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Numerical Studies for Solving Fractional Integro-Differential Equations by using Least Squares Method and Bernstein Polynomials

Oyedepo T1, Taiwo OA1*, Abubakar JU2 and Ogunwobi ZO2

1Department of Mathematics, Faculty of Physical Sciences, University of Ilorin, Nigeria

2Department of Mathematical Sciences, Onabisi Onabanjo University, Ago Iwoye, Ogun State, Nigeria

*Corresponding Author:
Taiwo OA
Department of Mathematics, Faculty of Physical Sciences
University of Ilorin, Nigeria
Tel: +234 813 535 9408
E-mail: [email protected]

Received Date: November 10, 2016; Accepted Date: December 23, 2016; Published Date: December 28, 2016

Citation: Oyedepo T, Taiwo OA, Abubakar JU, Ogunwobi ZO (2016) Numerical Studies for Solving Fractional Integro-Differential Equations by using Least Squares Method and Bernstein Polynomials. Fluid Mech Open Acc 3: 142.

Copyright: © 2016 Oyedepo T, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper, two numerical methods for solving fractional integro-differential equations are proposed. The fractional derivative is considered in the Caputo sense. The proposed methods are least squares method aid of Bernstein polynomials function as the basis. The proposed method reduces this type of equation into systems to the solution of system of linear algebraic equations. To demonstrate the accuracy and applicability of the presented methods some test examples are provided. Numerical results show that this approach is easy to implement and accurate when applied to fractional integro-differential equations. We show that the method is effective and has high convergenc rate.

Keywords

Bernstein polynomials; Numerical studies; Convergency rate; Gaussian elimination method

Introduction

The fractional calculus has a long history from 30 September 1695, when the derivative of order α=1/2 has been described by Leibniz [1-4]. The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Gr¨unwald, Letnikov and Riemann. There are many interesting books about fractional calculus and fractional differential equations [5-6]. The use of fractional differentiation for the mathematical modeling of real world physical problems has been wide spread in recent years, e.g., the modeling of earthquake, the fluid dynamic traffic model with fractional derivatives, measurement of viscoelastic material properties, etc. Derivatives of non-integer order are defined in different ways, e.g., Riemann–Liouville, Gr¨unwald– Letnikow, Caputo and Generalized Functions Approach [4]. In this work we focus attention on Caputo’s definition which turns out to be more useful in real-life applications since it is coupled with initial conditions having a clear physical meaning.

Furthermore the use of numerical method for solving fractional integro-differential equations cannot be over emphasized, for, it is of great importance to Mathematician, Engineers and Physicists. In recent years, much attention has been given for the solutions of fractional differential and integro-differential equations [7]. Proposed an efficient method for solving systems of fractional integro-differential equations using adomian decomposition method (ADM). Munkhammar JD proposed a numerical solution of fractional integro-differential equations by collocation method [8]. He JH used the Adomian Decomposition Method to solve fractional Integro-differential equations. ADM requires the construction of Adomian polynomials which are somehow difficult to obtain [9]. Homotopy Perturbation and Homotopy Analysis methods were applied to solve initial value problems of fractional order by Lanczos C [10]. These authors decomposed the given problems into basically two parts using linear and nonlinear operators. The basic assumption was that the solutions of the problem could be expressed as series of polynomials. The truncated parts of these polynomials are then solved to get the approximate solutions of the problems [11]. Employed application of the fractional differential transform method (FDTM) to fractional-order integrodifferential equations with nonlocal boundary conditions [11] also gives some application of nonlinear fractional differential equations and their approximation [12]. Presented numerical approximation of fractional integro-differential equations by an Iterative Decomposition Method (IDM). In the work, approximate solution of each problem is presented as a rapidly convergent series of easily computable terms [13] applied least square method for treating nonlinear fourth order integrodifferential equations [14,16-18] applied an efficient method for solving fractional differential equations using Bernstine polynomials [15] applied least squares method and shifted Chebyshev polynomial for solving fractional integro-differential equations. In his work he used shifted Chebyshev polynomial of the first kind as basis function.

In this work, using the idea of Momani and Qaralleh [15], we proposed an alternative method, called Standard and perturbed least square methods by Bernstein polynomial as basis function [19-22].

In this work, we are concerned with the numerical solution of the following linear fractional integro-differential equation by standard and perturbed least square methods using Bernstein polynomial as basis function

equation (1)

With the following supplementary conditions:

u(i)(0)=δi, n-1<α ≤n, n∈N (2)

Where Dα u(x) indicates the α th Caputo fractional derivative of u(x), f(x),

K(x, t) are given smooth functions, x and t are real variables varying [0,1] and u(x) is the unknown function to be determined.

Some Relevant Basic Definitions

Definition 1

A real function(x), x>0 [Mohammed (2004)], is said to be in the space Cμ, μ∈R if there

Exists a real number P>μ such that f(x)=xp f (x), where f1(x) ∈C [0,1].

Definition 2

A function (x), x>0 is said to be in the space equation if fm μ

Definition 3

The left sided Riemann–Liouvill fractional integral operator of the order μ3 0 of a function f∈Cμ, μ3−1, is defined as

equation (3)

jα f(x) = f (x) (4)

Definition 4

Let equation. Then the Caputo fractional derivate of f (x) is defined as

equation (5)

Hence, we have the following properties:

equation

D jα f(x) = f(x), x >0, m−1< α ≤ m

DC = 0,C is the constant

equation

Where [α] denoted the smallest integer greater than or equal to α and N0= {0, 1.2,…}

Definition 5

An integral equation is an equation in which the unknown function y(x) appears under an integral signs [4] and [5].

A standard integral equation y(x) is of the form:

equation (6)

Where g(x) and y(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a function of two variable x and t called the kernel or the nucleolus of the integral equation. The function y(x) that will be determined appears under the integral sign and also appears inside and outside the integral sign as well. It is to be noted that the limits of integration g(x) and h(x) may be both variables, constants or mixed.

Definition 6

An integro-differential equation is an equation in which the unknown function y(x) appears under an integral sign and contain ordinary derivatives y(n)(x) as well. A standard integro-differential equation is of the form:

equation (7)

Integral equations and integro-differential equations are classified into distinct types according to limits of integration and the kernel K(x; t) are as prescribed before.

1. If the limits of the integration are fixed, then the integral equation is called a Fredholm integral equation and is of the form:

equation (8)

2. If at least one limits is a variable, then the equation is called a Volterra integral equation and is given as:

equation (9)

Definition 7

Bernstein basis polynomials: A Bernstein polynomial [8] of degree n is defined by

equation (10)

where,

equation (11)

Often, for mathematical convenience, we see Bi,n (x)=0if ‹0 or i›n.

Definition 8

Bernstein polynomials: A linear combination Bernstein basis polynomials

equation (12)

is the Bernstein polynomial of degree n where, ai i= 0,1,2,….. are constants.

Definition 9

Shifted Chebyshev polynomial of the first kind denoted by Tn(x) is denoted by the following [11]:

equation (13)

and the recurrence relation is given by

equation (14)

With the initial condition

equation (15)

Definition 10

In this work, we defined absolute error as:

equation (16)

Demonstration of the Proposed Methods

In this section, we demonstrated the two proposed methods mentioned above.

Standard Least Squares Method (SLM)

The standard least square method with Bernstein polynomials as basis function is applied to find the numerical solution of fractional Integro-differential equation given in equation (1). This method is based on approximating the unknown function u(x) by assuming an approximation solution of the form defined in equation (12).

Thus, substituting equation (12) into equation (1), we obtained

equation (17)

Hence, the residual equation is obtained as

equation (18)

Thus, we minimized equation (18) as

equation (19)

Where w(x) is the positive weight function defined in the interval, [a,b], thus, equation (19) is given

equation (20)

We obtained the values of ai(i3 0) by finding the minimum value of S as:

equation (21)

A. Applying equation (21) into equation (20) for various values of i(i3 0);

We obtained (n+i) algebraic system of equations in (n+i) unknown constants a′iS. The systems of equations are then solved by Gaussian elimination method. The results of the unknown constants obtained are then substituted back into the approximate solution given by equation (12) to get the required approximation for the appropriate order.

Perturbed Least Squares Method (PLM)

The basic idea of the method as conceived by [10], is the substitution of equation (12) into a slightly perturbed equation (1) to obtain

equation (22)

Where, equation (23)

And equation is the shifted Chebyshev polynomials defined in equation (13) and τn is a free tau parameter to be determined along with ai(i30). Equations (12) and (23) are substituted into equation (22) to get

equation (24)

Hence the residual equation is defined as

equation (25)

Thus we minimized equation (24) by denoting

equation (26)

Where all the parameters involved are mention above, thus, equation (24) is given

equation (27)

We obtained the values of ai(i3 0) and τn by finding the minimum value of S as:

equation (28)

and equation (29)

Applying equation (29) and (28) into equation (27) for various values of i(i3 0)

To obtain (n+2) algebraic system of equations in (n+2) unknown constants a′iS. The systems of equations are then solved by Gaussian elimination method. The results of the unknown constants obtained and then substituted back into the approximate solution given by equation (12) to get the required approximation.

Remark 1: The convergence and stability of the method were discussed in Taiwo (1991) while the existence and uniqueness of solution have been proved by Adeniyi (1991).

Numerical Examples

In this section, we demonstrated the proposed methods discussed above on some examples.

Example 1: Consider the following fractional Integro-differential equation:

equation (30)

Subject to u(0)=0 with exact solution B(x)=x2−x (31)

We have solved the above problem for value of n=3 in order to compare the results obtained with the exact solution. Also graphical representations of the result are presented.

For case n=3.

Thus, the approximate solution given in equation (12) becomes

equation (32)

Hence expanding equation (32) further, we have

u3(x) = a0 (1 − 2x + 3x2) + a1 (3x − 6x2+3x3) + a1 (3x2 − 3x3) + a3x3 (33)

Substituting (33) into (31), we have

equation (34)

Applying the Caputo properties on equation (34), we have

equation (35)

Simplifying equation (35), we have

equation (36)

Hence, the residual equation is defined as

equation (37)

In order to minimize equation (37), let

equation (38)

equation (39)

Applying equation (21) and (20) on equation (39) , we have the following equations

equation (40)

equation (41)

equation (42)

equation

equation (43)

Integrating equations (40 - 43) with respect to x over the interval [0,1], we have four algebraic linear equations in 4 unknown constants. These algebraic linear equations are as follows:

equation

The values obtained solving equations (44 - 47) by Gaussian elimination method are as follows:

a0 = − 3.487888303×10−5,

a1 = −0.3333012074,

a2 = −3.3333014828,

a3 = −3.552432041×10−5

These values are then substituted into equation (33), after simplifying we have the approximate solution as

u3 (x)=−0.00003487888303−0.9997989856x+0.999798159x2+0.000 0001808x3 (48)

Demonstration of Perturbed Least Square Method

Example 1

Consider the following fractional Integro-differential equation:

equation (49)

Subject to u(0)=0, with exact solution U(x)=x2−x (50)

Solution

We have solved the problem above for the case n=3 with perturbation term Ha(x) given in equation (23). Thus, equation (50) becomes

equation (51)

Where equation (52)

and equation (53)

The approximate solution given in equation (12) becomes

equation (54)

Hence, expanding equation (54) further, we have

equation (55)

Substituting equations (53) and (55) into equation (51) we have

equation (56)

Applying the Caputo properties on equation (56) we have

equation (57)

Simplifying equation (57), we have

equation (58)

Hence, the residual equation is defined as

equation

In order to minimize equation (59), let

equation (60)

equation (61)

Applying equation (28) and (29) on equation (61), we have the following equations :

equation (62)

equation (63)

equation (64)

equation (65)

equation (66)

Integrating the equations (62-66) with respect to x over the interval [0,1], we have we algebraic linear equations in 5 unknown constants. These algebraic linear equations are as follows:

equation (67)

equation (68)

equation

The values obtained solving equations (67-71) by Gaussian elimination method, are as follows:

equation

The values are then substituted into equation (55), after simplification we the approximate solution as

u3 (x) = − 0.0001080032206 − 0.9998062943x + 0.999757556x2 + 0.000013067x3 (72)

Tables of Results

Numerical results of Example 1 (Table 1)

X Exact Solution Approximate Solution of standard least squares method (SLM ) Approximate solution of perturbed
(PLM)
Absolute error of standard least squares method (SLM) Absolute error of perturbed least squares method
(PLM)
0.0 0.00 -0.00000348788 -0.00001080032 3.4878E-5 1.0800E-4
0.1 -0.09 -0.09001679563 -0.09009104407 1.6795E-5 9.1044E-5
0.2 -0.16 -0.16000274820 -0.16007885540 2.7482E-6 7.8855E-5
0.3 -0.21 -0.20999273540 -0.21007135870 7.2646E-6 7.1358E-5
0.4 -0.24 -0.23998675610 -0.24006847560 1.3243E-5 6.8475E-5
0.5 -0.25 -0.24998480930 -0.25007012800 1.5190E-5 7.0128E-5
0.6 -0.24 -0.23998689400 -0.24007623710 1.3106E-5 7.6237E-5
0.7 -0.25 -0.20999300890 -0.21008672480 6.9911E-6 8.6724E-5
0.8 -0.16 -0.16000315300 -0.16010151250 3.1530E-6 1.0151E-4
0.9 -0.09 -0.09001732530 -0.09012052186 1.7325E-5 1.2052E-4
1.0 0.00 -0.00000355247 -0.00000143674 3.5524E-5 1.4367E-4

Table 1: Numerical results of Example 1.

Examples 2

Consider the following fractional Integro-differential equation:

equation (73)

Subject to u(0)=0 with the exact equation U(x)=x−x3 (74)

Numerical results of Example 2 (Table 2)

X Exact Solution Approximate solution of standard least squares method (SLM) Approximate solution of perturbed least squares method (PLM) Absolute error of  standard least squares method (SLM) Absolute error perturbed least squares method (PLM)
0.0 0.00 0.00010284974 0.00010996755 1.0284E-4 1.0996E-4
0.1 0.099 0.09906303620 0.09907034368 6.3036E-5 0.7034E-5
0.2 0.192 0.19202565970 0.19203342150 2.5659E-5 3.3421E-5
0.3 0.273 0.27299313320 0.27300160370 6.8668E-6 1.6037E-6
0.4 0.336 0.33596787000 0.33597729340 3.2130E-5 2.2706E-5
0.5 0.375 0.37495228320 0.37496289360 4.7716E-5 3.7106E-5
0.6 0.384 0.38394878610 0.38396080680 5.1213E-5 3.9193E-5
0.7 0.357 0.35695979180 0.35697343630 4.0208E-5 2.6563E-5
0.8 0.288 0.28798771360 0.28800318470 1.2286E-5 3.1847E-6
0.9 0.171 0.17103496450 1.71052455200 3.4964E-5 5.2455E-5
1.0 0.00 0.00010395790 0.00012365050 1.0395E-5 1.2365E-4

Table 2: Numerical results of Example 2.

Graphical Representation of the Two Methods (Figures 1-4)

fluid-mechanics-results

Figure 1: Numerical results of Example 1.

fluid-mechanics-example2

Figure 2: Numerical results of Example 2.

fluid-mechanics-error

Figure 3: Error of Example 1.

fluid-mechanics-error2

Figure 4: Error of Example 2.

Conclusion

In this paper, least square method with the aid of Bernstein polynomials was successfully deduced for solving fractional integrodifferential equations. The numerical results in the tables and graph show that the present method provides highly accurate numerical Solutions for solving these types of equations.

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