Solutions For System Of Equations Research Articles

Analysis of a Hybrid Coupled System of ψ-Caputo Fractional Derivatives with Generalized Slit-Strips-Type Integral Boundary Conditions and Impulses

In the current paper, we analyzed the existence and uniqueness of a solution for a coupled system of impulsive hybrid fractional differential equations involving ψ-Caputo fractional derivatives with generalized slit-strips-type integral boundary conditions. We also study the Ulam–Hyers stability for the considered system. For the existence and uniqueness of the solution, we use the Banach contraction principle. With the help of Schaefer’s fixed-point theorem and some assumptions, we also obtain at least one solution of the mentioned system. Finally, the main results are verified with an appropriate example.

Variational approach to instantaneous and noninstantaneous impulsive system of differential equations

In this paper, the existence and multiplicity of solutions for a coupled system of differential equations with instantaneous and noninstantaneous impulses are studied. By the virtue of variational methods, some new existence theorems of solutions are obtained. In addition, two examples are given to demonstrate our main results.

Novel soliton solutions for the fractional three-wave resonant interaction equations

Abstract In this article, we obtained new infinite sets of exact soliton solutions for the nonlinear evolution system of three-wave resonant interaction equations. The solved system contains the non-zero second-order dispersion coefficients, the non-zero phase velocity mismatch, and the conformable fractional time derivative of order between zero and one. The solution method is a constructed ansatz that consists of linear combinations of the tan and cotan hyperbolic functions with complex coefficients. We stated clear systematic steps toward writing an exact soliton solution for the studied system. To show the efficiency of this method, we introduced some numerical examples on each obtained set of solutions. The computations showed that similar solutions can be obtained if one replaces the tan and cotan hyperbolic functions with the tan and cotan trigonometric functions. The new obtained fractional solutions could be useful in studying the broad applications of triad resonances in plasma physics and in nonlinear optics.

Cayley Hash Values of Brauer Messages and Some of Their Applications in the Solutions of Systems of Differential Equations

Cayley hash values are defined by paths of some oriented graphs (quivers) called Cayley graphs, whose vertices and arrows are given by elements of a group H. On the other hand, Brauer messages are obtained by concatenating words associated with multisets constituting some configurations called Brauer configurations. These configurations define some oriented graphs named Brauer quivers which induce a particular class of bound quiver algebras named Brauer configuration algebras. Elements of multisets in Brauer configurations can be seen as letters of the Brauer messages. This paper proves that each point (x,y)∈V=R\{0}×R\{0} has an associated Brauer configuration algebra ΛB(x,y) induced by a Brauer configuration B(x,y). Additionally, the Brauer configuration algebras associated with points in a subset of the form (⌊(x)⌋,⌈(x)⌉]×(⌊(y)⌋,⌈(y)⌉]⊂V have the same dimension. We give an analysis of Cayley hash values associated with Brauer messages M(B(x,y)) defined by a semigroup generated by some appropriated matrices A0,A1,A2∈GL(2,R) over a commutative ring R. As an application, we use Brauer messages M(B(x,y)) to construct explicit solutions for systems of linear and nonlinear differential equations of the form X″(t)+MX(t)=0 and X′(t)−X2(t)N(t)=N(t) for some suitable square matrices, M and N(t). Python routines to compute Cayley hash values of Brauer messages are also included.

A fixed point index approach to Krasnosel’skiĭ-Precup fixed point theorem in cones and applications

We present an alternative approach to the vector version of Krasnosel’skiĭ compression–expansion fixed point theorem due to Precup, which is based on the fixed point index. It allows us to obtain new general versions of this fixed point theorem and also multiplicity results. We emphasize that all of them are coexistence fixed point theorems for operator systems, that means that every component of the fixed points obtained is non-trivial. Finally, these coexistence fixed point theorems are applied to obtain results concerning the existence of positive solutions for systems of Hammerstein integral equations and radially symmetric solutions of (p1,p2)-Laplacian systems.

Existence of infinitely many positive radial solutions for an iterative system of nonlinear elliptic equations on an exterior domain

Existence of infinitely many positive radial solutions for an iterative system of nonlinear elliptic equations on an exterior domain

Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach

Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.

A Lanczos-type procedure for tensors

The solution of linear non-autonomous ordinary differential equation systems (also known as the time-ordered exponential) is a computationally challenging problem arising in a variety of applications. In this work, we present and study a new framework for the computation of bilinear forms involving the time-ordered exponential. Such a framework is based on an extension of the non-Hermitian Lanczos algorithm to 4-mode tensors. Detailed results concerning its theoretical properties are presented. Moreover, computational results performed on real-world problems confirm the effectiveness of our approach.

Discovery of new characterizations of parameter distributions in a semi- empirical mass model: an investigation by dividing an overdetermined system into numerous balanced subsystems* *Supported by the National Natural Science Foundation of China (11975209, U2032211, 12075287), the Physics Research and Development Program of Zhengzhou University (32410017), the Project of Youth Backbone Teachers of Colleges and Universities of Henan Province (2017GGJS008)

We propose and test a new method of estimating the model parameters of the phenomenological Bethe-Weizsäcker mass formula. Based on the Monte Carlo sampling of a large dataset, we obtain, for the first time, a Cauchy-type parameter distribution formed by the exact solutions of linear equation systems. Using the maximum likelihood estimation, the location and scale parameters are evaluated. The estimated results are compared with those obtained by solving overdetermined systems, e.g., the solutions of the traditional least-squares method. Parameter correlations and uncertainty propagation are briefly discussed. As expected, it is also found that improvements in theoretical modeling (e.g., considering microscopic corrections) decrease the parameter and propagation uncertainties.

Decomposition-based multiobjective optimization for nonlinear equation systems with many and infinitely many roots

Although the development of Pareto-dominance-based multiobjective optimization algorithms has enabled the solution of nonlinear equation systems, few studies have been conducted on the use of multiobjective optimization techniques for the solution of nonlinear equation systems. Accordingly, in this study, we use decomposition-based multiobjective optimization to solve nonlinear equation system, an attempt is made at deploying the decomposition-based multiobjective optimization to solve nonlinear equation systems, including those with infinite roots. In our novel approach, a given system is transformed into a bi-objective optimization problem using reference points. An improved decomposition-based multiobjective optimization is then applied to solve a transformed bi-objective optimization problem. To ensure this optimization suits the characteristics of the problem, we develop an adaptive multiobjective differential evolution and local search approach. The roots of the original nonlinear equation system can then be identified, together with the Pareto-optimal solutions of the transformed problem. We conducted extensive experiments to compare the performance of our novel approach with that of 16 state-of-the-art algorithms in solving 30 nonlinear equation systems derived from industrial applications. The results clearly show that our novel approach achieves better performance with a shorter execution time than that of the selected single-objective-based evolutionary approaches or Pareto-dominance-based multiobjective optimization algorithms.

Uniqueness and stability of coupled sequential fractional differential equations with boundary conditions

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Caputo-type sequential fractional differential equations supplemented with boundary conditions. The existence of solutions is derived by applying the Leray–Schauder alternative, while the uniqueness of solutions is established via Banach’s contraction principle. Moreover, some necessary conditions for the Hyers–Ulam-type stability to the solutions of the boundary value problem are developed. Finally the results are supported by example.

Reliability model of the security subsystem countering to the impact of typed cyber-physical attacks

The article's main contribution is the description of the process of the security subsystem countering the impact of typed cyber-physical attacks as a model of end states in continuous time. The input parameters of the model are the flow intensities of typed cyber-physical attacks, the flow intensities of possible cyber-immune reactions, and the set of probabilities of neutralization of cyber-physical attacks. The set of admissible states of the info-communication system is described taking into account possible variants of the development of the modeled process. The initial parameters of the model are the probabilities of the studied system in the appropriate states at a particular moment. The dynamics of the info-communication system's life cycle are embodied in the form of a matrix of transient probabilities. The mentioned matrix connects the initial parameters in the form of a system of Chapman's equations. The article presents a computationally efficient concept based on Gershgorin's theorems to solve such a system of equations with given initiating values. Based on the presented scientific results, the article proposes the concept of calculating the time to failure as an indicator of the reliability of the info-communication system operating under the probable impact of typical cyber-physical attacks. The adequacy of the model and concepts presented in the article is proved by comparing a statically representative amount of empirical and simulated data. We emphasize that the main contribution of the research is the description of the process of the security subsystem countering the impact of typed cyber-physical attacks as a model of end states in continuous time. Based on the created model, the concept of computationally efficient solution of Chapman's equation system based on Gershgorin's theorems and calculating time to failure as an indicator of the reliability of the info-communication system operating under the probable impact of typed cyber-physical attacks are formalized. These models and concepts are the highlights of the research.

On a Coupled System of Fractional Differential Equations via the Generalized Proportional Fractional Derivatives

This work investigates the existence and uniqueness of solutions for a coupled system of fractional differential equations with three-point generalized fractional integral boundary conditions within generalized proportional fractional derivatives of the Riemann-Liouville type. By using the Schauder and Banach fixed point theorems, we study the existence and uniqueness of solutions for the aforesaid system. Finally, we present an example to validate our theoretical outcomes.

AAQAL: A Machine Learning-Based Tool for Performance Optimization of Parallel SPMV Computations Using Block CSR

The sparse matrix–vector product (SpMV), considered one of the seven dwarfs (numerical methods of significance), is essential in high-performance real-world scientific and analytical applications requiring solution of large sparse linear equation systems, where SpMV is a key computing operation. As the sparsity patterns of sparse matrices are unknown before runtime, we used machine learning-based performance optimization of the SpMV kernel by exploiting the structure of the sparse matrices using the Block Compressed Sparse Row (BCSR) storage format. As the structure of sparse matrices varies across application domains, optimizing the block size is important for reducing the overall execution time. Manual allocation of block sizes is error prone and time consuming. Thus, we propose AAQAL, a data-driven, machine learning-based tool that automates the process of data distribution and selection of near-optimal block sizes based on the structure of the matrix. We trained and tested the tool using different machine learning methods—decision tree, random forest, gradient boosting, ridge regressor, and AdaBoost—and nearly 700 real-world matrices from 43 application domains, including computer vision, robotics, and computational fluid dynamics. AAQAL achieved 93.47% of the maximum attainable performance with a substantial difference compared to in practice manual or random selection of block sizes. This is the first attempt at exploiting matrix structure using BCSR, to select optimal block sizes for the SpMV computations using machine learning techniques.

Operator level limit of the circular Jacobi β-ensemble

We prove an operator level limit for the circular Jacobi [Formula: see text]-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal [Formula: see text]-ensemble.

Randomized Newton’s Method for Solving Differential Equations Based on the Neural Network Discretization

We develop a randomized Newton’s method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton’s method randomly chooses equations from the overdetermined nonlinear system resulting from the neural network discretization and solves the nonlinear system adaptively. We theoretically prove that the randomized Newton’s method has a quadratic convergence locally. We also apply this new method to various numerical examples, from one to high-dimensional differential equations, to verify its feasibility and efficiency. Moreover, the randomized Newton’s method can allow the neural network to “learn” multiple solutions for nonlinear systems of differential equations, such as pattern formation problems, and provides an alternative way to study the solution structure of nonlinear differential equations overall.

AN EXISTENCE SOLUTION FOR A COUPLED SYSTEM WITH LAPLACIAN OPERATOR AND HILFER DERIVATIVES

In this paper, we study the existence of solutions for a coupled system of fractional differential equations with nonlocal integro multi point boundary conditions by using the Laplacian operator and the Hilfer derivatives. The presented results are obtained by the fixed point theorems of Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such problem is considered.

Solvability of a system of integral equations in two variables in the weighted Sobolev space $W^{1,1}_\omega(a,b)$ using a generalized measure of noncompactness

In this paper, we deal with the existence of solutions for a coupled system of integral equations in the Cartesian product of weighted Sobolev spaces E = Wω1,1 (a,b) x Wω1,1 (a,b). The results were achieved by equipping the space E with the vector-valued norms and using the measure of noncompactness constructed in [F.P. Najafabad, J.J. Nieto, H.A. Kayvanloo, Measure of noncompactness on weighted Sobolev space with an application to some nonlinear convolution type integral equations, J. Fixed Point Theory Appl., 22(3), 75, 2020] to applicate the generalized Darbo’s fixed point theorem [J.R. Graef, J. Henderson, and A. Ouahab, Topological Methods for Differential Equations and Inclusions, CRC Press, Boca Raton, FL, 2018].

Models for IMPATT Diode Analysis and Optimization

Some of nonlinear models for high-power pulsed IMPATT diode simulation and analysis is presented. These models are suitable for the analysis of the different operational modes of the oscillator. Its take into account the main electric and thermal phenomena in the semiconductor structure and the functional dependence of the equation coefficients on the electrical field and temperature. The first model is a precise one, which describes all important electrical phenomena on the basis of the continuity equations and Poisson equation and it is correct until 300 GHz. The second approximate mathematical model suitable for the analysis of IMPATT diode stationary operation oscillator and for optimization of internal structure of the diode. This model is based on the continuity equation system solution by reducing the boundary problem for the differential partial equations to a system of the ordinary differential equations. The temperature distribution in the semiconductor structure is obtained using the special thermal model of the IMPATT diode, which is based on the numerical solution of the non-linear thermal conductivity equation. The described models can be applied for analysis, optimization and practical design of pulsed-mode millimetric IMPATT diodes. Its can be also utilized for diode thermal regime estimation, for the proper selection of feed-pulse shape and amplitude, and for the development of the different type of complex doping-profile high-power pulsed millimetric IMPATT diodes with improved characteristics.

On Results Between Matrix Division and Some other Matrix Operations

In this study, the relationships between matrix division and some matrix properties such as transpose, simplification, and expansion are discussed. Unless otherwise stated, our matrices are real number matrices. The relationship between our definitions of column co-divisor and row co-divisor of a matrix on another matrix is examined. Equality and differences of these connections are observed. The validity of the provided features is discussed under which conditions. These features have been determined. Examples are given for features that do not. Some of the obtained properties bring different solutions to the solution of linear equation systems, which are encountered in the literature and are more used. This led us to examine the relationship between the defined division operation and transpose. Investigations on this subject enabled the presentation of new definitions, lemmas and theorems. Examination of these offered a broader perspective on the given mathematical expressions.