View of NEW FUNCTIONAL TAYLOR SERIES METHOD FOR SOLVING FRACTIONAL RICCATI DIFFERENTIAL EQUATION

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

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Vol.04,Special Issue 04, ²^nd Conference (ICIRSTM) April 2019, Available Online: www.ajeee.co.in/index.php/AJEEE

NEW FUNCTIONAL TAYLOR SERIES METHOD FOR SOLVING FRACTIONAL RICCATI DIFFERENTIAL EQUATION

Deeksha Sharma

Modi institute of management and technology, Kota 1 INTRODUCTION

Fractional calculus has recently been applied in various areas of engineering, science, applied mathematics and bio engineering. It is a branch of mathematical analysis applied to the study of integrals and derivatives of arbitrary order, not only fractional but also real.

Commonly these fractional integrals and derivatives are not known for many scientists and up to recent years have been used only in a pure mathematical context. The fractional calculus has found diverse applications in various scientific and technological fields [1], [2]

such as thermal engineering, acoustics, electromagnetism, diffusion, turbulence, signal processing and many other physical processes. Fractional differential equations have also been applied in modelling many physical, engineering problems and also in non-linear dynamics [3], [4]. The Riccati differential equation is named after the Italian nobleman count Jacopo Francesco Riccati (1676-1754). The book of Reid [5] includes the main theories of Riccati equation, with implementations to random processes, optimal control and diffusion problems [6]. Fractional derivatives have also been used in anomalous diffusion description, where the derivatives can explain sub – diffusive and super – diffusive phenomena observed in real systems. Many applications of fractional derivatives are electromagnetic theory, circuit theory, biology etc. Many definitions of fractional derivatives and integrals can be found in books [1], [7], [8]. While studying fractional derivatives they do not have a clear significance. The other difficulty arises is their complex-integro – differential definition, which makes a simple manipulation with standard integer operators, a complex operation should be made carefully.

Power series is used in the study of elementary functions. In physics, chemistry and many other sciences this power expansions has allowed scientist to make an approximate study of many systems, neglecting higher order terms around the equilibrium point. Power series helps us to linearize a problem which guaranties easy analysis. Power series expansion has been widely used in computational science obtaining an easy approximate of a function [9], numerical schemes to integrate a Cauchy problem [10], or gaining knowledge about the singularities of a function by comparing two different taylor series expansions around different points [11], [12], [13]. Fractional Taylor series has been developed for Riemann-Liouville derivative [14] and in the present work a similar study has been made for caputo fractional derivative defined as –

f ( x ) =

– ∫ _– dt (1)

Where, the fractional integral is defined as – f ( x ) =

∫ – f ( t ) dt ( 2 )

The use of this definition of the fractional derivative is justified since it has good physical properties [2]. Theorem 2.1 [15] allows to obtain a fractional power series for a function in terms of its caputo fractional derivatives evaluated at a, which is in some sense the initial point of the independent variable x. The ideal situation would be to obtain a similar series with the derivatives evaluated at any other point a, so the expansion can be constructed independently from the starting point a.

2 ANALYSIS OF THE METHOD

We consider the following fractional Riccati differential equation A(t) B(t) y C(t) , 0 α 1 (3) Subject to the initial condition

(0) = , k = 0, 1, 2, 3 ... n – 1

Where, = ⁄ , α is the order of fractional derivative, A(t), B(t) and C(t) are real functions of t and ’s are constants.

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Theorem – Let α (0, 1] and f(x) a continuous function in [a, b] satisfying the following conditions :

(1) j = 1, 2, 3..., 8 , f Є C ([a , b]) and f Є ([a , b]) (2) f(x) is continuous on [a , b]

Let Є (a, b] Then x Є [a, b], we know that from fractional Taylor series expansion [15]

f(x) = f( f( )

f ( )

f( (x , ,a ) ( 4 ) Where, , are the differences given by

= [ ]

[ ]

= [ ]

= [ ] (5)

and H = ( x – a ) , L = ( – a ) , ( x , ,a ) is the remainder term

Proof : Since f ( x ) and f ( x ) for j = 1 , 2 , 3 ,4 fulfil the conditions of theorem 2.1 [15 ] , there must be fractional power series expansions for f(x) , f(x), f(x) , f(x) , f(x), and using them at x = , we have :

f ( a ) = f ( ) f (a )

f ( a )

f ( a ) = f ( ) f ( a )

f ( a )

f ( a ) = f ( ) f ( a )

f ( a )

f ( a ) f ( a ) (6)

f ( a ) = f ( ) f ( a ) f ( a )

f ( a )

f ( a ) = f ( ) f ( a )

f ( a )

Being the corresponding reminder terms of the fourth order series of every f (x) at x = .

Then substituting every series from equation (6) in the fourth order series of f (x) and grouping corresponding terms, equation (4) with differences (5) is obtained.

The reminder term results as a combination of the reminder term of expressions (6) and higher order terms with derivatives of order 5α, 6α, 7α and 8α evaluated at x = a.

Explicit form is omitted because of its complex form, but can be directly computed following indications given above. The method outlined above can be extended up to any order obtaining series similar to equation (4) for which the difference would be given by

= (k α ∑

( – )

And the corresponding series of order n would be

f ( x ) = ∑ f ( ) ( x , , a )

With (x, , a ) being the reminder term of order nα.

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As we know that from [15], Euler method uses the first order Taylor series to obtain the integration scheme. It is the easiest numerical integration method but the errors are usually very big so it is not frequently used. If we suppose that y (t) is a solution of problem (3), then according to the fractional Taylor series for y (t) at can be written as –

y ( ) = y ( ) y ( ) ( t , , 0 )

With = ( ) (7) Now using (3) in (7) we have the following equation

y ( ) = y ( )

(A ( t ) B ( t ) y C ( t ) ) (t , , 0 ) (8)

The fractional Euler method consists in approximate the function y (t) at , i = 0, 1, 2, 3,...,N neglecting the reminder term. Starting from the Initial point i = 0 assume the initial approximation y ( ) and solve for y ( ) using (8). Similarly solve for y ( ) using y ( ) in (8). In this manner find out the successive approximations and this process is repeated up to the desired degree of accuracy.

3 CONCLUSION

In this paper, the fractional Euler method is used to solve the fractional Riccati Differential equation. Successive approximations are obtained up to the desired degree of accuracy which can converge to the exact solution.

References

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