Solutions For System Of Equations Research Articles

Semi-analytical solutions of flow and heat transfer of fluid along expandable-stretching horizontal cylinder

In the presence of suction/injection, a mathematical formulation for laminar boundary layer flow and heat transfer of an incompressible viscous fluid along an expandable-stretching horizontal cylinder is provided. The partial differential equations governing system is reduced to a system of ordinary differential equations. With the aid of Mathematica, semi-analytical solutions for the resultant system of ordinary differential equations are obtained using the OHAM (Optimal Homotopy Asymptotic Method) and the Legendre-Galerkin method. It is found that the suction process increases surface stiffness and strength while the injection process reduces surface skin friction. Also, other flow and thermal factors are discussed graphically.

On the behaviour of coalgebras with side effects and algebras with effectful iteration

Abstract For every finitary monad $T$ on sets and every endofunctor $F$ on the category of $T$-algebras, we introduce the concept of an ffg-Elgot algebra for $F$, i.e. an algebra admitting coherent solutions for finite systems of recursive equations with effects represented by the monad $T$. The goal is to study the existence and construction of free ffg-Elgot algebras. To this end, we investigate the locally ffg fixed point $\varphi F$, i.e. the colimit of all $F$-coalgebras with free finitely generated carrier, which is shown to be the initial ffg-Elgot algebra. This is the technical foundation for our main result: the category of ffg-Elgot algebras is monadic over the category of $T$-algebras.

Hypothetical Learning Trajectory: To Determine The Solution of Linear Equation System in Elementary Row Operation

The background of this research is the learning obstacles that arise on the topic of determining the solution of a system of linear equations (SLE) in applying elementary row operations (ERO). From these findings, it is necessary to formulate a hypothetical learning trajectory (HLT) that can guide the flow of thinking in designing didactic situations in the learning process. HLT expect to anticipate learning obstacles (LO) that arise. This research is a preliminary design of a series of didactical design research. The steps taken in compiling the HLT are 1) reviewing the lesson plan, the methods used to convey the material to determine SST solutions, and the textbooks used; 2) reviewing students' thinking processes conceptually and procedurally regarding theoretically determining SLE solutions; 3) reviewing design plans didactic. HLT is formulated into five types of student responses. It integrates into didactic situations by explaining the relevance of the elimination method in high school with three operations on ERO and then by explaining matrix material before SPL material. From this didactic situation, it can be formulated in teaching material that can accommodate emerging LO.

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

BSMPT v2 a tool for the electroweak phase transition and the baryon asymmetry of the universe in extended Higgs Sectors

We present the C++ code BSMPT v2 which is an extension of the previous code BSMPT for the calculation of the strength of the electroweak phase transition in extended Higgs sectors. The new version BSMPT v2 includes the features of BSMPT and extends the already implemented models (the 2-Higgs-Doublet model (2HDM) in its CP-conserving and CP-violating versions and the Next-to-2HDM) by the Complex Singlet Extension of the Standard Model (CxSM). The major upgrade is the implementation of the computation of the baryon asymmetry of the Universe for the CP-violating 2HDM (C2HDM), which is performed in two different approximations. These changes and further smaller modifications are described in this manual. Additionally, a detailed explanation of the procedure for the implementation of new models is given, which has also changed with respect to the previous version. Program summaryProgram Title:BSMPTCPC Library link to program files:https://doi.org/10.17632/sjtp7bb33t.1Licensing provisions: GPL-3.0 LicenseProgramming Language:C++14Nature of problem: Non-minimal extended Higgs sector models provide non-trivial vacuum structures which allow for a strong first order electroweak phase transition. Such a phase transition is one of the three Sakharov conditions that are required for a dynamical generation of the observed baryon asymmetry of the universe (BAU) through an electroweak phase transition. The actual calculation of the electroweak baryogenesis requires the solution of the quantum transport equation system describing the non-thermal equilibrium state of the early universe during the phase transition. BSMPT v2 provides a numerical tool to investigate the vacuum structure of the one-loop effective potential at finite temperature including thermal masses, for an arbitrary extended Higgs sector. It allows for the computation of the strength of the electroweak phase transition for the implemented models. For the CP-violating 2HDM (C2HDM) also the generated BAU is calculated. For the latter task BSMPT v2 has two different approaches implemented for the formulation of the quantum transport equations, given by the FH approach based on the semi-classical force and the vacuum expectation value insertion method VIA.Solution Method: Numerical minimization of the one-loop effective potential including thermal masses, at finite temperature with three different numerical minimizers, GSL, cmaes and NLopt, in order to determine the relevant parameters required for the phase transition dynamics at the critical temperature and the criticial field configuration. Furthermore, with the updated version BSMPT v2 it is possible to numerically solve for the C2HDM the system of coupled differential transport equations and calculate the BAU for this model.Additional comments including restrictions and unusual features: The BSM extensions are restricted to the Higgs sectors. New gauge bosons and fermions would require an adaption in the thermal corrections of the one-loop potential, which is not implemented in BSMPT v2. At present, the BAU is only calculated for the C2HDM. In the computation of the BAU the wall velocity is an input parameter and assumed to be small, for the VEV configuration a kink profile is assumed, the nucleation temperature is approximated by the critical temperature.

Some properties of implicit impulsive coupled system via \u03c6-Hilfer fractional operator

The major goal of this work is investigating sufficient conditions for the existence and uniqueness of solutions for implicit impulsive coupled system of φ-Hilfer fractional differential equations (FDEs) with instantaneous impulses and terminal conditions. First, we derive equivalent fractional integral equations of the proposed system. Next, by employing some standard fixed point theorems such as Leray–Schauder alternative and Banach, we obtain the existence and uniqueness of solutions. Further, by mathematical analysis technique we investigate the Ulam–Hyers (UH) and generalized UH (GUH) stability of solutions. Finally, we provide a pertinent example to corroborate the results obtained.

Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a 2 × 2 block operator matrix defined on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

Weighted composite outer inverses

In order to extend and unify the definitions of W-weighted DMP, W-weighted MPD, W-weighted CMP and composite outer inverses, we present the weighted composite outer inverses. Precisely, the notions of MNOMP, MPMNO and MPMNOMP inverses are introduced as appropriate expressions involving the (M,N)-weighted (B,C)-inverse and Moore–Penrose inverse. Basic properties and a number of characterizations for the MNOMP, MPMNO or MPMNOMP inverse are discovered. Various representations and characterizations of weighted composite outer inverses are studied. General solutions for certain systems of linear equations are given in terms of weighted composite outer inverses. Numerical examples are presented on randomly generated matrices of various orders.

Measures of noncompactness and infinite systems of integral equations of Urysohn type in $L^\infty(\mathfrak{G})$

"In this article, applying the concept of measure of noncompactness, some fixed point theorems in the Fréchet space $L^\infty(\mathfrak{G})$ (where $\mathfrak{G}\subseteq \mathbb{R}^{\omega}$) have been proved. We handle our obtained consequences to inquiry the existence of solutions for infinite systems of Urysohn type integral equations. Our results extend some famous related results in the literature. Finally, to indicate the effectiveness of our results we present a genuine example."

Identification of Mathematics Prospective Teachers’ Conceptual Understanding in Determining Solutions of Linear Equation Systems

<p style="text-align:justify">This research is motivated by a linear equations system, which is the basis for studying necessary linear algebra materials, such as rank, range, linear independent/dependent, linear transformations, characteristic values and vectors. There are still prospective mathematics teachers who have difficulty solving linear equations system and understanding the form of row echelon and reduced row echelon forms. In this study, subjects were three prospective mathematics teachers from Swadaya Gunung Jati University Cirebon who were taking matrix algebra courses. This study aims to reveal the conceptual understanding of prospective mathematics teachers in determining the solution to systems of linear equations. The results show that there are still prospective mathematics teachers who only use memory about the properties and procedures in determining whether a matrix is said to be a row echelon form or a reduced row echelon form. Then, there is still weakness in building the algorithms' relationship due to the immature knowledge of the concepts. Researchers found that many prospective mathematics teachers were more comfortable solving problems that were performed procedurally. Further research is needed to determine how the mental construction process and mathematical conceptual knowledge of prospective mathematics teachers are through meaningful learning so that conceptual understanding is maximized.</p>

ON COUPLED SYSTEMS OF LIDSTONE-TYPE BOUNDARY VALUE PROBLEMS

This research concerns the existence and location of solutions for coupled system of differential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented.

Effective numerical technique for nonlinear Caputo-Fabrizio systems of fractional Volterra integro-differential equations in Hilbert space

The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives.

On the Colored and the Set-Theoretical Yang–Baxter Equations

This paper is related to several articles published in AXIOMS, SCI, etc. The main concepts of the current paper are the colored Yang–Baxter equation and the set-theoretical Yang–Baxter equation. The Euler formula, colagebra structures, and means play an important role in our study. We show that some new solutions for a certain system of equations lead to colored Yang–Baxter operators, which are related to an Euler formula for matrices, and the set-theoretical solutions to the Yang–Baxter equation are related to means. A new coalgebra is obtained and studied.

Effect of moving boundaries on the modeling of heat and mass transfer from an evaporating spherical drop

The effect of unsteadiness of the energy and vapor transport within the gas phase from an evaporating drop is studied by solving a moving boundary problem taking into account the effect of drop temperature variation and radius shrinking. The effect of convection is also taken into account in a simplified way by means of the film theory approach, which yields a double moving boundary problem. A proper change of the reference system leads to the numerical solution of a partial differential equation system with fixed boundaries. A comparison with the commonly adopted quasi-steady model allows to point out the effect of ambient temperature and pressure, convection and chemical species, by quantifying the discrepancies between the two predictions for sixteen different compounds, Reynolds number ranging between 0 and 20 and gas pressure up to 20 bar. The simplified approach used in this paper was chosen to maintain the same simplifying assumptions of the widely used quasi-steady model, with the only exception of the drop shrinking. Therefore, the discrepancies between the two predictions can solely be ascribed to the unsteadiness caused by the interface movement, allowing to quantitatively point out this specific effect.

Geometric Interpretation and Manifold Structure of Markov Matrices

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Andreyevich Markov. Every Markov matrix gives a linear equation system, and the solution of this equation system gives us a subset of $\mathbb{R}_{n}^{n}$. This paper presents the new manifold structure on the set of the Markov matrices. In addition, this paper presents the set of Markov matrices is drawable, and this gives geometrical interpretation to Markov matrices. For the proof, we use the one-to-one corresponding among $n \times n$ Markov matrices, the solution of linear equation system from derived Markov property, and the set of $(n-1)$-polytopes.

Existence of solutions for a coupled system of fractional differential equations by means of topological degree theory

This paper investigates the existence of solutions for a coupled system of fractional differential equations. The existence is proved by using the topological degree theory, and an example is given to show the applicability of our main result.

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

This paper is concerned with the investigation of series solutions of some time fractional nonlinear systems of partial differential equations. The symmetries and reductions of the discussed time fractional nonlinear systems are presented in a recent work (J. Math. Phys. 57 (2016) 101504). But, their explicit solutions are investigated in the present study by using power series expansion technique. Moreover, the reported solutions are interpreted graphically to highlight the importance of present study.

The existence and forms of solutions for several systems of the Fermat-type difference-differential equations

The sets of functional equations f m 1 + g n 1 = R 1 , f m 2 + g n 2 = R 2 , can be regarded as the Fermat-type system, where m1,m2,n1,n2 are positive integers, R1,R2 are nonzero polynomials. The purpose of this paper is to study the existence of solutions for several Fermat-type systems of the difference-differential equations [ f 1 ′ ( z ) ] 2 + f 2 ( z + c ) 2 = R 1 ( z ) , [ f 2 ′ ( z ) ] 2 + f 1 ( z + c ) 2 = R 2 ( z ) , [ f 1 ′ ( z ) ] 2 + [ f 2 ( z + c ) − f 1 ( z ) ] 2 = R 1 ( z ) , [ f 2 ′ ( z ) ] 2 + [ f 1 ( z + c ) − f 2 ( z ) ] 2 = R 2 ( z ) , and [ f 1 ( z ) ] 2 + [ f 2 ( z + c ) − f 1 ( z ) ] 2 = R 1 ( z ) , [ f 2 ( z ) ] 2 + [ f 1 ( z + c ) − f 2 ( z ) ] 2 = R 2 ( z ) . Our results about the existence and the forms of solutions for these Fermat types systems are some improvements of the previous theorems given by Gao, Liu, Yang, Cao, Qi. Meanwhile, we give some examples to utilize the existence and forms of solutions for such systems in each case.

Positive Stepanov-like almost automorphic solutions for systems of nonlinear delay integral equations

In this article we show the existence of positive Stepanov-like almost automorphic solutions for systems of nonlinear delay integral equations. To do this, we apply the well-known Guo-Krasnosel'skii fixed point theorem for cone expansion and compression.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/43/abstr.html

A Self-Adjoint Coupled System of Nonlinear Ordinary Differential Equations with Nonlocal Multi-Point Boundary Conditions on an Arbitrary Domain

The main object of this paper is to investigate the existence of solutions for a self-adjoint coupled system of nonlinear second-order ordinary differential equations equipped with nonlocal multi-point coupled boundary conditions on an arbitrary domain. We apply the Leray–Schauder alternative, the Schauder fixed point theorem and the Banach contraction mapping principle in order to derive the main results, which are then well-illustrated with the aid of several examples. Some potential directions for related further researches are also indicated.