Homogeneous Linear Equations Research Articles

POINTWISE ESTIMATES OF SOLUTIONS TO WEIGHTED POROUS MEDIUM AND FAST DIFFUSION EQUATIONS VIA WEIGHTED RIESZ POTENTIALS

Problems related to the study of the properties of solutions of partial differential equations have attracted the attention of many authors in recent decades. The main qualitative properties of solutions of homogeneous linear elliptic equations of the second-order divergent type with measurable coefficients without lower-order terms are already known from the results of De Giorgi, Nash, and Moser. These results are generalized by Serrin, Ladyzhenska and Uraltseva, Aronson and Serrin, and Trudinger for wide classes of elliptic and parabolic equations with lower-order terms from the corresponding $ L^{q} $-classes. Analogous results for evolution equations with $ p \,-$Laplacian appeared much later. The first significant transition to the $ p \,-$Laplace equation with the measure $~\mu~$ in the right-hand side was achieved by Kilpelainen and Maly. They established point estimates of the solutions in terms of the nonlinear Wolff potential. These results were later extended by \linebreak Trudinger and Wang and Laboutin to nonlinear and subelliptic quasilinear equations. Irregularly elliptic and inhomogeneous parabolic equations without/or with singular lower terms have been studied for a long time. The first results in this direction were obtained by Fabes, Kenig and Separioni and Gutierrez for a weighted linear elliptic equation with weight representing $ A_{2} $ of the Mackenhaupt class. In recent decades, there has been a growing interest in parabolic and elliptic equations due to their application in modeling nonlinear physical processes occurring in heterogeneous media. Also, these equations are interesting because a general qualitative theory has not been constructed for them. Among the researchers who obtained the first significant results, we note Di Benedetto E., Bogelein V., Ivanov A. V., Duzaar F., Gianazza U., Vespri V..

On the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations in terms of (p,q)-order

On the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations in terms of (p,q)-order

Дифференцирование линейных алгебр с единицей над полем

Linear algebras over a given field arise when studying various problems of algebra, analysis and geometry. The operation of differentiation, which originated in mathematical analysis, was transferred to the theory of linear algebras over a field, as well as to the theory of rings. The set of all differentiations of a linear algebra themselves form a linear algebra. This algebra is called the algebra of differentiations. At the same time, this algebra admits the structure of a Lie algebra. If the algebra whose differentiations are considered is finite-dimensional, then its Lie algebra of differentiations will also be finite-dimensional. Therefore, the­re is a natural problem of determining the dimension of the Lie algebras of the differentiations of the linear algebra under consideration or to obtain an estimate from above of the dimension of the algebra of differentiations. To solve these problems, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differen­tia­tion. Evaluation of the rank of this system allows us to obtain an estimate from below of the rank of the matrix under consideration.

Analytical free vibration solutions of rectangular edge-cracked plates by the finite integral transform method

Analytical free vibration solutions of rectangular edge-cracked thin plates are obtained by the finite integral transform (FIT) method. While the previous studies focus on the mechanical behaviors of intact plates, this study establishes an FIT-based new scheme for analyzing the free vibration characteristics of rectangular thin plates with an edge crack. The plates may have any combined conditions of simply supported, clamped, and free boundaries. In specific solution scheme, an edge-cracked plate is decomposed into four subdomains, and a double cosine FIT is imposed on the fourth-order partial differential governing equations of each subdomain, which yields the relationship between the transformed modal deflections and naturally defined unknowns. By substituting the exterior boundary conditions of the plate, continuity conditions across the subdomains, and free conditions at the crack into the inverse transforms, the natural frequencies and unknowns are successively determined via the established homogeneous linear algebraic equations. The modal deflections are provided by further substituting the obtained unknowns into the inverse transforms. Comprehensive benchmark results for the edge-cracked plates under representative boundary conditions are shown. The results are well verified by other solution methods. A parametric study quantitatively reveals the effects of boundary conditions and crack length on the natural frequencies of edge-cracked plates. Due to the rigorous but easy-to-implement mathematical derivations, this study presents a novel solid way for exploring analytical solutions of vibration problems.

Matching variables to equations in infinite linear equation systems

A fundamental result in linear algebra states that if a homogenous linear equation system has only the trivial solution, then there are at most as many variables as equations. We prove the following generalisation of this phenomenon. If a possibly infinite homogenous linear equation system with finitely many variables in each equation has only the trivial solution, then there exists an injection from the variables to the equations that maps each variable to an equation in which it appears.

Linear quaternion-valued difference equations: Representation of solutions, controllability, and observability

In this paper, we present the fundamental theory of linear quaternion-valued difference equations. Firstly, we derive general solutions for linear homogeneous equations and give the algorithm for calculating the fundamental matrix in the case of the diagonalizable form and Jordan form. Secondly, we apply the variation of the constant formula and Z transformation to study general solutions of linear nonhomogeneous equations. We obtain the representation of solutions in the case of quaternion and complex numbers. Thirdly, we adopt the ideas from the Gram matrix and the rank of the criteria to establish sufficient and necessary conditions to guarantee that linear quaternion-valued difference equations are controllable and observable in the sense of quaternion-valued and complex numbers, respectively. In addition, a direct method to solve the control function and duality is also given. Finally, we illustrate our theoretical results with some examples.

Wong type oscillation criteria for nonlinear impulsive differential equations

We present Wong‐type oscillation criteria for nonlinear impulsive differential equations having discontinuous solutions and involving both negative and positive coefficients. We use a technique that involves the use of a nonprincipal solution of the associated linear homogeneous equation. The existence of principal and nonpricipal solutions was recently obtained by the present authors. As in special cases, we have superlinear and sublinear Emden–Fowler equations under impulse effects. It is shown that the oscillatory behavior may change due to impulses. An example is also given to illustrate the importance of the results.

Exact solutions of fractional oscillation systems with pure delay

In this paper we study the exact solutions of a class of fractional delay differential equations. We consider the fractional derivative of the order between 1 and 2 in the sense of Caputo. In the first part, we introduce two novel matrix functions, namely, the generalized cosine-type and sine-type delay Mittag-Leffler matrix functions. Then we obtain the explicit solutions for the linear homogeneous equations subjecting to corresponding initial conditions, by means of undetermined coefficients. In the second part, we first obtain a particular solution by means of the Laplace transform for the inhomogeneous equations with null initial conditions. Then we give an analytical representation of the general solution of the inhomogeneous equations through the sum of its particular solution and the general solution of the corresponding homogeneous equation.

Quantum Linear System Algorithm for General Matrices in System Identification

Solving linear systems of equations is one of the most common and basic problems in classical identification systems. Given a coefficient matrix A and a vector b, the ultimate task is to find the solution x such that . Based on the technique of the singular value estimation, the paper proposes a modified quantum scheme to obtain the quantum state corresponding to the solution of the linear system of equations in poly time for a general dimensional A, which is superior to existing quantum algorithms, where is the condition number, r is the rank of matrix A and is the precision parameter. Meanwhile, we also design a quantum circuit for the homogeneous linear equations and achieve an exponential improvement. The coefficient matrix A in our scheme is a sparsity-independent and non-square matrix, which can be applied in more general situations. Our research provides a universal quantum linear system solver and can enrich the research scope of quantum computation.

The natural frequency of a mast with an arbitrary number of panels

Introduction. The article addresses a spatial model of a statically definable mast truss consisting of four identical planar trusses with a crosswise grid system and a base with four supports at the corners. The authors solve the problem of deriving the analytical dependence between the bottom vibration frequency of the mast truss and the number of panels, mass, linear dimensions of its construction and properties of the material.
 
 Materials and methods. To calculate the values of forces, arising in the rods of a mast truss with an arbitrary number of panels, and analyze the obtained results, the induction method and operators of the Maple computer system for mathematics were used. The problem of deriving the analytical dependence between the bottom frequency of vibrations of the mast truss and its parameters is solved using the Dunkerley method, which generates the bottom estimate of the natural frequency. The rigidity of the truss structure is calculated according to the Maxwell – Mohr formula. To calculate the common members of sequences of coefficients, homogeneous linear recurrent equations are derived and solved in the frequency formula.
 
 Results. A formula is obtained for estimating the first frequency of natural vibrations of a truss. The formula coefficients have the form of polynomials of no higher than the fourth order. The accuracy of the calculation formula, obtained using the Dunkerley method, is estimated by the comparison with the first frequency, obtained through the numerical calculation of the entire spectrum of natural frequencies.
 
 Conclusions. The analysis of the analytical results and their comparison with the numerical ones shows high accuracy of the derived formula. The authors have identified a dependence, whereby an increase in the number of mast truss panels boosts the accuracy of the bottom estimate of the natural frequency.

Протокол обміну повідомленнями на основі лінійних функцій і операторів

Introduction.Safety of human activity is required for almost every enterpriseinstitution, organization, bank, etc. Therefore, it is extremely important to have a possibility to transform the information in such a way that it becomes inaccessible to the malicious user. Purpose. The purpose of this article is to describe a protocol, based on linear function and operators properties, that allows to transmit the data in fast and secure way between two subscribers – sender and receiver. Methods. The simple message exchange protocol is based on the properties of operators in linear space and the complexity of algorithms for solving systems of linear homogeneous Diophantine equations in the set of natural numbers. Results. An application to transmit a message from sender to receiver in a secure way was developed. Conclusion. The complexity of the developed algorithm belongs to the polynomial class and complexity of decryption belongs to Exptime

INVESTIGATION OF THE NORMAL VIBRATION MODES STABILITY IN SOME ESSENTIALLY NONLINEAR SYSTEMS

In the paper stability of nonlinear normal modes is analyzed by two approaches. One of them is the method of Ince algebraization, when a new independent variable associated with the unperturbed solution is introduced in the problem. In this case equations in variations are transformed to equations with singular points. The problem of determination of solutions corresponding to boundaries of the stability/ instability regions is reduced here to the problem of determination of functions that have singularity at the mentioned points. Such solutions can be obtained in the form of power series, which coefficients are satisfying a system of homogeneous linear algebraic equations. The condition ensuring the existence non-trivial solutions for such systems determines the boundaries between the stability / instability regions in the system parameter space. An advantage of the Ince algebraization is that we do not use the time-presentation of the solution when studying its stability. Other approach to investigating steady state stability is associated with the classical Lyapunov definition of stability. The analytical-numerical test proposed in the paper can be applied to a stability problem when the problem has no analytical solution. It also allows to obtain boundaries between the stability / instability regions in the system parameter space. In the present paper the first approach is used to analyze stability of normal vibration modes in the system of connected oscillators on the essentially nonlinear elastic support, and the second one is used to analyze stability of a horizontal vibration mode in the so-called stochastic absorber.

A probabilistic point of view on peak effects in linear difference equations

It is known from the literature that solutions of homogeneous linear stable difference equations may experience large deviations, or peaks, from the nonzero initial conditions at finite time instants. While the problem has been studied from a deterministic standpoint, not much is known about the probability of occurrence of such event when both the initial conditions and the coefficients of the equation have random nature. In this paper, by exploiting results on the volume of the Schur domain, we are able to compute the probability for deviations to occur. This turns out to be very close to unity, even for equations of low degree. Hence, we claim that “solutions of stable difference equations probably experience peak”. Then, we make use of tools from statistical learning to address other issues such as evaluation of the mean magnitude and maximum value of peak.

Buckling optimization of composite rectangular plates by sequential permutation search with bending-twisting correction

Buckling problem of composite plates is of importance since it is commonly encountered in thin-walled structural components in engineering. Buckling resistance design of composite plates involves the design of laminate stacking sequence exhibiting high dimensional discrete variables owing to the combination and permutation of ply orientations of layers. In this work, a sequential permutation search (SPS) optimization algorithm is presented for the stacking sequence design of composite rectangular plates accounting for the inherent physical-mechanical behavior of composite laminates. Owing to that the governing equation for buckling of composite rectangular plates is a linear homogeneous equation on bending stiffness parameters, the convexity of flexural lamination parameters and sensitivity of bending stiffness are utilized to construct sensitive detection techniques and linear search procedures. Moreover, to increase the robustness of SPS, a sign optimization algorithm (SOA) is coupled in SPS to regulate bending-twisting coupling effects. The SPS algorithm is applied to the buckling resistance design of composite rectangular plates exhibiting various boundaries under axial compression, where the Rayleigh-Ritz method is developed to perform the buckling analysis. Optimal results of SPS are compared against layerwise optimization approach (LOA) and genetic algorithm (GA), illustrating its reliability and efficiency.

Analysis of positive fractional-order neutral time-delay systems

In this paper, we consider an initial value problem for linear matrix coefficient systems of the fractional-order neutral differential equations with two incommensurate constant delays in Caputo’s sense. Firstly, we introduce the exact analytical representation of solutions to linear homogeneous and non-homogeneous neutral fractional-order differential-difference equations system by means of newly defined delayed Mittag–Leffler type matrix functions. Secondly, a criterion on the positivity of a class of fractional-order linear homogeneous time-delay systems has been proposed. Furthermore, we prove the global existence and uniqueness of solutions to non-linear fractional neutral delay differential equations system using the contraction mapping principle in a weighted space of continuous functions with regard to classical Mittag–Leffler functions. In addition, Ulam–Hyers stability results of solutions are attained based on fixed-point approach. Finally, we present an example to demonstrate the applicability of our theoretical results.

Fundamental frequency maximization of composite rectangular plates by sequential permutation search algorithm

Due to the classical thin plate theory based governing equations of composite rectangular plates for dynamic problems are linear homogeneous equations on bending stiffness coefficients, a general sequential permutation search (SPS) algorithm is presented to optimize stacking sequences of composite laminates by exploiting convexity of lamination parameters, sensitivity of bending stiffness, and bending-twisting coupling stiffness feature. Numerical examples are conducted for 8-layer composite rectangular plates with different boundaries to maximize the fundamental frequency evaluated by the Rayleigh–Ritz method, and the results are compared with those of layerwise optimization approach and genetic algorithm, demonstrating robustness and efficiency of SPS.

Solutions to a Simplified Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy

A 1-parameter initial boundary value problem (IBVP) for a linear homogeneous degenerate wave equation (JODEA, 28(1), 1) in a space-time rectangle is considered. The origin of degeneracy is the power law coefficient function with respect to the spatial distance to the symmetry line of the rectangle, the exponent being the only parameter of the problem, ranging in (0,1) and (1,2) and producing the weak and strong degeneracy respectively. In the case of weak degeneracy separation of variables is used in the rectangle to obtain the unique bounded continuous solution to the IBVP, having the continuous flux. In the case of strong degeneracy the IBVP splits into the two derived IBVPs posed respectively in left and right half-rectangles and solved separately using separation of variables. Continuous matching of the obtained left and right families of bounded solutions to the IBVPs results in a linear integro-differential equation of convolution type. The Laplace transformation is used to solve the equation and obtain a family of bounded solutions to the IBVP, having the continuous flux and depending on one undetermined function.

Automated Discovery of Geometric Theorems Based on Vector Equations

Automated discovery of geometric theorems has attracted considerable attention from the research community. In this paper, a new method is proposed to discover geometric theorems automatically. This method first generates vector equations based on given geometric relations about a geometric figure and then transforms the vector equations into a system of homogeneous linear equations; after computing the determinants of the coefficient matrices corresponding to the system of equations, the elimination method is applied to obtain a large number of geometric relationships. The test on more than 200 geometric problems shows that the geometric relationships discovered automatically by the proposed method are of obvious geometric meaning.

F-Operators for the Construction of Closed Form Solutions to Linear Homogenous PDEs with Variable Coefficients

A computational framework for the construction of solutions to linear homogenous partial differential equations (PDEs) with variable coefficients is developed in this paper. The considered class of PDEs reads: ∂p∂t−∑j=0m∑r=0njajrtxr∂jp∂xj=0 F-operators are introduced and used to transform the original PDE into the image PDE. Factorization of the solution into rational and exponential parts enables us to construct analytic solutions without direct integrations. A number of computational examples are used to demonstrate the efficiency of the proposed scheme.

A full probabilistic solution of a stochastic red blood cells model using RVT technique

Mathematical modeling is one of the most interesting ways to express the problem that describe the dynamics of the number of red blood cells count in the bloodstream of the human body. This problem has been deterministically solved based on continuous or discrete differential models. However, the stochastic models of this problem are rarely available and inadequate. This paper is organized to solve the random homogeneous linear second-order difference equation that describes a stochastic discrete red blood cells model. A complete probabilistic solution of this problem is conducted via applying the random variable transformation technique. This is achieved by deriving the first probability density function of the solution processes. The probabilistic behavior of the steady-state case (when time tends to infinity) is also studied. Moreover, the fundamental statistical measures, related to the stochastic solutions, such as the mean, the variance and the confidence intervals, are obtained. For the sake of clarity, numerical results (for pre-assigned distributions to the model parameters and initial conditions) with conclusions are presented.