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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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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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.

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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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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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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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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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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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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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 .

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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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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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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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]

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**

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]

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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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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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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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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