The numerical **solution** of fractional partial **differential** **equations** has been developed in several ways by using the Finite Difference method (Chen, Liu & Burrage 2008, Murio 2008, Hu & Zhang 2012, Sweilam, Khader & Mahdy 2012, Tadjeran 2007, Tadjeran & Meerschaert 2007), the Adomian Decomposition method (Dhaigude & Birajdar 2012, Di- ethelm & Ford 2002), the Predictor–Corrector method (Diethelm & Ford 2002), the Finite Element method (Deng 2008, Jiang & Ma 2013), and Numerical Quadrature (Diethelm 1997, Murio 2008). The majority of these numerical methods either use the Gr¨ unwald–Letnikov approximation or the L1 scheme to approximate the fractional deriva- tive. However, there are other techniques used to approximate the fractional derivative such as the Spline method (Pedas & Tamme 2011, Li 2012) and the Collocation method (Rawashdeh 2006, Hesameddini & Asadollahifard 2016).

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functions. For example, f (σ)(0) = σ(0), f (σ) ′ = f (σ ′′ ) (this time using second-order derivatives) defines the function f (σ) = (σ(0), σ(2), σ(4), . . . ). In these examples, it is easy to see that the stream **differential** **equations** have a unique **solution**. But how about τ (0) = 0, τ ′ = f (τ )? A moment’s thought reveals that this equation has several solutions, e.g. τ = (0, 0, 0, . . .) and τ = (0, 0, 1, 1, 1, . . .). But what is the difference between this equation and the previous ones? How can we ensure the existence of unique solutions? Which classes of streams can be defined using a finite amount of information? These questions have been studied by several authors in many different contexts in recent years, and have led to notions such as rational streams, context-free streams, and new insights into automatic and regular sequences.

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To study the techniques for solving **differential** **equations** based on numerical approximations were developed before programmable computers existed. **Differential** **equations** are an important part of many areas of mathematics, from fluid dynamics to celestial mechanics. Many mathematicians have studied the nature of these **equations** for hundreds of years and there are many well-developed **solution** techniques. Except for a few special cases, **differential** **equations** cannot normally be solved analytically. Instead, there are many numerical methods which have been developed to provide solutions. Keywords: **Differential** **Equations**, Fluid Dynamics, Numerical Method.

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methods have been used to solve fractional **differential** **equations**. There are some further method, such as operational method,the Adomian decomposition method(ADM)[6], the homotopy perturbation method(HPM)[7, 10], the generalized **differential** transformation method(GDTM)[9]. In this work, we approximate the exact **solution** by use of Radial Basis Functions method(RBFs).We present the advantages of using the RBFs especially where in the data points are scattered. Radial basis function(RBF) is one of the most popular basis for construction of meshless methods. It is (conditionally) positive defi- nite, rotationally and translationally invariant.Over the last 27 years, RBF methods have become an important tool for the interpolation of scattered data and for solving partial **differential** **equations**[10].

Hybrid system is a dynamic system that exhibits both continuous and discrete dynamic behavior. The hybrid **differential** **equations** have a wide range of applications in science and engineering. The hybrid systems are devoted to modeling, design, and validation of interactive systems of computer programs and continuous systems. Hybrid fuzzy **differential** **equations** (HFDEs) is considered by Kim et al. [11]. In the present paper it is shown that the example presented by Kim et al. in the Case I is not very accurate and in the Case II, is incorrect. Namely, the exact **solution** proposed by the authors in the Case II are not solutions of the given HFDE. The correct exact **solution** is also presented here, together with some results for characterizing solutions of FDEs under Hukuhara differentiability by an equivalent system of ODEs. Then, the homotopy analysis method (HAM) is applied to obtained the series **solution** of the HFDEs. Finally, we illustrate our approach by a numerical example. Keywords : Fuzzy **differential** **equations**; Homotopy analysis method; Approximate **solution**.

The objective of this article is to discuss a variation of the direct Taylor series (DTS) algorithm for the **solution** of first- and higher-order **differential** **equations**. We show that not only this algorithm remains accurate away from the initial point, evaluation of the higher derivatives that are needed for accuracies com- parable to the RK, ABM, and Milne methods are indeed quite simple. Finally, the accuracy and ease of application of the DTS method are explicitly demon- strated by considering several important second-order linear and nonlinear dif- ferential **equations** of mathematical physics and comparing their solutions using the fourth-order DTS, RK, ABM, and Milne methods.

Fuzzy fractional **differential** **equations** have been introduced via general differentiability. Variational iteration method has been applied to obtain approximate fuzzy **solution**. Although the example given in this paper is a fuzzy fractional **differential** equation, it might also be applicable to fuzzy fractional partial **differential** **equations**.

**differential** transform method (DTM) to solve the Neutral functional-**differential** equation with proportional delay. Firstly Zhou [21] introduced **Differential** transform Method on different type of nonlinear **differential** **equations** and has shown various remarkable results of this method. Using **differential** transformation method, a closed form series **solution** or an approximate **solution** can be obtained. The **differential** transform method obtains an analytical **solution** in the form of a polynomial. It is different from the traditional high order Taylor’s series method, which requires symbolic competition of the necessary derivatives of the data functions. The Taylor series method is computationally expansive for large orders in terms of time. This method produced solutions in the form of polynomials and avoids large computational work and round off error. In present time, much nonlinear type of ODEs is easily solved by DTM .This method has been successfully applied to solve many types of nonlinear problems in science and engineering [22-24]. Recently, many adaptive numerical methods have been used which are very effective for these problems [26-29]. In this work the proposed DTM method is analytically applied to Neutral functional-**differential** equation with proportional delays. Several examples are given to verify the efficiency and compatibility of the proposed method.

Because x - 0 is not a point of maximum deviation of the optimal **solution** to the continuous problem, this **solution** is optimal on the wider range X £ x £ 2, where X s -0.76860078 (to eight figures). On this wider range, the maximum deviation is taken at three points, so that the problem is no longer s i n g u l a r However, the **solution** to the dual of the linear programming formulation of the discrete problem on the pornts of maximum deviation is still degenerate.

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In this article, we are presenting numerical solutions of first order **differential** **equations** arising in various applications of science and engineering using some classical numerical methods. We are considering only such practical problems which contain **differential** **equations** of the first order. Picard’s and Taylor series methods are used for solving such type of problems.

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Picard’s and Taylor’s series methods are powerful mathematical tools for solving linear and nonlinear **differential** **equations**. It is concluded that Picard’s and Taylor’s series methods gives more accurate solutions, which are much closer to exact solutions, for solving first order **differential** **equations** arising in some applications of sciences and engineering.

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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 .

has a **solution** provided f is continuous and satisﬁes a Lipschitz condition by C. Corduneanu [2]. The deﬁnition given here generalizes that of Aumann [1] for set- valued mappings. Kaleva [3] discussed the properties of diﬀerentiable fuzzy set-valued mappings and gave the existence and uniqueness theorem for a **solution** of the fuzzy diﬀerential equation x (t) = f (t,x(t)) when f satisﬁes the Lipschitz condition. Also,

In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathemat- ical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of **solution** accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.

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with d j as above is simply the power series representation of the Mittag-Leffer function E Lh n ( ) , and hence the required convergence of the series follows immediately. Therefore, we may apply Weissinger’s Fixed Point Theorem and deduce the uniqueness of the **solution** of our initial value problem (1-2)

In this paper, we study the following methods. Murali Krishna’s method[1,2,3] for Non-Homogeneous First Order **Differential** **Equations** and formation of the **differential** equation by eliminating parameter in short methods. We consider z as dependent variables x and y are independent variables. The first order partial derivatives of z with respect to x and y are ∂x ∂z , ∂z ∂y which are denoted by p, q respectively. The equation d dx 2 y 2 + a 1 dy dx + a 2 y = X is called

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In this paper, the Generalized **Differential** Transformation Method (GDTM) for approximating the **solution** of systems of linear volterra integro-**differential** **equations** of fractional of fractional is implemented. The fractional derivative is considered in the Caputo sense. The approximate solutions are calculated in the form of a convergent series with easily quantifiable workings. Numerical results show that this approach is easy to implement and accurate when applied to systems integro-**differential** **equations**.

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It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional **differential** **equations** on terms of special functions. Recently, there are some papers dealing with the existence and multiplicity of **solution** to the nonlinear fractional **differential** **equations** boundary value problems, see [8]-[14].

Abstract A semi-analytical **solution** is presented for three dimensional elastic analysis of finitelylong, simply supported, orthotropic, laminated cylindrical panels with piezoelectric layers subjected to outer pressure and electrostatic excitation. Both the direct and inverse piezoelectric effects are investigated. The **solution** is obtained through reducing the highly coupled partial **differential** **equations** (PDE's) of equilibrium to ordinary **differential** **equations** (ODE's) with variable coefficients by means of trigonometric function expansion in longitudinal and circumferential directions. The resulting ODE's are solved by dividing the radial domain into some finite subdivisions and imposing necessary continuity conditions between the adjacent sub-layers. Some numerical examples are presented for the stress distribution and electric responses due to outer pressure in both sensorial and actuating states. Also, the effect of geometric properties on the sensitivity and actuating power of the structure are investigated.

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In this paper, we study the existence, uniqueness and stability **solution** of integro-**differential** **equations** of second order with the operators by using both method Picard approximation and Banach fixed point theorem.These investigations lead us to improving and extending the above method. . Thus the integro- **differential** **equations** of second order with the operators are more general and

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