Numerical Solution of Nonlinear Fredholm Integro-Differential Equations using

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

Numerical Solution of Nonlinear Fredholm Integro-Differential Equations using Leibnitz-Haar Wavelet Collocation Method S. C. Shiralashetti1, a, R. A. Mundewadi1 1 P. G. Department of Studies in Mathematics, Karnatak University, Dharwad-580003, India a

shiralashettisc@gmail.com DOI 10.13140/RG.2.2.31444.19848

Keywords: Leibnitz-Haar wavelet collocation method, operational matrix, nonlinear Fredholm integro-differential equations.

ABSTRACT. In this work, we present a Leibnitz-Haar wavelet collocation method for solving nonlinear Fredholm integro-differential equation of the second kind. Haar wavelet and its Operational matrix are utilized to convert the differential equations into a system of algebraic equations, solving these equations using MATLAB to compute the required Haar coefficients. The numerical results obtained by the present method have been compared with those obtained by [3, 4] of the illustrative examples, which shows the efficiency of the method.

1. Introduction. In recent years, there has been a growing interest in the integro-differential equations (IDEs) which are a combination of differential and Fredholm-Volterra integral equations. IDEs play an important role in many branches of linear and nonlinear functional analysis and their applications in the theory of engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and electrostatics. The mentioned integro-differential equations are usually difficult to solve analytically, so a numerical method is required. There are several numerical methods for approximating the solution of nonlinear Fredholm integro-differential equations are known and many different basic functions have been used [1-4].

Wavelets theory is a relatively new and an emerging tool in applied mathematical research area. It has been applied in a wide range of engineering disciplines; particularly, signal analysis for waveform representation and segmentations, time-frequency analysis and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators.

Moreover, wavelets establish a connection with fast numerical algorithms [5, 6]. Since from 1991 the various types of wavelet method have been applied for the numerical solution of different kinds of integral equations, a detailed survey on these papers can be found in [7]. The solutions are often quite complicated and the advantages of the wavelet method get lost. Therefore, any kind of simplification is welcome. One possibility for it is to make use of the Haar wavelets, which are mathematically the simplest wavelets. Haar wavelet methods are applied for different type of problems in [8-14]. Sirajul-Islam et al. [15], V. Mishra et al. [13] and Lepik [16] and Tamme [17] have applied the Haar wavelet method for solving nonlinear Fredholm integro-differential equations. In the present work, a Leibnitz-Haar wavelet collocation method for solving nonlinear Fredholm integro-differential equations of the second kind is proposed.

The article is organized as follows: In Section 2, the basic formulation of Haar wavelets and its operational matrix is given. Section 3 is devoted to the method of solution. In section 4, we report our numerical results and demonstrated the accuracy of the proposed scheme. Conclusion is discussed in section 5.

2. Properties of Haar Wavelets.

2.1. Haar wavelets MMSE Journal. Open Access www.mmse.xyz

108

Mechanics, Materials Science & Engineering, September 2016

The scaling function

ISSN 2412-5954

for the family of the Haar wavelet is defined as

(1)

The Haar Wavelet family for

is defined as,

(2)

where where

J is the level of resolution;

,

is the translation parameter.

Maximum level of resolution is . The index minimal values

then

in (2) is calculated using

. The maximal value of

is

Let us define the collocation points which has the dimension

. In case of .

, Haar coefficient matrix . For instance,

, then we have

Any function f(x) which is square integrable in the interval (0, 1) can be expressed as an infinite sum of Haar wavelets as

MMSE Journal. Open Access www.mmse.xyz

109

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

(3)

The above series terminates at finite terms if f(x) is piecewise constant or can be approximated as piecewise constant during each subinterval. Given a function f(x) L2(R) a multi-resolution analysis (MRA) of L2(R) produces a sequence of subspaces such that the projections of f(x) onto these spaces give finer and finer approximations of the function f(x) as 2.2. Operational Matrix of Haar Wavelet The operational matrix P which is an N square matrix is defined by

(4)

often, we need the integrals

(5)

For corresponds to the function analytically; we get

, with the help of (2) these integrals can be calculated

=

(6)

=

(7)

In general, the operational matrix of integration of

order is given as

=

(8)

MMSE Journal. Open Access www.mmse.xyz

110

Mechanics, Materials Science & Engineering, September 2016

For instance, =3

ISSN 2412-5954

N = 16, then we have

and

3. Method of Solution In this section, we present a Leibnitz-Haar wavelet collocation method (LHWCM) for solving nonlinear Fredholm integro-differential equations of the second kind,

,

(9)

where K(x, t, u(t)) is a nonlinear function defined on [0, 1 0, 1] are the known function K(x, t, u(t)) is called the kernel of the integral equation and f(x) is also a known function while the unknown function u(x) represents the solution of the integral equation.

MMSE Journal. Open Access www.mmse.xyz

111

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

Leibnitz rule: The conversion of the integral equation into an equivalent differential equation. The conversion is achieved by using the well-known Leibnitz rule [17] for differentiation of integrals.

Let,

(10)

Then differentiation of the integral in (10) exists and is given by

(11)

For Fredholm, If g(x) =0 and h(x) =1 where 0 & 1 are fixed constants, then the Leibnitz rule (11) reduces to

.

(12)

A numerical computation procedure is as follows: Step 1: Differentiating (9) thrice w.r.t x, using Leibnitz rule (11) we get, (13) (14) (15) subject to initial conditions,

(16)

Step 2: Applying Haar wavelet collocation method, Let us assume that,

(17)

Step 3: By integrating (17) four times and using (16), we get,

(18) MMSE Journal. Open Access www.mmse.xyz

112

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

(19)

(20)

(21)

Step 4: Substituting (17) (21) in the differential equation (15), which reduces to the nonlinear system of N equations with N coefficients Substituting Haar coefficients in (21) to obtain the required solution of equation (9).

4. Illustrative Examples In this section we consider the some of the examples to demonstrate the capability of the method in section 3 and error function is presented to verify the accuracy and efficiency of the following numerical results:

where

and

are the exact and approximate solution respectively.

Example 4.1. Let us first consider the Nonlinear Fredholm Integro-differential equation [3],

(22)

with initial conditions

Which has the exact solution

.

We applied the Haar wavelet technique presented in section 3 and solved (22) as follows.

Successively differentiating (22) w.r.to x and using Leibnitz rule (11) which reduces to the differential equation.

Let us first consider the given

differentiating w.r.to x

MMSE Journal. Open Access www.mmse.xyz

113

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

,

again differentiating twice w.r.to x

, .

(23)

Next, consider

differentiating w.r.to x using Leibnitz rule (11),

again differentiating twice w.r.to x using Leibnitz rule (11),

.

(24)

Substituting (23) and (24) in (15), we get the differential equation uiv ( x) 0

(25)

assume that,

(26)

integrating (26) twice, we get

MMSE Journal. Open Access www.mmse.xyz

114

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

(27) (28) (29) (30)

again integrating (30) twice, we get

(31) (32) (33) (34)

Substituting (26), (28), (30), (32) and (34) in the differential equation (25), we get the system of N equations with N unknowns

(35)

N values the coefficients are zero. Substituting these coefficients in (34) and obtained the required LHWCM solutions, which gives the exact solutions is presented in fig 1. This justifies the efficiency of the LHWCM. Example 4.2. Next, consider the Nonlinear Fredholm Integro-differential equation [3],

(36)

with initial conditions

Which has the exact solution

.

Differentiating (36) twice w.r.t x and Using Leibnitz rule (11), its equivalent differential equation, MMSE Journal. Open Access www.mmse.xyz

115

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

(37)

Assume that, (38)

Integrating (38) thrice, we get

(39) (40) (41)

Substituting (38) - (41) in the differential equation (37), we get the system of N equations with N unknowns

(42)

N values the coefficients are zero. Substituting these coefficients in (41) and obtained the required LHWCM solutions, which gives the exact solutions is presented in fig 2. This justifies the efficiency of the LHWCM. Example 4.3. Now, consider the Nonlinear Fredholm Integro-differential equation [4],

(43)

with initial conditions

Which has the exact solution

.

Differentiating (43) w.r.t x and Using Leibnitz rule (11), its equivalent differential equation, (44) assume that, MMSE Journal. Open Access www.mmse.xyz

116

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

(45)

Integrating (45) twice, we get

(46) (47)

Substituting (45) - (47) in the differential equation (44), we get the system of N equations with N unknowns

(48)

N values the coefficients are zero. Substituting these coefficients in (47) and obtained the required LHWCM solutions, which gives the exact solutions is presented in fig 3. This justifies the efficiency of the LHWCM.

Fig. 1. Comparison of LHWCM with exact solution for N = 64 of example 1.

MMSE Journal. Open Access www.mmse.xyz

117

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

Fig. 2. Comparison of LHWCM with exact solution for N = 64 of example 2.

Fig. 3. Comparison of LHWCM with exact solution for N = 64 of example 3.

Summary. The aim of this work is to apply the Leibnitz-Haar wavelet collocation method for solving nonlinear Fredholm integro-differential equations of the second kind. Nonlinear integro-differential equations are usually difficult to solve analytically. In many cases, it is required to obtain the approximate solutions, for this purpose the presented method is proposed, which gives the exact ones.

Our present method avoids the tedious work, it minimizes the computational calculus and supplies quantitatively reliable results. Using Leibnitz rule, converts integral equations into differential equations with initial conditions. The Haar wavelet function and its operational matrix were employed to solve the resultant differential equations. The results obtained by the proposed method have been compared with the method [3, 4]. The illustrative examples have been included to justify the efficiency and which confirms plausibility of new technique.

Acknowledgement F. 14-2/2008(NS/PE), dated-19/06/2012 and F. No. 14-2/2012(NS/PE), dated 22/01/2013.

MMSE Journal. Open Access www.mmse.xyz

118

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

References [1] Y. Ordokhani, S. Davaei, Application of the Bernstein Polynomials for Solving the Nonlinear Fredholm Integro-Differential Equations, J. Appl. Math. Bio. 1(2) (2011) 13-31.

[2] O. A. Arqub, M. Al-Smadi, N. Shawagfeh , Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method, Appl. Math. Comp. 219 (2013) 8938 8948.

[3] A. Shahsavaran, A. Shahsavaran, Application of lagrange interpolation for Nonlinear integro differential equations, Appl. Math. Sci. 6 (2012) 887 892.

[4] F. Mirzaee, E. Hadadiyan, A collocation method to the solution of Nonlinear fredholmhammerstein integral and integro-differential equations, J. Hyper. 2 (1) (2013) 72-86.

[5] C. K. Chui, Wavelets: A Mathematical Tool for Signal Analysis, SIAM. Philadelphia. PA. 1997.

[6] G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Commun. Pure Appl. Math. 44 (1991) 141 183.

Lepik, E. Tamme, Application of the Haar wavelets for solution of linear integral Equations, Ant. Turk Dynam. Sys. Appl. Proce. (2005) 395 407.

[8] N. M. Bujurke, S. C. Shiralashetti, C. S. Salimath, Numerical solution of stiff systems from nonlinear dynamics using single term haar wavelet series, Inter. Jour. Nonlin. Dynam. 51 (2008) 595605.

[9] N. M. Bujurke, S. C. Shiralashetti, C. S. Salimath, Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets, J. Comp. Appl. Math. 219 (2008) 90-101.

[10] N. M. Bujurke, S. C. Shiralashetti, C. S. Salimath, An Application of Single Term Haar Wavelet Series in the Solution of non-linear oscillator Equations, J. Comp. Appl. Math. 227 (2010) 234-244.

[11] N. M. Bujurke, C. S. Salimath, R. B. Kudenatti, S. C. Shiralashetti, A Fast Wavelet- Multigrid method to solve elliptic partial differential equations, Appl. Math. Comp. 185 (2007) 667-680.

[12] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of nonlinear integral equation, Appl. Math. Comp. 167(2) (2005) 1119 1129.

[13] V. Mishra, H. Kaur, R. C. Mittal, Haar wavelet algorithm for solving certain differential, integral and integro-differential equations, Int. J. of Appl. Math. Mech. 8 (6) (2012) 69-82.

[14] S. Islam, I. Aziz, B. Sarler, The numerical solution of second order boundary value problems by collocation method with the Haar wavelets, Math. comp. Model. 52 (2010) 1577-1590.

[15] S. Islam, I. Aziz, M. Fayyaz, A new approach for numerical solution of integro-differential equations via Haar wavelets, Int. J. Comp. Math, 90 (9) (2013) 1971-1989.

-differential equations, Appl. Math. Comp 176 (2006) 324-333.

[17] A. M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, Springer, 2011.

Cite the paper C. Shiralashetti & R. A. Mundewadi (2016). Numerical Solution of Nonlinear Fredholm IntegroDifferential Equations using Leibnitz-Haar Wavelet Collocation Method. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.31444.19848

MMSE Journal. Open Access www.mmse.xyz

119