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The Objects of This Report

1. Deals with heat conduction

2. Drive new equations based on Legendre polynomials

3. This report based on new study with its chats

Abstract

In this report for the mechanical engineering, it develop a new scheme for

numerical solutions of the fractional two- dimensional heat conduction equation on

a rectangular plane. It main aim is to generalize the Legendre operational matrices

of derivatives and integrals to the three dimensional case. By the use of these

operational matrices, it reduce the corresponding fractional order partial

differential equations to a system of easily solvable algebraic equations. The

method is applied to solve several problems. The results It obtain are compared

with the exact solutions and It find that the error is negligible.

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Introduction

The diffusion equation is of great importance in many engineering problemssuch as heat conduction, chemical diffusion, fluid flow, mass transfer,

refrigeration and traffic analysis and so on. After the development of fractional

derivatives it is found that most of these phenomena can. It'll be explained by

fractional order partial differential equations (FPDEs), see for example [1,2] and

the references quoted therein.This report consider the problem in generalized

form as:

Where C1and C2are generalized constants, 0 < 1, t [0, 1], x [0, 1]

and y [0, 1]. Conventionally various methods such as smoothed partial

hydrodynamic method [3], meshless method [4,2], homotopy perturbation

method [5,6], Tau method [7], method of local radial functions [8], Sinc

Legendre collocation method [9,10] are used for the solutions of such type of

problems. Recently some approximate solutions for integer order heat

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conduction equations are obtained by Exp-function method [11], variational

iteration method and energy balance method [12,13]. These methods are

very efficient and provide very good approximations to the solutions but due

to high computational complexities these methods are not so easy to apply to

fractional order partial differential equations in higher dimensions.

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It need an easy and efficient method to solve such type of problems.

More recently, the techniques based on operational matrices are

extensively used for approximate solutions of a wide class of differentialequations as it'll as partial differential equations [14,15] and references

quoted therein. The technique based on the operational matrices is simple

and provides high accuracy but up to now this technique is used only to solve

partial differential equations (PDEs) with only two variables. It generalize the

technique to solve PDEs with three variables.

It use Legendre polynomials and develop new matrices of fractional order

differentiations and integrations to solve the corresponding fractional

order partial differential equations without actually discretizing the

problem.

It method reduces the FPDEs to a system of easily solvable algebraic

equations of Sylvester type which can be easily solved by any

computational software. Generally, large systems of algebraic equations

may lead to greater computational complexity and large storage

requirements. HoItver It technique is simple and reduces the

computational complexity of the resulting algebraic system.

It is worthwhile to mention that, the method based on using the

operational matrix of orthogonal functions for solving FPDEs is computer

oriented. It use Matlab to perform necessary calculations. The article is

organized as follows.

It begin by introducing some necessary definitions and mathematical

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-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 0 0.2 0.4 0.6 0.8-0.2 1

x

P

preliminaries of the fractional calculus and Legendre polynomials as show in

Fig.1 which are required for establishing main results.

The Legendre operational matrices of fractional derivatives and

fractional integrals are obtained.

The devoted to the application of the Legendre operational matrices of

fractional derivatives and fractional integrals to solve the transient state

time fractional heat conduction equation on a rectangular Plane also in the

same section the proposed method is applied to several examples.

Fig.1 Shown Legendre polynomials

Preliminaries

For convenience, this section summarizes some concepts, definitions

and basic results from fractional calculus.

Where

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Definition 2.1 . Given an interval [a, b] R. The RiemannLiouville

fractional order integral of a

f u n c t i o n (L1[a, b], R) order

Ris defined bprovided that the integral on the right hand side exists.

Definition 2.2. Caputo Derivative: For a given function (x) Cn[a, b], the

Caputo fractional order derivative is defined as:

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Hence, it follows that

Where n=[] + 1

2.1. The shifted Legendre polynomials

The Legendre polynomials defined on [1, 1] are given by the following recurrence

relation:

Which implies that any f (x) C [0, 1] can be approximated by Legendre

polynomials as follows:

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In vector notation, it writeas:

f (x) KT PM(x), (5)

Where M = m + 1, K is the coefficient vector and P M (x) is M terms

function vector. The notion was extended to the two- dimensional space

and the two-dimensional Legendre polynomials of order M are defined as

a product function of twoLegendre polynomials

Pn(x, y) = Pa(x) Pb(y), n = Ma + b + 1, a = 0, 1, 2, . . , m, b = 0, 1, 2, . . , m.

(6)

The orthogonality condition of Pn(x, y) is

Any f (x, y) C ([0, 1] [0, 1]) can be approximated by the polynomials Pn(x, y) as

follows:

For simplicity, It use the notation Cn = Cab wheren = Ma + b + 1, and rewrite

(7)as follows:

f(x, y) CnPn(x, y)=KM 2 (x, y) (8)

In vector notation,whereKM 2is the 1 M 2 coefficientrow vectorand (x,

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T

y)is the M 2 1 column vectoroffunctionsdefined by

(x, y)=11(x, y) 1M (x, y) 21(x, y) 2M (x, y) MM (x,

y) (9)

Where i+1,j+1(x, y) = Pi(x)Pj(y), i, j = 0, 1, 2, . . . , m.

1.1Three-dimensional Legendrepolynomials

It generalize the notion to the case of the three-dimensional space and

define as show in Fig.2 Legendre polynomials of order M as the product of

Legendre polynomials of the form:

P(abc)(t, x, y) = Pa(t)Pb(x)Pc(y), a = 0, 1, 2, . . . , m, b = 0, 1, 2, . . . , m, c

= 0, 1, 2, . . . , m. (10)

The orthogonality relation for P(abc)(x, y, t) is given by

Any f (x, y, t) C([0, 1] [0, 1] [0, 1]) can be approximated by P(abc)

1.2 Function approximation with three-dimensional Legendre polynomials

Where C abc can be obtained by the relation:

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For simplicity, It use the notation as C(an)= C(abc), where n = Mb + c +1.

Hence,(11)can be rewritten :

Where K is the coefficient matrix and is a function vector (t) is the one-

dimensional related to x, y and . Legendre function vector related to the

variable t.

2.3. Error analysis

In this section, we provide an analytic expression for the error as show

in Fig.3 of approximation of a sufficiently smooth function(x, y, t) ,

where = [a, b] [c, d] [e, f ].

g(x, y, t) g(M,M,M)(x, y, t)2 g(x, y, t) Q (M,M,M)(x, y, t)2. (14)

The in equation(14)also holds if Q(M,M,M)(x, y, t) is the interpolating

polynomial of the function g at points (xi, yj, tk) where

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Fig. 2.The approximate solution of example 1 at different values of t (t = 0.2,

t = 0.4, t = 0.6, t = 0.8), where M = 7, = 1 and = 2. The dots represent the

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