Multidimensional Differential Equations Research Articles

Constructing Multidimensional Difference Equations From a State-Space Representation Using the Generalized Cayley-Hamilton Theorem

Constructing Multidimensional Difference Equations From a State-Space Representation Using the Generalized Cayley-Hamilton Theorem

RESEARCH OF COMPUTATIONAL MATHEMATICAL MODELS FOR TECHNICAL SYSTEMS

In the theory of analysis and synthesis of technical systems, mathematical modelling and optimization of multilayer systems containing sources of physical fields occupy an important place. This is due to the fact that their state is described by means of boundary value problems with multidimensional differential equations. To solve the boundary value problems and implement the process of optimizing the technical parameters of the modelled systems, it is necessary to conduct interdisciplinary studies of computational and applied optimization mathematical models. Fulfilment of the conditions for the existence of a single solution to boundary value problems by default is possible only when the object of study is a single-layer material under the action of load sources. If it is necessary to calculate and optimize the technical parameters of a multilayer material subjected to load sources, then it is impossible to immediately guarantee the correctness of the calculated and applied optimization mathematical models, since it is necessary to obtain the conditions for the existence and uniqueness of solutions to boundary value problems with systems of differential equations. Maximizing the technical parameters of load sources and averaging the characteristics of material layers will lead to approximate values of the objective function and technical parameters of the modelled system, which leads to irrational consumption of energy and heat resources and uncontrolled losses, and useless losses of the test material in the technological process. The article presents the conditions for the correctness of multipoint boundary value problems with multidimensional differential equations describing the state of a multilayer material under thermal action. It is advisable to use these studies to substantiate the correctness of other technical and biotechnological systems, which will increase the accuracy of the implementation of applied optimization problems of economic and mathematical modelling.

Inverse Problem for Whitham Type Multi-Dimensional Differential Equation with Impulse Effects

Inverse Problem for Whitham Type Multi-Dimensional Differential Equation with Impulse Effects

On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems

The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Three physical examples illustrate this point.

Existence and stability results on multidimensional fractional-order systems

The novelties of this paper is the study of existence and uniqueness results for a class of incommensurate fractional differential equation systems with general multiorders using the weighted infinity norm with respect to the classical Mittag-Leffler function via the contraction mapping principle. The lack of stability results in the Ulam–Hyers sense on fractional multidimensional differential equations motivates us to generalize our theory to stability analysis based on fixed point approach.

Сечения производящего ряда решения разностного уравнения в симплициальном конусе

We consider a multidimensional difference equation in a simplicial lattice cone with coefficients from a field of characteristic zero and sections of a generating series of a solution to the Cauchy problem for such equations. We use properties of the shift and projection operators on the integer lattice <math xmlns='http://www.w3.org/1998/Math/MathML'><msup><mi>Z</mi><mi>n</mi></msup></math> to find a recurrence relation (difference equation with polynomial coefficients) for the section of the generating series. This formula allows us to find a generating series of a solution to the Cauchy problem in the lattice cone through a generating series of its initial data and a right-side function of the difference equation. We derived an integral representation for sections of the holomorphic function, whose coefficients satisfy the difference equation with complex coefficients. Finally, we propose a system of differential equations for sections that represent D-finite functions of two complex variables.

A preservative splitting approximation of the solution of a variable coefficient quenching problem

This paper studies the numerical solution of a two-dimensional quenching type nonlinear reaction-diffusion problem via dimensional splitting. The variable coefficient differential equation considered possesses a nonlinear forcing term, and may lead to strong quenching singularities that have profound multiphysics and engineering applications to the energy industry. Our investigations focus on the construction and preservative properties of a semi-adaptive Peaceman-Rachford procedure for solving the aforementioned problem. In the study, the multidimensional differential equation is decomposed into single dimensional subequations so that the computational efficiency can be effectively raised. Temporal adaptation is implemented through arc-length estimations of the rate-of-change of the numerical solution. The positivity, monotonicity and localized linear stability of the variable step splitting scheme are analyzed. These ensure key preservations of underlying physical features of the computed approximations for applications. Computational experiments are presented to illustrate our results as well as to demonstrate the viability and accuracy of the splitting method for solving singular quenching-combustion problems.

Recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation

In this paper, we study the sections of the generating series for solutions to a linear multidimensional difference equation with constant coefficients and find recurrent relations for these sections. As a consequence, a multidimensional analogue of Moivre's theorem on the rationality of sections of the generating series depending on the form of the initial data of the Cauchy problem for a multidimensional difference equation is proved. For problems on the number of paths on an integer lattice, it is shown that the sections of their generating series represent the well-known sequences of polynomials (Fibonacci, Pell, etc.) with a suitable choice of steps.

Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation

In this work, an explicit solution of the initial-boundary value problem for a multidimensional time-fractional differential equation is constructed. The possibility of obtaining this equation from an integro-differential wave equation with a Mittag–Leffler–type memory kernel is shown. An explicit solution to the problem under consideration is obtained using the Laplace and Fourier transforms, the properties of the Fox H-functions and the convolution theorem.

Symplectic-Structure-Preserving Uncertain Differential Equations

Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-path distributions and expected values of solutions are given. Then, structure preserving uncertain differential equations, uncertain Hamiltonian systems driven by Liu processes, which possess a kind of uncertain symplectic structures, are presented. A symplectic scheme with six-order accuracy and a Yao-Chen algorithm are applied to design an algorithm to solve uncertain Hamiltonian systems. At last, numerical experiments are given to investigate four uncertain Hamiltonian systems, which highlight the efficiency of our algorithm.

Theoretical and methodological approach to the study of technological processes

The article investigates some issues of complex systems analysis and synthesis that contain local, discrete sources of temperature fields. The emphasis of the author's research lies in the calculation and optimization of the parameters of the laser action on the embryo. The biotechnological process is described by the boundary value problem of a non-stationary, multidimensional differential equation of thermal conductivity which satisfies the boundary conditions of heat flux, beginning and end of the laser action process. The author proposes an algorithm for calculation and optimization of the control parameters of laser action on multilayer microbiological material. The object of the study is the embryo under the action of a laser beam for fission. As a first approximation, at the stage of calculation and optimization of the parameters of the biotechnological process, the embryo is considered as a homogeneous body. The values of thermophysical characteristics are calculated by the method of expert evaluation of the parameters of the emitters. The correctness of the boundary value problem of the process of laser action on the embryo is proved by the author using the method of pseudodifferential operators. Seeking the solution of the boundary value problem in the form of the series, the analytical function of the temperature field distribution is obtained using the Fourier method of the separated variables in the article. Using the method of indeterminate coefficients, the author found the temperature of the laser action on the embryo. Using the approximate gradient method of finding local extrema and the method of directed search of local extrema, it is possible to obtain rational values of optimized parameters of the biotechnological process. The author outlines possible approaches for optimization of technical parameters of laser emitters. In his opinion, the results of research can be considered fundamental for the calculation and optimization of the parameters of the laser action on the embryo, taking into account the multilayer structure of the latter.

Uniqueness of the Solution of the Dirichlet Problem for a Multidimensional Differential Equation of Fractional Order

Uniqueness of the Solution of the Dirichlet Problem for a Multidimensional Differential Equation of Fractional Order

Uniqueness of the Solution of the Dirichlet Problem for a Multidimensional Differential Equation of Fractional Order

Исследуется уравнение в частных производных дробного порядка с произвольным числом независимых переменных. При целых значениях порядков дробных производных исследуемое уравнение переходит в линейное эллиптическое уравнение второго порядка с оператором Лапласа в главной части. Рассмотрена задача Дирихле в многомерной области. Доказан принцип экстремума для исследуемого уравнения и единственность решения рассматриваемой задачи в области, ограниченной или неограниченной. Библиография: 16 названий.

Оптимізація біотехнологічних систем

The article considers the issues of software and hardware development for the automatization of the design of multilayer biotechnological systems that contain sources of thermal load, including sources of scanned laser radiation. The object of the authors' research is a multilayer microbiological material under the action of laser radiation sources. The purpose of the study is to develop principles for building software and hardware to improve the quality of the action process by taking into account the multilayer structure of the material under study and the technical parameters of the emitters. The authors present the main optimization problem of finding rational parameters of laser beam action on multilayer microbiological material taking into account restrictions on the resulting temperature field of the material to control the use of technical resources of emitters and ensure the viability of segments of microbiological material. It should be stated that to calculate and optimize the temperature of the laser, as well as the technical parameters of the radiators, in connection with the multilayer structure of the object, the authors propose to use a system of non-stationary, inhomogeneous, multidimensional differential equations of thermal conductivity with appropriate initial, boundary and conjugation conditions. layers of microbiological material. These features require significant time in the implementation of the procedure for solving a series of boundary value problems and providing an iterative process of finding rational parameters of thermal exposure. Therefore, it is advisable to implement this procedure on a specialized network analog or hybrid model. This will significantly reduce the time to solve edge problems at each iteration, which will increase the accuracy of solving the entire problem of finding laser parameters.

Solution of multidimensional system with XY differentiation of probability density equations for identification of silver nanoparticles on fibers

The article presents the results of compiling and solving a system of multidimensional differential correlation equations of probability densities for identifying colloidal silver nanoparticles on polyester fibers with multidimensional correlation components of Raman polarization spectra. A method is proposed for increasing the accuracy and speed of identification of silver nanoparticles on polyester fibers, taking into account the longitudinal and transverse polarization of laser radiation over the entire range of the Raman spectrum with the analysis of two peaks sequentially and in order at the same time along the X-side and along the Y-side of the fibers. To implement the method, a program was developed in the Mathcad software. When solving the system using the nonlinear quadratic and differential equations with respect to XY, the probability density equations of distribution ellipses obtained very high accuracy p0 and p1 up to 10−16.

Stability in p-th moment of multi-dimensional uncertain differential equation

Stability in p-th moment of multi-dimensional uncertain differential equation

The Cauchy Problem for Multidimensional Difference Equations in Lattice Cones

We consider a variant of the Cauchy problem for a multidimensional difference equation with constant coefficients, which connected with a lattice path problem in enumerative combinatorial analysis. We obtained a formula in which generating function of the solution to the Cauchy problem is expressed in terms of generating functions of the Cauchy data and a formula expressing solution to the Cauchy problem through its fundamental solution and Cauchy data

Regarding new numerical solution of fractional Schistosomiasis disease arising in biological phenomena

Abstract In this paper, we study to find the numerical solution of fractional Schistosomiasis disease by using a numerical method. Fractional Schistosomiasis disease model is used to symbolize a parasitic disease caused by trematode flukes of the genus Schistosoma. The physical behaviour of results obtained by using q-homotopy analyses transform method (q-HATM) in terms of plots for different fractional-order is captured. The results obtained by using considered method is more effective and easy to apply in order to examine the nature of multi-dimensional differential equations of fractional order arising in biological disease.

Нахождение особых решений многомерных дифференциальных уравнений типа Клеро в частных производных с тригонометрическими функциями

В работе изучается класс дифференциальных уравнений типа Клеро в частных производных первого порядка, которые представляют собой многомерное обобщение обыкновенного дифференциального уравнения Клеро на случай, когда искомая функция зависит от многих переменных. Известно, что общее решение дифференциального уравнения типа Клеро в частных производных представляет собой семейство интегральных (гипер-) плоскостей. Помимо общего решения, могут существовать частные решения, а в некоторых частных случаях удается найти особое (сингулярное) решение.

An efficient technique for two-dimensional fractional order biological population model

In this paper, we find the solutions for two-dimensional biological population model having fractional order using fractional natural decomposition method (FNDM). The proposed method is a graceful blend of decomposition scheme with natural transform, and three examples are considered to validate and illustrate its efficiency. The nature of FNDM solution has been captured for distinct arbitrary order. In order to illustrate the proficiency and reliability of the considered scheme, the numerical simulation has been presented. The obtained results illuminate that the considered method is easy to apply and more effective to examine the nature of multi-dimensional differential equations of fractional order arisen in connected areas of science and technology.