Inhomogeneous Differential Equations Research Articles

New results for drift estimation in inhomogeneous stochastic differential equations

New results for drift estimation in inhomogeneous stochastic differential equations

Formal solutions of some family of inhomogeneous nonlinear partial differential equations, Part 2: Summability

In this article, we investigate the summability of the formal power series solutions in time of a class of inhomogeneous nonlinear partial differential equations in two variables, whose corresponding Newton polygon admits a unique positive slope k, the latter being determined by the highest spatial-derivative order of the initial equation. We give in particular a necessary and sufficient condition for the k-summability of the solutions in a given direction, and we illustrate this result by some examples. This condition generalizes the ones already given by the second author in Remy (2016, 2020, 2021 [25,26], 2022, 2023). In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof of our main result.

Semi-Classical Limit and Quantum Corrections in Non-Coincidence Power-Law f(Q)-Cosmology

Within the framework of symmetric teleparallel fQ-gravity, using a connection defined in the non-coincidence gauge, we derive the Wheeler–DeWitt equation of quantum cosmology. The gravitational field equation in fQ-gravity permits a minisuperspace description, rendering the Wheeler–DeWitt equation a single inhomogeneous partial differential equation. We use the power-law fQ=f0Qμ model, and with the application of linear quantum observables, we calculate the wave function of the universe. Finally, we investigate the effects of the quantum correction terms in the semi-classical limit.

Investigation of the dynamics of transverse oscillations of a vertical rod under gravity, friction, and thermal expansion

The paper considers the mathematical formulation of the problem of transverse oscillations of a vertical rod under gravity, friction and external pulse effect, leading to thermal expansion of the rod. The dynamics of the system under consideration corresponds to the behavior of a fuel element (FE) in a pulsed reactor and is related to the dynamic stability of the processes occurring in it.The FE dynamics is described by inhomogeneous linear differential equation of the fourth order with non-constant coefficients. Initial boundary conditions are not smooth since they correspond to the instant heating of the part of the FE surface exposed to neutron pulse radiation. The general solution of the homogeneous linear equation can be found using the concept of generalized functions expressed as a series expansion in terms of coordinate eigenfunctions dependent on p parameter. The p parameter is related with the FE mass and at some certain values results in bifurcation points of the boundary value problem for eigenfunctions. The partial solution can be found in several ways: using the Fourier transform, the method of Green's function, and in terms of series expansion by eigenfunctions.Eigenfunction expansion coefficients in an explicit form have been obtained and the numerical solution accuracy has been estimated for some important particular cases. The results of the obtained exact solutions appear to be very close to the results of the ANSYS numerical estimations. In the future the obtained exact solutions will be used as input data to simulate self-consistent system dynamics of more than 400 FE. Therefore, the advantage of the exact solution in practical use is that its calculation is much faster as compared with the finite element method done with ANSYS.

Analysis of fractional Euler-Bernoulli bending beams using Green’s function method

This paper deals with the effect of fractional order on the fractional Euler-Bernoulli beams model. This model is based on the elastic curve that can be approximated using tangent lines and the osculating circle. We introduce a general form of the fractional Green’s function for the highest-order term 1<α≤2, which gives a unique solution for the geometrical fractional bending beams with an inhomogeneous fractional differential equation. The analytical closed-form responses were obtained by inverse problem technique and integration over Green’s function kernel. The theory and properties of fractional Green’s function method are described by generalizing classical theory to the initial and two-point fractional boundary value problems. Static analysis of the numerical examples, such as determinate and indeterminate beams with typical loads and beam property boundary conditions, are presented and discussed. The results were verified using the direct integration method. The fractional conventional ratio indicates the beam's deformation deviation from its classical state. The analysis results reveal that changing the fractional parameter significantly impacts the deflection behavior of flexural beams due to the beam’s characteristics and the fractional order. The behavior of the fractional beam is identical to the classical state when the fractional parameter is an integer number.

Analytical method for solving linear abstract differential equations of order [formula omitted]

In this paper, the concept of point-wise independent set of closed operators is introduced. The new concept together with the use of Hahn–Banach Theorem enable us to reduce an inhomogeneous linear abstract differential equation of order n namely, Anu(n)(t)+An−1u(n−1)(t)+⋯+A1u′(t)+A∘u(t)=f(t), into an inhomogeneous linear ordinary differential equation with constant coefficients of order n. The involved operators in the main abstract equation are assumed to be closed and linear on a given Banach space while the unknown function is assumed to be n-times differentiable.

Higher order contribution to ion-acoustic Korteweg–de Vries (KdV) soliton in unmagnetized quantum plasmas with arbitrary degeneracy of electrons

The propagation of nonlinear ion-acoustic Korteweg–de Vries (KdV) solitons and dressed solitons (with higher order contribution term) in unmagnetized quantum plasmas with arbitrary degeneracy of electrons is investigated using quantum hydrodynamic model. The reductive perturbation method is used to derive the first order nonlinear ion-acoustic KdV equation and also higher order contribution of inhomogeneous differential equation is obtained for the dressed ion-acoustic wave soliton in arbitrary degenerate electron plasmas. Moreover, the higher order inhomogeneous differential equation is solved (nonsecular solution is obtained) using the renormalization method. The conditions for the validity of the higher order correction are also discussed in detail for ion-acoustic solitons in unmagnetized arbitrary degenerate electrons plasma case. The effects of chosen values of electron temperature, equilibrium fugacity and corresponding electron density of a physical quantum plasma system and quantum diffraction parameter H (for two conditions i.e., H < 2 or H > 2) on the amplitude and width of the KdV and dressed ion-acoustic wave solitons for both electrostatic potential hump (compressive) or dip (rarefactive) structures are investigated. The numerical plots are also presented for illustration with numerical values of a physical partially degenerate electrons plasma system exist in the astrophysical or laboratory conditions.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

Investigation of Thermo-Electrical Instabilities in a Semiconductor as 2D Dynamical Systems

A semiconducting sample placed in cryogenic media with applied electric field generates low frequency oscillations of electric current and sample temperature and known to be thermo-electrical instabilities. Although observation of current oscillations on oscilloscope is possible, change of sample temperature cannot be detected experimentally. Description of the phenomenon through mathematical equations helps to understand relationship of the two variables as well as their connection to deep trap behavior that are involved in supporting the instability. Mathematical model for thermo-electrical instabilities in an n type semiconductor based on the two deep trap level model with non-degenerate electron statistics has been introduced in order to investigate the unique relationship between the change in time of both electric current flowing through a semiconductor sample and the sample temperature. The 3D dynamical system of nonlinear inhomogeneous ordinary differential equations has been investigated as component 2D dynamical systems (n,T), (n,n&amp;lt;sub&amp;gt;t&amp;lt;/sub&amp;gt;) and (n&amp;lt;sub&amp;gt;t&amp;lt;/sub&amp;gt;,T) for local behavior at isolated equilibrium and at points on individual trajectories, where n, n&amp;lt;sub&amp;gt;t&amp;lt;/sub&amp;gt; and T are free electron concentration at conduction band, electron concentration at deep traps and temperature of a semiconductor sample accordingly. Each of the planar systems is expressed in canonical form and investigated as a Cauchy problem with a set of appropriate initial values. This paper presents investigation results of phase trajectories of the planar systems depending on a single parameter – the temperature of cooling media T&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;. Based on obtained calculation results of time sequences of the three variables n, n&amp;lt;sub&amp;gt;t&amp;lt;/sub&amp;gt; and T, phase differences among these variables have been determined for different values of T&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;. It has been established that the change in sample temperature lags behind change in current and this lag increases with T&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;. Clearly defined correlations among systems (n,T), (n,n&amp;lt;sub&amp;gt;t&amp;lt;/sub&amp;gt;) and (n&amp;lt;sub&amp;gt;t&amp;lt;/sub&amp;gt;,T) are seen, being the result of balance between field aided and thermal ionization mechanisms for charge carrier generation and recombination processes. Thermal and field assisted generation mechanisms compete with one another in achieving steady non equilibrium state in the system depending on temperature of cooling media T&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;.

IMPROVEMENT OF THE METHODOLOGY FOR OPTIMIZING THE PARAMETERS OF COMPLEX SYSTEMS UNDER THE INFLUENCE OF THERMAL LOAD SOURCES

The article investigates some aspects of solving the problem of constructing mathematical models for optimizing the parameters of local laser action on multilayer microbiological systems. The constraints on the desired laser parameters and on the temperature field in the material subject to laser fission are set. Verification of compliance with the restrictions on not exceeding the maximum value of the temperature field requires multiple calculations of the corresponding temperature field in the microbiological material. Control over the temperature field limits ensures the viability of its parts during the biotechnological process of laser fission. To construct adequate optimization mathematical models, the author substantiates the adequacy of the computational mathematical models describing the state of a microbiological system under the action of laser radiation sources. A multi-point boundary value problem with a system of inhomogeneous differential heat conduction equations for a multilayer microbiological medium is investigated and the correctness of this problem is substantiated for minor perturbations of the right-hand side of the differential equation. The results obtained in the article guarantee the adequacy of applied optimization mathematical models for finding rational values of technical parameters of laser emitters. In order to improve the accuracy of optimization of the technical parameters of laser emitters, the article presents a mathematical modeling of the preparatory stage of embryo defrosting. The presented mathematical models and methods of their implementation are necessary to improve the quality of embryo defrosting. Improvement of the methodology for solving applied biotechnological problems will certainly lead to the complication of mathematical models, but it will increase the accuracy of calculation and optimization of technical parameters of the biotechnological process of laser embryo division.

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.

Creep of a closed cylindrical tank at hydrostatic pressure

Objective. When constructing the resolving relations of the theory of shells, the validity of the basic assumptions about the material of the structure under consideration is assumed, which is considered homogeneous, isotropic and viscoelastic, i.e. obeying the Maxwell-Gurevich law. The subject to study is a polymer cylindrical shell, rigidly clamped at the base and subject to internal hydrostatic pressure. Method. The problem is reduced to an inhomogeneous differential equation of the fourth degree with respect to the displacement of the middle surface w along the z axis. Since the closed form representation of the solution to this equation is extremely difficult, the search for it is presented numerically, in particular, using the grid method. The creep strain components ε*x, ε*θ, γ*xθ were determined as a linear approximation of the velocity by the Runge-Kutta method. Result. In the process of calculating the shell using moment theory, it was found that as a result of shell creep, tangential stresses increased by more than 12 percent. Conclusion. The proposed technique makes it possible to simulate changes in the mechanical properties of the shell (for example, indirect heterogeneity) caused by the influence of physical fields.

The inhomogeneous complex partial differential equations for bi-polyanalytic functions

&lt;abstract&gt;&lt;p&gt;In this paper, we study a Riemann-Hilbert problem related to inhomogeneous complex partial differential operators of higher order on the unit disk. Applying the Cauchy-Pompeiu formula, we find out the solvable conditions and obtain the representation of the solutions. Then, we investigate the boundary value problems for bi-polyanalytic functions with the Dirichlet and Riemann-Hilbert boundary conditions, obtain the specific solution and the solvable conditions, and extend the conclusion to the corresponding higher-order problems. Therefore, we obtain the solution to the half-Neumann problem of higher order for bi-polyanalytic functions.&lt;/p&gt;&lt;/abstract&gt;

THERMAL ANALYSIS OF BI-DIRECTIONAL FUNCTIONALLY GRADED PLATES SUBJECTED TO HEAT GENERATION

The two-dimensional steady-state heat transfer problem is analyzed under the most general boundary conditions in a heterogeneous environment. It is assumed that the internal heat generation varies arbitrarily in two directions, while the thermal conductivity coefficient of the material is bi-directly (thickness-length) graded with the Mori-Tanaka model. The linear inhomogeneous partial differential equations produced under these conditions may not be solved by conventional methods. The governing equation of the system is numerically solved with high accuracy using the 2D pseudospectral Chebyshev method. Benchmark problems are used to check the accuracy of the method. It has been observed that the effect of volumetric heat generation on the temperature distribution is more than the heat transmission coefficient. Besides, this method is simple and effective and can be easily applied to such problems.

Advances in mathematical analysis for solving inhomogeneous scalar differential equation

&lt;p&gt;This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model's parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term.&lt;/p&gt;

Research on the distribution of heat fluxes during infrared drying of grain material

Purpose. To develop mathematical models of unsteady heat and mass transfer during the infrared (IR) drying process, and to create an algorithm for calculating the variable power of IR energy input to ensure quality drying of thermolabile material. Methods. The methods of mathematical modeling used to describe the unsteady drying process of thermolabile material in a dryer are based on the formalized description in the form of nonlinear inhomogeneous differential equations, taking into account the change in the power of IR energy input during the drying process. Results. A dependency for calculating the perceivable power of the IR source by the grain, based on the given process mode parameters, has been obtained. The drying kinetics of the grain, derived from formula (10), more accurately characterizes the process from a physical point of view than the drying equation (8) with a constant coefficient kck. It was established that three characteristic periods of the drying process can be traced in the graphical dependencies: I – heating; II – constant drying rate; III – falling drying rate. The calculation of heat fluxes for downward cross-ventilation of the grain layer, where air washes over IR heaters and reflectors and heats up to 35 °C, is presented. It was found that in the implementation of the process with a downward airflow, it is necessary to supply 20–25 % less radiant energy to the material at the same drying exposures according to the graphical dependencies. Conclusions. A mathematical description of the unsteady drying process of thermolabile material in a dryer, considering the change in the power of IR energy input during drying, has been formulated. An algorithm for calculating parameters that can be used for process identification and optimization of the operating parameters of existing grain dryers has been developed. Keywords: drying, grain material, mathematical model, diffusive transfer, moisture content, temperature, speed.

Параболічні крайові задачі математичної фізики в кусково-однорідному клиновидному циліндрично-круговому півпросторі з порожниною

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous by the radial variable r wedge-shaped by the angular variable φ cylindrical-circular half-space with a cavity were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of the 1st kind (Dirichlet) and the 2nd kind (Neumann) and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a finite integral Fourier transform on the Cartesian semiaxis (0; +∞) by an applicative variable z and a Weber hybrid integral transform type on the polar axis (R0; +∞) with n points of conjugation by a radial variable were used to construct solutions of investigated boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The consistent application of inverse integral transforms to the obtained solution in the space of images restores the solutions of the considered parabolic boundary value problems through their integral image in an explicit form in the space of the originals. At the same time, the main solutions to the problems were obtained in an explicit form.

Cauchy Problem for a System of Linear Inhomogeneous Differential Equations of the First Order with Degenerate Matrix at the Derivatives and Pulsed Actions at Fixed Times

Cauchy Problem for a System of Linear Inhomogeneous Differential Equations of the First Order with Degenerate Matrix at the Derivatives and Pulsed Actions at Fixed Times

Phase Function Methods for Second Order Inhomogeneous Linear Ordinary Differential Equations

Phase Function Methods for Second Order Inhomogeneous Linear Ordinary Differential Equations

A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations

A computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.