspects of the symmetry of the concepts of integrability and differentiability. Namely, it is shown that, in the absence of a sufficiently differentiable resolving function for a fractional differential equation, the problem of the existence of a solution can be solved by an integrated resolving function of the equation.

Mechanical Analysis and Optimal Design of Wave Energy Device

At present, the global energy problem is serious. Wave energy is the kinetic energy and potential energy of waves on the ocean surface. Because of its high energy density, wide distribution, and long continuous operation of the device, it has become the focus of academic research. The key problem to be solved urgently for the large-scale application of wave energy is how to optimize the energy conversion efficiency of wave energy devices. This paper starts from the Newtonian dynamic equation, analyzes the motion state of the wave energy device in different wave environments, and studies the optimization and control of its power output. This paper first adopts the analysis method of first whole and then isolation, and establishes a one-dimensional coordinate system with the zero point of sea level. Then, the force analysis of the system is carried out, the dynamic differential equation is established, and the initial draft of the system is solved through the initial balance equation. Since the dynamic equation is a second-order linear ordinary differential inhomogeneous linear equation, the fourth-order Runge is used. The Kutta method is used to solve the problem, and the numerical solution for the position and velocity of the float is obtained. Secondly, the movable coordinate system is established with the float as the reference system, and the force analysis of the oscillator is carried out to obtain its relative dynamic equation. Then, the initial relative position of the oscillator is solved through the initial balance equation, and the relative dynamic differential equation is obtained by using the fourth-order Runge-Kutta method to obtain the numerical solution of the relative position and relative velocity of the oscillator with respect to time. Finally, the numerical solution of the absolute position and absolute velocity of the oscillator with respect to time is obtained through coordinate transformation.

Real representations of powers of real matrices and its applications

We give real representations of (or ) based on for a real square matrix A, where is the projection to the generalized eigenspace associated with an imaginary eigenvalue μ of A. Our method is based on the spectral decomposition theorem. As applications, we can easily obtain realifications of representations of solutions of inhomogeneous linear difference equations with constant coefficients.

Mean first‐passage time of cell migration in confined domains

A number of key biological processes involving cell migration in which cells traverse boundaries separating well‐defined tissues can be modeled in terms of mean first‐passage time (MFPT) problems in confined domains. Motivated by this scenario, we consider suitable three‐dimensional domains on which MFPT functions , fulfilling a Poisson‐like equation and different boundary conditions on the surface enclosing , are studied. By extending methods coming from potential theory, the calculation of boils down to dealing with inhomogeneous linear integral equations having singular kernels on . The latter are solved compactly and yield consistent 's for several domains and homogeneous boundary conditions. Moreover, the integral equation approach allows us to analyze the MFPT with mixed (Dirichlet‐Neumann) boundary conditions on for the case of a closed spherical surface with Dirichlet conditions on most of , except for a small complementary surface domain with Neumann conditions.

A fixed point theorem and Ulam stability of a general linear functional equation in random normed spaces

We prove a very general fixed point theorem in the space of functions taking values in a random normed space (RN-space). Next, we show several of its consequences and, among others, we present applications of it in proving Ulam stability results for the general inhomogeneous linear functional equation with several variables in the class of functions f mapping a vector space X into an RN-space. Particular cases of the equation are for instance the functional equations of Cauchy, Jensen, Jordan–von Neumann, Drygas, Fréchet, Popoviciu, the polynomials, the monomials, the p-Wright affine functions, and several others. We also show how to use the theorem to study the approximate eigenvalues and eigenvectors of some linear operators.

A rational-expansion-based method to compute Gabor coefficients of 2D indicator functions supported on polygonal domain

We propose a method to compute Gabor coefficients of a two-dimensional (2D) indicator function supported on a polygonal domain by means of rational expansion of the Faddeeva function and by solving second-order linear difference equations. This method has the following three attractive features: (1) the problem of computing Gabor coefficients is formulated as the calculation of a sequence of integrals with a uniform structure, (2) a rational expansion based on fast Fourier transform (FFT) is used to approximate the Faddeeva function on the entire complex plane, (3) second-order inhomogeneous linear difference equations are derived for previous integrals and they are solved stably with Olver’s algorithm. Numerical quadrature to compute Gabor coefficients is avoided. Numerical examples show this rational-expansion-based method significantly outperforms numerical quadrature in terms of computation time while maintaining accuracy.

The higher-order nonlinearity and parametric effects on dust-ion-acoustic shock waves

In the case of obliquely propagated small-amplitude shock waves, in this study, we investigate the impacts of higher-order nonlinearity as well as various parameters (such as dust concentrations, viscosity, trapping parameters, etc) on the shock wave structures. The considered magnetized plasma system consists of three components, such as inertial positive ions (mobile), trapped electrons, and immobile negatively charged dust particles. The modified Burgers equation with a dominating dissipative term (in which the viscous effect is significant) is derived initially to examine the lower-order nonlinear and dissipative effects, and then, to the best of our knowledge, the modified Burgers-type linear inhomogeneous equation is derived for the first time to observe the higher-order nonlinear effects on shock waves while the plasma contains trapped electrons. The reductive perturbation method is used for the derivation of the equations, whereas the Abel’s theorem and the method of variation of parameters are used for adding the higher-order effect. From the theoretical investigation, we observe that the higher-order nonlinearity has an increasing effect on the shock amplitude. Furthermore, the viscosity and dust concentration increase the shock width and the phase speed, respectively.

Non-homogeneous boundary value problems of the Kawahara equation posed on a finite interval

Considered in this paper is the initial boundary value problem (IBVP) of the Kawahara equation, a class of the fifth order KdV equation, posed on a finite interval, (0.1)ut+ux+βuxxx+uxxxxx+uux=0,0<x<L,t>0,u(x,0)=ϕ(x), subject to the non-homogeneous boundary conditions (0.2)Bju=hj(t),j=1,2,3,4,5t>0where Bju=∑k=04(ajk∂xku(0,t)+bjk∂xku(L,t)),j=1,2,3,4,5and ajk,bjk(k=0,1,2,3,4andj=1,2,3,4,5) are real constants. Under some general assumptions imposed on the coefficients ajk,bjk, the IBVP (0.1)–(0.2) is shown to be locally well posed in the space Hs(0,L) for any s≥0 with naturally compatible ϕ∈Hs(0,L) and boundary values hj,j=1,2,3,4,5 belonging to some appropriate spaces with optimal regularity. The sharp Kato smoothing properties (due to Kenig, Ponce and Vega (Kenig et al., 1991; Kenig et al., 1993)) of the pure initial value problem (IVP) of the linear inhomogeneous fifth order KdV equation posed on the whole line R, vt+βvxxx+vxxxxx=g(x,t),v(x,0)=ψ(x),x,t∈R,have played an important role in establishing the well-posedness of the IBVP (0.1)–(0.2).

Monotone methods in counterparty risk models with nonlinear Black–Scholes-type equations

A nonlinear Black–Scholes-type equation is studied within counterparty risk models. The classical hypothesis on the uniform Lipschitz-continuity of the nonlinear reaction function allows for an equivalent transformation of the semilinear Black–Scholes equation into a standard parabolic problem with a monotone nonlinear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most computational power for computing an approximate solution to the inhomogeneous linear diffusion equation, we discuss also this question and suggest several solution methods, including those based on Monte Carlo and finite differences/elements.

Vertical vibration of a rigid strip footing on a half-space of unsaturated porous soil

This study presents an analytical solution to the compliance coefficient of rigid strip footings lying on an unsaturated porous soil half-space when subjected to vertical vibration. Using the Fourier transform techniques, the complex interaction between unsaturated soils and rigid strip foundations is transformed into a mixed boundary-value problem based on the developed framework of Biot’s model. The wavenumber domain auxiliary function and boundary conditions are used to convert the governing equations to a pair of dual integral equations. To yield the compliance function, the integral equations are transformed into a system of inhomogeneous linear equations by utilizing hypergeometric polynomials and Bessel functions. The proposed analytical solution is verified against existing solutions to interactions between foundations and saturated soil or single-phase soil by adjusting the parameters in the unsaturated soil model. Additionally, the impacts of soil skeleton saturation, intrinsic permeability, and Poisson’s ratio of unsaturated soil on the dynamic compliance coefficient of rigid strip footings are examined. It is shown that the Poisson’s ratio of the soil skeleton and the soil saturation at critical saturation have a significant effect on the dynamic compliance coefficient, but have a negligible effect on the intrinsic permeability.

GOURSAT PROBLEM FOR THE ADJOINT MODEL TELEGRAPH EQUATION WITH THE RATE a(s, t) IN THE UPPER HALF-PLANE

In the upper half-plane, the classical solution and correctness criterion of the Goursat problem for a linear inhomogeneous adjoint model telegraph equation with variable rate a(s,t) are found explicitly. An explicit formula is obtained for the classical solution of this Goursat problem, unique and stable with respect to the right hand side of the equation and Goursat data. This formula contains implicit characteristic functions of the equation. In the case of a homogeneous conjugate model telegraph equation, the classical solution of this Goursat problem is the Riemann function in all linear mixed (initial-boundary) problems for an inhomogeneous model telegraph equation with variable rate a(s,t). This Riemann function has been calculated by us. A correctness criterion according to Hadamard (necessary and sufficient conditions) of its unique and stable on the right-hand side of the equation and the Goursat data solvability is found. This criterion consists of smoothness requirements on the righthand side of the equation and two Goursat data. The smoothness requirements on the right side of the equation are the condition of continuity of the right-side and the corresponding integral smoothness conditions on the right side of the equation and on the Goursat data – their twice continuous differentiability in the upper half-plane.

Strong Solution and Optimal Control Problems for a Class of Fractional Linear Equations

In this paper, we examine the unique solvability (in the sense of strong solutions) of the Cauchy problem for a linear inhomogeneous equation in a Banach space solved with respect to the Caputo fractional derivative. We assume that the operator acting on the unknown function in the right-hand side of the equation generates an analytic resolving operator family for the corresponding homogeneous equation. We obtain a representation of a strong solution of the Cauchy problem and examine the solvability of optimal control problems with a convex, lower semicontinuous, lower bounded, coercive functional for the equation considered. The general results obtained are used to prove the existence of an optimal control in problems with specific functionals. Abstract results obtained for a control system described by an equation in a Banach space are illustrated by examples of optimal control problems for a fractional equation whose special cases are the subdiffusion equation and the diffusion wave equation.

The solution of nonlinear direct and inverse problems for beam by means of the Trefftz functions

Mathematical modelling makes it possible to analyse physical phenomena which are frequently described by non-linear differential equations. Those equations can be solved by a number of numerical methods, but not all are efficient. The paper presents an approximate solution of the nonlinear problems for beam vibrations. The main aim of this work is to develop an efficient method for solving inverse nonlinear and time-dependent problems. The main idea of this method is a decomposition of a nonlinear operator into a linear and nonlinear part. Next, the Picard’s iterations are used. In each iteration a linear combination of the Trefftz functions for the linear operator is used. The nonlinearity is treated as an inhomogeneity for the linear operator. Hence, in each iteration a linear inhomogeneous equation is solved. The presented numerical examples confirm efficiency of the method in finding a stable solution of inverse nonlinear problems, even with noisy data.