Numerical Solution for Fractional Integro-differential Equations Using Haar Wavelets
A.Padmanabha Reddy1, Manjula S. H2., C. Sateesha3
1,2,3Department of Studies in Mathematics, V. S. K. University, Ballari, Karnataka,India.
Abstract: In this paper, we solved the fractional integro-differential equations using Haar wavelets. Many problems of science and engineering are modeling by the fractional integro-differential equations. For such problems we constructed the algorithm based on Haar wavelets using collocation method. Some test problems are solved and compared results with other numerical methods to show the accuracy and efficiency of the proposed method.
Keywords: Fractional integro-differential equations, Haar wavelets, Collocation method.
The fractional integro-differential equations arise in many fields of science and engineering viz. thermal systems, turbulence, ?uid ?ow, Brownian motion, mechanics, viscoelastic, image processing mathematical modeling of heat conduction in materials with memory1. However, these equations are found in combined conduction, convection and radiation problems 2 and other areas of applications.
The wavelets have many applications in science and engineering since 1910. Many families of wavelets are exist viz. Haar, Daubechies, Coi?et, Symlet, biorthogonal spline, etc. Alfred Haar3 had invented the Haar wavelets, which are yielded from pairs of piecewise constant functions. These functions are orthogonal, symmetric, compactly supported and have explicit expression for each scaling function 4. Due to these properties Haar wavelet is using as a mathematical tool in the numerical solution of many problems. Chen and Hsiao 5,6 have derived the operational matrix method to solve problems of dynamical systems. The method is improved by the Lepik7 through integration of Haar wavelets, where as Haar wavelets with a collocation method is applied to the many problems of higher order ordinary differential equations, Integral and integro differential equations by Siraj ul Islam8, Fazal et. al.9,10, Reddy et al.11,12, Mishra et al.13 etc. We motivated by these works and constructed the algorithm to solve considered problems using Haar wavelet collocation method(HWCM).
Fractional integro-differential equations are solved by many researchers with various numerical methods. R.C Mittal et al.14 have employed the adomian decomposition method, Ordokhani et al. 15 have applied the Bessel polynomial method(BPM), Mohammed 16 had used the least squares method with the aid of shifted Chebyshev polynomial, Awawdeh et al.17 have employed the Homotopy analysis method and C. S .Singh et al.18 have used the Legendre scaling basis functions.
We considered the following types of linear fractional integro-differential equations defined over (0,1)
Fractional Fredholm integro-differential equation
Fractional Volterra integro-differential equation
here , is the highest order of derivative of are the given functions, is the unknown function to be determined.
In this paper, section 2 contains the Haar wavelets and their integrals of fractional order, method of solution is shown in section 3, experimental problems are inserted in section 4 and conclusion is given in section 5.
2. Haar wavelets and their integrals of fractional order
Haar functions forms the basis for The Haar function defined in the interval is divided into subintervals of equal length where the maximum level of resolution. Two more parameters are required i.e. dilatation and translation to construct Haar wavelet 7,12. When ,
For we have father wavelet
For we have father wavelet
We can integrate the Haar functions several times and notations are given by
If is fractional and then using gamma function Eq. (6) is reduced to
and for , we get
If is natural number and , then Eq. (6) is given as
For , Eq. (6) becomes
By the Haar multiresolution analysis property, any finite energy function defined in the interval i.e.can be decomposed as infinite sum of Haar wavelets
here, are Haar wavelet coefficients. If is piecewise constant or approximated by piecewise constant during each subinterval then infinite series can be terminate as finite series
3.Method of Solution
Haar functions are piecewise constant and discontinuous in the interval So that differentiation of these functions at their point of discontinuity does not exist but we can integrate the functions many times. By this reason we expand highest fractional order derivative in the fractional integro-differential equation into Haar functions, through the integration we get other derivatives.
Express highest order derivative in Eq.(1) or Eq.(2) in terms of Haar functions for given resolution
Substitute Eq. (13) and (14) in Eq. (1) or Eq. (2).
Discritize the obtained equation at collocation points where is the grid point given by we get algebraic system.
Calculate the Haar coefficients and obtain the Haar solution for unknown function
4. Numerical Experiments
We solved the experimental problems of fractional integro-differential equations, whose exact solution is known. We showed the solutions, numerically and graphically using Matlab software.
Example 1: Let us take the linear fractional Fredholm integro-differential equation15:
with initial conditions Exact solution of the problem is We shown the exact and approximate solutions with J=4 in Figure 1. Absolute errors obtained by HWCM for J=1 are compared with the BPM15 in table 1.
Figure1: Approximate and exact solution for Example 1 with J=4.
Table 1: Absolute errors for Example 1.t HWCM
0.0 0.00E-00 5.81E-04 2.04E-05 2.51E-06
0.1 0.00E-00 9.12E-06 1.85E-05 6.45E-06
0.2 0.00E-00 5.30E-04 6.49E-05 1.95E-05
0.3 0.00E-00 1.00E-3 1.13E-04 3.28E-05
0.4 0.00E-00 1.50E-03 1.60E-04 4.49E-05
0.5 0.00E-00 2.00E-03 2.03E-04 5.55E-05
0.6 0.00E-00 2.40E-03 2.39E-04 6.46E-05
0.7 5.55E-17 2.70E-03 2.68E-04 7.22E-05
0.8 0.00E-00 3.10E-03 2.91E-04 7.81E-05
0.9 1.11E-16 3.40E-03 3.10E-04 8.23E-05
1.0 2.22E-16 3.70E-03 3.28E-04 8.58E-05
Example 2: We considered the linear fractional Volterra integro-differential equation14 :
with initial condition Exact solution of the problem is Approximate and exact solution with J=3 is shown in Figure 2. Absolute errors obtained for J=3 is inserted in Table 2.
Figure2: Approximate and exact solution for Example 2 with J=3.
Table 2: Absolute errors for Example 2.t HWCM
5.ConclusionWe used the Haar wavelet collocation method to solve linear fractional integro-differential equations of Fredholm and Volterra types. Applicability of the method is tested on few examples. We got least absolute errors for small number of collocation points. Comparison of numerical solutions with other method revealed that proposed method has efficiency and accuracy.
Podulbny, Fractional differential equations. Academic Press, 1999.
F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics. A. Carpinteri and F Mainardi (Eds.), Fractals and fractional calculus in continuum mechanics, pp. 291–348. New York: Springer-Verlag, 1997.
A. Haar, Zur theoric der orthogonalen Funktionsysteme, Math. Annal. 69 (1991) 331–371.
K.D. Ruch and J.V. Patrik, Wavelet theory: an elementary approach with applications, John wiely and sons, New Jersey, USA, 2009.
C.F. Chen and C.H. Hsiao, Wavelet approach to optimizing dynamic system, IEEE Proc. Control Theory Appl. 146 (1999) 213–219.
C.F. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEEE Proc. Control Theory Appl. 144 (1997) 87–94.
U. Lepik and H. Hein, Haar wavelets with applications, vol. 4, Springer, 2014.
Siraj-ul-Islam, A. Imran and S. Bozidar, The numerical solution of second order boundary value problems by collocation method with the Haar wavelets, Math. Comp. Model. 52 (2010) 1577–1590.
Fazal-i-Haq and A. Arshed, Numerical solution of fourth order boundary value problems using Haar wavelets, Appl. Math. Sci. 63 (2011) 3131–3146.
Fazal-i-Haq, A. Arshed and H. Iltaf , Solution of sixth-order boundary value problems by collocation method using Haar wavelets, Appl. Math. Sci. 43 (2012) 5729–5735.
A. Padmanabha Reddy, S. H. Manjula and C. Sateesha, A numerical approach to solve eighth order boundary value problems by Haar wavelet collocation method, Math. Model. Eng. Probl. 3 (2016), 108?114.
A. Padmanabha Reddy, S. H. Manjula and C. Sateesha, Solution for nth order mixed Fredholm-Volterra integro-differential equations using Haar wavelets, Aryabhatta. J. Math. Inf., 9(2017), 975-7139.
V. Mishra, H. Kaur, and R. C. Mittal, Haar wavelet algorithm for solving certain differential, integral and integro-differential equations, Int. J. Appl. Math. Mech., 8(2012), 69-82.
R. C. Mittal and R. Nigam, Solution of fractional integro-differential equations by Adomian decomposition method, Int. J. Appl. Math. and Mech. 4(2008), 87-94.
Y. Ordokhani and H. Dehestani, Numerical solution of linear Fredholm-Volterra integro-differential equations of fractional order, Wl. J. Mod. Simu. 12(2016), 204-216.
D. Sh. Mohammed , Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomial, Math. Prob. Eng., (2014) 1-5.
F. Awawdeh, E. A. Rawashdeh and H. M. Jaradat, Analytic solution of fractional integro-di?erential equations, Ann. Uni. Craiova, Math. and Comp. Sci. Series, 38(2011), 1-10.
C. S. Singh, H. Singh, V. K. Singh and O. P. Singh, Fractional order operational matrix methods for fractional singular integro differential equations, applied mathematical modelling , 40, 23-24, December 2016, 10705-10718.