Solutions For System Of Equations Research Articles

Stability of Synchronous Slowly Oscillating Periodic Solutions for Systems of Delay Differential Equations with Coupled Nonlinearity

We study stability of so-called synchronous slowly oscillating periodic solutions (SOPSs) for a system of identical delay differential equations (DDEs) with linear decay and nonlinear delayed negative feedback that are coupled through their nonlinear term. Under a row sum condition on the coupling matrix, existence of a unique SOPS for the corresponding scalar DDE implies existence of a unique synchronous SOPS for the coupled DDEs. However, stability of the SOPS for the scalar DDE does not generally imply stability of the synchronous SOPS for the coupled DDEs. We obtain an explicit formula, depending only on the spectrum of the coupling matrix, the strength of the linear decay and the values of the nonlinear negative feedback function near plus/minus infinity, that determines the stability of the synchronous SOPS in the asymptotic regime where the nonlinear term is heavily weighted. We also treat the special cases of so-called weakly coupled systems, near uniformly coupled systems, and doubly nonnegative coupled systems, in the aforementioned asymptotic regime. Our approach is to estimate the characteristic (Floquet) multipliers for the synchronous SOPS. We first reduce the analysis of the multidimensional variational equation to the analysis of a family of scalar variational-type equations, and then establish limits for an associated family of monodromy-type operators. We illustrate our results with examples of systems of DDEs with mean-field coupling and systems of DDEs arranged in a ring.

Existence of positive solutions for p-Laplacian systems involving left and right fractional derivatives

PurposeThe paper deals with the existence of positive solutions for a coupled system of nonlinear fractional differential equations with p-Laplacian operator and involving both right Riemann–Liouville and left Caputo-type fractional derivatives. The existence results are obtained by the help of Guo–Krasnosel'skii fixed-point theorem on a cone in the sublinear case. In addition, an example is included to illustrate the main results.Design/methodology/approachFixed-point theorems.FindingsNo finding.Originality/valueThe obtained results are original.

Using Trust Region Method with BFGS Technique for Solving Nonlinear Systems of Equations

The major aim of this paper is to find a solution for systems of non-linear equations, a new method is suggested by combining the trust region method (TRM) with BFGS technique, which will help us to find the solution of the non-linear system in less time and less effort, by using BFGS in a new equation based on [Xiangrong Li, Bopeng Wang & Wujie Hu]. Then the convergence of the new algorithm was provided and also calculate the numerical results. By comparing the suggested method with three other methods in terms of time, functions evaluation and number of iterations, the numerical result will tell us that our method is the best.

A Study of Fuzzy Methods for Solving System of Fuzzy Differential Equations

This paper is devoted to obtain an analytical solution for first-order fuzzy differential equations and system of fuzzy differential equations by different methods by considering the type of generalized Hukuhara differentiability of a solution and without embedding them to crisp equations. Moreover, the fuzzy solutions of a second-order fuzzy differential equation by considering the type of differentiability are obtained using reduction to a system of fuzzy differential equations. The effectiveness and efficiency of the approaches are illustrated by solving several practical examples such as Newton’s law of cooling, the mathematical models for the distribution of a drug in the human body and the fuzzy forced harmonic oscillator problem.

Nonexistence of Global Solutions for Coupled System of Pseudoparabolic Equations with Variable Exponents and Weak Memories

The most important behavior for evolution system is the blow-up phenomena because of its wide applications in modern science. The article discusses the finite time blowup that arise under an appropriate conditions. The nonsolvability of boundary value problem for damped pseudoparabolic differential equations with variable exponents is investigated. Such problem has been previously studied in the case if p and q are constants. New here is the case of variables of nonlinearity p and q which make the problem has a scientific interest.

Existence and Ulam–Hyers stability results of a coupled system of [formula omitted]-Hilfer sequential fractional differential equations

The major goal of this work is to investigate sufficient conditions of existence, uniqueness, and Ulam–Hyers stability of solutions for a coupled system of ψ-Hilfer sequential fractional differential equations with two-point boundary conditions. Some standard fixed point theorems of Leray–Schauder alternative and Banach are applied to establish these results. A pertinent example is provided to corroboration the results obtained.

Monotone iterative solutions for a coupled system of p-Laplacian differential equations involving the Riemann\u2013Liouville fractional derivative

Applying the monotone iterative technique and the method of upper and lower solutions, we investigate the existence of extremal solutions for a nonlinear system of p-Laplacian differential equations with nonlocal coupled integral boundary conditions. We present a numerical example to illustrate the main result.

A generalization of Darbo's theorem with application to the solvability of systems of integral-differential equations in Sobolev spaces

‎In this article‎, ‎we introduce the notion of $(alpha,beta)$-generalized Meir-Keeler condensing operator in a‎ ‎Banach space‎, ‎a characterization using strictly L-functions and provide an extension of Darbo's fixed point theorem associated with measures‎ of noncompactness‎. ‎Then‎, ‎we establish some results on the existence of coupled fixed points for a‎ ‎class of condensing operators in Banach spaces‎. ‎As an application‎, ‎we study the‎ ‎problem of existence of entire solutions for a general system of nonlinear integral-differential equations in a Sobolev space‎. ‎Further‎, an example is presented to verify the effectiveness and applicability of our main results‎.

A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations

Inspired by the large number of applications for symmetric nonlinear equations, this article will suggest two optimal choices for the modified Polak–Ribiére–Polyak (PRP) conjugate gradient (CG) method by minimizing the measure function of the search direction matrix and combining the proposed direction with the default Newton direction. In addition, the corresponding PRP parameters are incorporated with the Li and Fukushima approximate gradient to propose two robust CG-type algorithms for finding solutions for large-scale systems of symmetric nonlinear equations. We have also demonstrated the global convergence of the suggested algorithms using some classical assumptions. Finally, we demonstrated the numerical advantages of the proposed algorithms compared to some of the existing methods for nonlinear symmetric equations.

Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations

Linear systems of fractional differential equations have been studied from various points of view: applications to electric circuit theory, approximate solutions by numerical methods, and recently exact solutions by analytical methods. We discover here that, to obtain a fully closed-form solution in all cases, it is necessary to introduce a new type of Mittag-Leffler function involving triple series, and also to construct the associated fractional calculus operators, which we introduce and study in this paper. We then complete the rigorous analytical solutions for the aforesaid systems of fractional differential equations. As a consequence, comparing the solutions found here with the vector-matrix solutions known in the literature, we obtain explicit formulae for the elements of the 2×2 matrix Mittag-Leffler function.

Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type

This article is mainly concerned with the existence and the forms of entire solutions for several systems of the second-order partial differential difference equations of Fermat type α ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 1 z 1 , z 2 / ∂ z 2 2 n 1 + f 2 z 1 + c 1 , z 2 + c 2 m 1 = 1 α ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 2 z 1 , z 2 / ∂ z 2 2 n 2 + f 1 z 1 + c 1 , z 2 + c 2 m 2 = 1 and ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 2 + f 2 z 1 + c 1 , z 2 + c 2 2 = 1 ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 2 + f 1 z 1 + c 1 , z 2 + c 2 2 = 1 . Our results about the existence and the forms of solutions for these systems generalize the previous theorems given by Xu and Cao, Gao, Liu, and Yang. In addition, we give some examples to explain the existence of solutions of this system in each case.

Existence and decay of solution to coupled system of viscoelastic wave equations with strong damping in Rn

In this paper, we establish a general decay rate properties of solutions for a coupled system of viscoelastic wave equations in IRn under some assumptions on g1; g2 and linear forcing terms. We exploit a density function to introduce weighted spaces for solutions and using an appropriate perturbed energy method. The questions of global existence in the nonlinear cases is also proved in Sobolev spaces using the well known Galerkin method.

System of boundary random fractional differential equations via Hadamard derivative

Abstract We study the existence of solutions for random system of fractional differential equations with boundary nonlocal initial conditions. Our approach is based on random fixed point principles of Schaefer and Perov, combined with a vector approach that uses matrices that converge to zero. We prove existence and uniqueness results for these systems. Some examples are presented to illustrate the theory.

Symmetry of positive solutions for systems of fractional Hartree equations

<p style='text-indent:20px;'>In this paper, we deal with a system of fractional Hartree equations. By means of a direct method of moving planes, the radial symmetry and monotonicity of positive solutions are presented.</p>

Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables

<abstract><p>By making use of the Nevanlinna theory and difference Nevanlinna theory of several complex variables, we investigate some properties of the transcendental entire solutions for several systems of partial differential difference equations of Fermat type, and obtain some results about the existence and the forms of transcendental entire solutions of the above systems, which improve and generalize the previous results given by Cao, Gao, Liu <sup>[<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b24">24</xref>,<xref ref-type="bibr" rid="b39">39</xref>]</sup>. Some examples are given show that there exist some significant differences in the forms of transcendental entire solutions with finite order of the systems of equations with between several complex variables and a single complex variable.</p></abstract>

Stability and existence the solution for a coupled system of hybrid fractional differential equation with uniqueness

In this literature, we are investigating the existence and uniqueness of solutions for a coupled system of hybrid differential equations with p-Laplacian operator involving the fractional derivatives Caputo and Riemann–Liouville derivatives of various orders. For this purpose, the proposed problem will be converted into integral equations by using two Green functions for ], where . The existence of a solution is proved by using topological degree and Leray–Schauder fixed point theorems. Uniqueness of solution is obtained with the help of Banach principle. Some adequate conditions for Hyers–Ulam- stability of the solution also are investigated. To verify the results. An expressive example will be presented to clarify the results.

Some New Iterative Algorithms for Solving One-Dimensional Non-Linear Equations and Their Graphical Representation

Solving non-linear equation is perhaps one of the most difficult problems in all of numerical computations, especially in a diverse range of engineering applications. The convergence and performance characteristics can be highly sensitive to the initial guess of the solution for most numerical methods such as Newton's method. However, it is very difficult to select reasonable initial guess of the solution for most systems of non-linear equations. Besides, the computational efficiency is not high enough. Taking this into account, based on variational iteration technique, we develop some new iterative algorithms for solving one-dimensional non-linear equations. The convergence criteria of these iterative algorithms has also been discussed. The superiority of the proposed iterative algorithms is illustrated by solving some test examples and comparing them with other well-known existing iterative algorithms in literature. In the end, the graphical comparison of the proposed iterative algorithms with other well-known iterative algorithms have been made by means of polynomiographs of different complex polynomials which reflect the fractal behavior and dynamical aspects of the proposed iterative algorithms.

Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative

<abstract> In this paper, we establish the existence and uniqueness of solution for a nonlinear coupled system of implicit fractional differential equations including $ \psi $-Caputo fractional operator under nonlocal conditions. Schaefer's and Banach fixed-point theorems are applied to obtain the solvability results for the proposed system. Furthermore, we extend the results to investigate several types of Ulam stability for the proposed system by using classical tool of nonlinear analysis. Finally, an example is provided to illustrate the abstract results. </abstract>

Умови розв’язуваностi систем нелiнiйних рiвнянь в евклiдових просторах

The solution of many applied problems is to find a solution of nonlinear equations systems in finite- dimensional Euclidean spaces. The problem of finding the solution of a nonlinear system is divided into two problems: 1. The existence of a solution of a nonlinear equations system; in the case of nonunique of the solution, it is necessary to find the number of these solutions and their surroundings. 2. Finding the solution of a system of nonlinear equations with a given accuracy. Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence and estimates of the solution accuracy. In contrast to problem 2, for problem 1 there is no general algorithm for solving this task, there are no constructive conditions for the existence of a solution of a nonlinear equations system in Euclidean spaces. In this article, in finite-dimensional Euclidean spaces, the constructive conditions for the existence of a solution of nonlinear systems of polynomial form are found. The connection of these conditions with the linear polynomial interpolant of the minimum norm, generated by a scalar product with Gaussian measure and the conditions of its existence, is given.

Solving System of Integro-Differential Equations using a New Hybrid Semi-Analytical Method

This paper presents a new hybrid method that combines Kharrat-Toma transform technique with the homotopy perturbation method for the solutions of linear and nonlinear initial value problems represented by integro-differential equations systems with initial conditions. Where, the advantages of this hybrid approach are rapid convergence to an approximate or exact solution, reduces the computational steps and integrals which making it a highly efficient and applicable method compared with the classical methods. Therefore, this proposed new hybrid approach (Kharrat-Toma transform and homotopy perturbation method) can be apply to find the analytical solutions of integro-differential equations systems arising in many applications. To show the accuracy and effectiveness of the presented technique several examples are introduced.