system of differential equations

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A. System of linear ordinary differential equations of first

A. System of linear ordinary differential equations of first

Abstract— A Finite Transfer numerical approximation method is presented to solve a system of linear ordinary differential equations with boundary conditions. It is applied to determine the structural behaviour of the problem of a spatially curved beam element. The approach of this boundary value problem yields a unique system of differential equations. A Runge-Kutta scheme is chosen to obtain Finite Transfer expressions. The use of a recurrence strategy in these equations permits to relate both ends in the domain where boundary conditions are defined. Example of a Catenary shaped arch is provided for validation.
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II. Differential Equations for modeling the Cartpole System

II. Differential Equations for modeling the Cartpole System

Abstract -The prevalence of differential equations as a mathematical technique has refined the fields of control theory and constrained optimization due to the newfound ability to accurately model chaotic, unbalanced systems. However, in recent research, systems are increasingly more nonlinear and difficult to model using Differential Equations only. Thus, a newer technique is to use policy iteration and Reinforcement Learning, techniques that center around an action and reward sequence for a controller. Reinforcement Learning (RL) can be applied to control theory problems since a system can robustly apply RL in a dynamic
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Differential inequalities for a finite system of hybrid Caputo fractional differential equations

Differential inequalities for a finite system of hybrid Caputo fractional differential equations

Abstract. In this paper, some basic fractional differential inequalities for a finite system of an initial value problem of hybrid fractional differential equations involving derivatives are proved with a linear perturbation of second type. An existence and a comparison theorem for the considered hybrid fractional differential have also been established. Keywords: hybrid differential equation; differential inequalities; existence theorem; comparison result.

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Mathematical thinking in differential equations through a computer algebra system

Mathematical thinking in differential equations through a computer algebra system

In this research, mathematical thinking powers were given attention including specializing and generalizing; conjecturing and convincing; imagining and expressing; stressing and ignoring; extending and restricting; classifying and characterizing; changing, varying, reversing and altering; and selecting, comparing, sorting and organizing. The distinct effects of a computer algebra system on the development of mathematical knowledge in differential equations are highlighted through the use of mathematical thinking powers that are not commonly used in the pen and paper environment, such as changing, comparing, sorting, organizing and imagining the graphs. The mathematics used in a computer algebra system is different mathematics to that which is available with pen and paper algorithms. Differential equations at the undergraduate level which includes topics on first order differential equations, second differential equations with constant coefficients, and Laplace transforms (see Appendix I). Maxima is a CAS that is open source software without any limitations to install on many computers. It is free and the language used is close to the language of mathematics. The population was chosen from one public university in Malaysia and the participants were chosen from the Faculty of Chemical Engineering because the lecturer was familiar with the research and objectives of teaching experiment methodology. The demographic characteristics of the participants in the intervention sessions and interviews were not considered in this research. The intervention sessions were conducted over 11 weeks in one academic semester to cover the first order and second order differential equations with constant coefficients, and Laplace transforms. The same examination for all the engineering students in differential equations, SSCE1793, is the biggest limitation in this research.
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Random semilinear system of differential equations with impulses

Random semilinear system of differential equations with impulses

Unfortunately, in most cases the available data for the description and evaluation of parameters of a dynamic system are inaccurate, imprecise, or confusing. In other words, evaluation of parameters of a dynamical system is not without uncertainties. Differential equations with random coefficients are used as models in many different applications. This is due to a combination of uncertainties, complexities, and ignorance on our part which inevitably cloud our mathematical modeling process (e.g., Kampé de Feriet [], Becus [] and their references). This interest is due to the fact that there are many applications of this theory to various applied fields such as control theory, statistics, biological sciences, and others. For a discussion of such applications, one may consult the books [–] and the papers [–], and the references therein.
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Periodicity in a System of Differential Equations with Finite Delay

Periodicity in a System of Differential Equations with Finite Delay

Floquet theory offers a lot of results on the periodicity of the system (1.1) when τ = 0. In [16], the author extended Floquet theory to non- autonomous linear systems of the form z 0 = A(x)z, where A : C → C is an ω− periodic function in the complex variable x, whose solutions are meromorphic. There are however no corresponding results for system (1.1). The qualitative properties of the scalar version of (1.1) have been studied in [4]. Therefore, in this paper by using the notion of the fundamental solution coupled with Floquet theory we prove the existence and uniqueness of solutions of (1.1).
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A coupled system of fractional differential equations on the half-line

A coupled system of fractional differential equations on the half-line

Fractional calculus has recently evolved as an excellent tool for mathematical modeling owing to its widespread applications in the fields of engineering, physics, electrodynam- ics of complex medium, photoelasticity, etc; one can see [1–12] and the references cited therein. Meanwhile, relevant theory of fractional differential and integral equations has been established, and the research on fractional differential equations for boundary value problems is in a stage of rapid development.

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On Efficient Method for System of Fractional Differential Equations

On Efficient Method for System of Fractional Differential Equations

13 A. Yıldırım, “Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3175–3180, 2008. 14 H. Koc¸ak, T. ¨ Ozis¸, and A. Yıldırım, “Homotopy perturbation method for the nonlinear dispersive Km,n,1 equations with fractional time derivatives,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20, no. 2, pp. 174–185, 2010.

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Unique solution for a new system of fractional differential equations

Unique solution for a new system of fractional differential equations

Recently, fractional differential systems have been increasingly used to describe problems in optical and thermal systems, rheology and materials and mechanics systems, signal processing and system identification, control, robotics, and other applications. Because of their deep realistic background and important role, people are paying more and more attention. For nonlinear fractional differential systems subject to different boundary con- ditions, there are many articles studying the existence or multiplicity of solutions or pos- itive solutions. But the unique results are very rare. In this paper, we study a system of fractional differential Eqs. (1.1). By constructing two functions e and h and using fixed point theorem of increasing Ψ -(h, e)-concave operators defined on ordered set P h,e , we
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An Algorithm for the Numerical Solution of System of Fractional Differential Equations

An Algorithm for the Numerical Solution of System of Fractional Differential Equations

derivatives, respectively. In section 5, we derive the fractional s method for the numerical solution of ordinary differential equations. The algorithm itself is presented in details in section 6. In section 7, we present three examples to show the efficiency and the simplicify of the algorithm .

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Variational iterative method: an appropriate numerical scheme for solving system of linear Volterra fuzzy integro differential equations

Variational iterative method: an appropriate numerical scheme for solving system of linear Volterra fuzzy integro differential equations

Now, we apply the proposed approximation technique by evaluating the system of lin- ear Volterra fuzzy integro-differential equations, our solutions of the variational iteration method and the demonstration are given in Sect. 4, convert to the numerical variational iteration algorithm is drawn in this section. Its application of the linear Volterra fuzzy integro-differential system is demonstrated.

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Attractivity for a k dimensional system of fractional functional differential equations and global attractivity for a k dimensional system of nonlinear fractional differential equations

Attractivity for a k dimensional system of fractional functional differential equations and global attractivity for a k dimensional system of nonlinear fractional differential equations

In this paper, we present some results for the attractivity of solutions for a k-dimensional system of fractional functional differential equations involving the Caputo fractional derivative by using the classical Schauder’s fixed-point theorem. Also, the global attractivity of solutions for a k-dimensional system of fractional differential equations involving Riemann-Liouville fractional derivative are obtained by using Krasnoselskii’s fixed-point theorem. We give two examples to illustrate our main results.

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EFFECTS OF THERMAL AND SOLUTAL STRATIFICATION ON MIXED CONVECTION FLOW ALONG A VERTICAL PLATE SATURATED WITH COUPLE STRESS FLUID

EFFECTS OF THERMAL AND SOLUTAL STRATIFICATION ON MIXED CONVECTION FLOW ALONG A VERTICAL PLATE SATURATED WITH COUPLE STRESS FLUID

The energy equation (9) and concentration (10) are coupled with the flow equation (8). Since the system of differential equations is nonlinear and non-homogenous, the closed form solutions are not obtained and hence the system is solved using the implicit Keller-box method (Cebeci and Bradshaw (1984)). This method has been proven to be adequate and give accurate results for boundary layer equations. In boundary conditions, the value of η at ∞ is replaced by a sufficiently large value of η where the temperature, concentration profiles approach zero and velocity approaches to one. For that the value is considered as η = 8 with a grid size of η of 0.01. The dimensionless velocity, temperature and concentration function are evaluated and are presented through plots, which are shown in Figs. 2-13. The effects of emerging parameters of the flow on dimensionless velocity, temperature and concentration are discussed. In throughout the computations the constant values are assumed as Re = 100, Pr = 0.71, Sc = 0.22 and  =0.1.
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On Stable Reconstruction of the Impact in the System of Ordinary Differential Equations

On Stable Reconstruction of the Impact in the System of Ordinary Differential Equations

The Upper Estimation, the Asymptotic Order of Accuracy It is known that there exist constant K20 > 0 such that the lower estimation of accuracy Dh in C[a,b] is of the form v1 h.. In view[r]

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Partially integrable nonlinear equations with one higher symmetry

Partially integrable nonlinear equations with one higher symmetry

For a long time there has been a general belief that ”if a partial differential equation or a system of differential equations has one nontrivial symmetry, then it has infinitely many” [1]. Indeed, this statement is true in the case of evolutionary equations, which right hand side is a homogeneous differential polynomial [2]. However, in 1991 Bakirov proposed an example putting this conjecture in doubt [3]. He found that the system

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Generalized Whittaker's equations for holonomic mechanical
systems

Generalized Whittaker's equations for holonomic mechanical systems

It is known [4] that canonical equations for a conservative holonomic system whose Hamiltonian is H are obtained by forming the first Pfaff’s system of differential equations of the diff[r]

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The eigenvalue problem for a coupled system of singular p Laplacian differential equations involving fractional differential integral conditions

The eigenvalue problem for a coupled system of singular p Laplacian differential equations involving fractional differential integral conditions

In this paper, we deal with a coupled system of singular p-Laplacian differential equations involving fractional differential-integral conditions. By employing Schauder’s fixed point theorem and the upper and lower solution method, we establish an eigenvalue interval for the existence of positive solutions. As an application an example is presented to illustrate the main results.

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Stochastic Differential Equations: Models and Numerics - Free Computer, Programming, Mathematics, Technical Books, Lecture Notes and Tutorials

Stochastic Differential Equations: Models and Numerics - Free Computer, Programming, Mathematics, Technical Books, Lecture Notes and Tutorials

We can state our problem in optimal control terms as the maximization of an objective function, the expected profit from selling electricity power during a given period, with respect to control functions, like the hourly turbined flow. Observe that the turbined flow is positive and smaller than a given maximum value, so it is natural to have a set of feasible controls, namely the set of those controls we can use in practice. In addition, our dynamical system evolves according to a given law, also called the dynamics, which here comes from a mass balance in the dam’s lake. This law tells us how the state variable, the amount of water in the lake, evolves with time according to the control we give. Since the volume in the lake cannot be negative, there exist additional constraints, known as state constraints, that have to be fulfilled in the optimal control problem.
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Continuous Genetic Algorithm : A Robust Method to Solve Higher Order Non-Linear Boundary Value Problem

Continuous Genetic Algorithm : A Robust Method to Solve Higher Order Non-Linear Boundary Value Problem

Genetic algorithm (GAs) is stochastic population-based search techniques. GAs operates on a population of potential solutions applying the principal of survival of the fittest to produce better approximation to a solution. The appeal of GAs comes from their simplicity and robustness as well as their power to discover good solutions for complex high dimensional global optimization problems that are very difficult to handle by more conventional techniques (Forrest and Mitchell, 1993). They have performed well in a number of diverse application such as the solution of ordinary differential equations [3], solution to system of nonlinear equations [5], application of genetic algorithm and CFD for flow control optimization [11], collision-free Cartesian path planning of robot manipulation [12], numerical solution of boundary value problem [7].
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Three-dimensional Magneto-thermo-elastic Analysis of Functionally Graded Truncated Conical Shells

Three-dimensional Magneto-thermo-elastic Analysis of Functionally Graded Truncated Conical Shells

Shadmehri et al. [3] proposed a semi-analytical approach to obtain the linear buckling response of conical composite shells under axial compression load. The principle of minimum total potential energy was used to obtain the governing equations and Ritz method was applied to solve them. Sofiyev et al. [4] studied the stability of three layered conical shell containing an FGM layer subjected to axial compressive load. The fundamental relations for stability and compatibility equations were transformed into a pair of time- dependent differential equations via Galerkin's method. Xu et al. [5] used the dynamic virtual work principal to derive non-linear equations of transverse motion of truncated conical shells. The Galerkin procedure was used to develop a system of equations for time functions which were solved by the harmonic balance method. Patel et al. [6] studied the thermo-elastic stability characteristics of cross-ply oval cylindrical/conical shells subjected to uniform temperature rise through non-linear static and finite element method. Zhang and Li [7] discussed the buckling behavior of functionally graded truncated conical shells subjected to normal impact loads and employed the Galerkin procedure and Runge-Kutta integration scheme to solve non-linear governing equations. Aghdam et al. [8] carried out bending analysis of moderately thick clamped FG conical panels subjected to uniform and non-uniform distributed loadings. The First Order Shear Deformation Theory (FSDT) was applied to derive the governing equations and Extended Kantorovich Method (EKM) was used to solve the equations. Wu et al. [9] presented the three-dimensional solution of laminated conical shells subjected to axisymmetric loadings using the method of perturbation. Petrovic [10] investigated stress analysis of a cylindrical pressure vessel loaded by axial and transverse forces on the free end of the nozzle applying the finite element method. Jabbari et al. [11] developed a general analysis of one-dimensional thermal stresses in a hollow thick cylinder made of functionally graded material, using the direct method to solve the governing equations. Eslami et al. [12] obtained a general solution for the one-dimensional
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