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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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