Applied Mathematical Sciences, Vol. 6, 2012, no. 32, 1563 - 1569
Properties of BPFs for Approximating the Solution of Nonlinear Fredholm Integro Differential Equation Ahmad Shahsavaran1 and Akbar Shahsavaran Islamic Azad University, Boroujerd Branch, Boroujerd, Iran Abstract We are interested in finding approximate solutions to nonlinear integro differential equation. This paper concentrates on approximating the exact solution by truncated series of Block Pulse Functions(BPFs) which gives desired accuracy in such problems. This theory supported by some numerical examples that shows efficiently and validity of the technique.
Keywords: Nonlinear integro differential equation; Block-Pulse Function; Collocation points
1. Introduction Modeling and analysis of physical phenomena in applied sciences often generates nonlinear mathematical problems. Nonlinearity may be an inner feature of the model, i.e., evolution equations with nonlinear terms, or of the problem, i.e., nonlinear boundary conditions. The interplay between applied sciences and mathematics then leads to the development of initial and/or boundary value problems for nonlinear partial differential or integral or integro differential equations modeling real physical systems. The theory and application of integral and integro differential equations is an important subject within applied mathematics. Integral and integro differential equations are used as mathematical models for many and varied physical situations, and also occur as reformulations of other mathematical problems. Since many physical problems are modeled by integral and integro differential equations, the numerical solutions of such equations have been highly studied by many authors. In recent years, numerous works have been focusing on the development of more advanced and efficient methods for integro differential equations such as Haar 1
Corresponding author. E-mail addresses:
[email protected] (Ahmad Shahsavaran),
[email protected] (Akbar Shahsavaran)
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Ahmad Shahsavaran and Akbar Shahsavaran
wavelets, homotopy perturbation method, lagrange functions, Taylor polynomials, Chebyshev polynomials, sine-cosine wavelets, Adomian decomposition method and so on [1-7]. In the present article, we are concerned with the application of BPFs to the numerical solution of a nonlinear Fredholm integro differential equation as 1 u (x) = 0 k(x, t)ψ(t, u(t))dt + f (x), 0 ≤ x ≤ 1, (1) u(0) = u0 , where, k, ψ ∈ L2 [0, 1)2 , f ∈ L2 [0, 1) are known functions. The method consists of expanding the solution by BPFs with unknown coefficients. The properties of BPFs together with the collocation method are then utilized to evaluate the unknown coefficients and find an approximate solution to Eq. (1). 2. BPFs and function approximation We define a k-set of BPFs as 1, i−1 ≤ t < ki , for all i = 1, 2, . . . , k k Bi (t) = 0, elsewhere the functions Br (t) are disjoint and orthogonal. That is, 0, i = j Bj (t)Bi (t) = Bi (t), i = j 0, i = j < Bi (t), Bj (t) > = 1 , i=j k
(2)
(3) (4)
Now we approximate z(t) as z(t) zk (t) =
k
zi Bi (t),
(5)
i=1
where, zi = k < z(t), Bi (t) >= k the matrix form
1 0
z(t)Bi (t)dt. Also, Eq.(5) can be restate in
zk (t) = zt B(t),
(6)
where, z = [z1 , z2 , . . . , zk ]t , B(t) = [B1 (t), B2 (t), . . . , Bk (t)]t and k is a power of 2. Also, K(x, t) ∈ L2 [0, 1)2 may be approximated in the matrix form as K(x, t) Bt (x)KB(t),
(7)
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Fredholm integro differential equation
11 where K = [Kij ]1≤i,j≤k and Kij = k 2 0 0 K(x, t)Bi (x)Bj (t)dxdt. Also the integration of BPFs is expandable into BPFs series: t B(x)dx = PB(t), (8) 0
the k-square matrix P is called the operational matrix of integration of the transform and is defined as follows:
Pk×k
⎛ 1 ⎜0 1 ⎜ ⎜0 = ⎜ 2k ⎜ .. ⎝. 0
2 1 0 .. .
2 2 1 .. .
... ... ... .. .
0 0 ...
⎞ 2 2⎟ ⎟ 2⎟ ⎟ .. ⎟ .⎠ 1
(9)
therefore, if we set
1
A=
B(t)Bt (t)dt,
0
by using disjoint property of BPFs we obtain for i = j, Aij =
1 0
Bi (t)Bj (t)dt = 0,
for i = j, Aij =
1
Bi2 (t)dt
0 1
= 0
= =
Bi (t)dt i k
i−1 k
1dt
1 , k
hence, A=
1 I, k
(10)
where I is the identity matrix of order k. Now we define z(t) = ψ(t, u(t)),
(11)
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Ahmad Shahsavaran and Akbar Shahsavaran
since u(t) =
t 0
u (x)dx + u0 ,
(12)
by using (1) and (11)-(12) we obtain
t z(t) = ψ t, u (x)dx + u0 0
t 1 k(x, t)z(t)dt + f (x) dx + u0 = ψ t, 0 0
t 1 t = ψ t, k(x, t)z(t)dtdx + f (x)dx + u0 . 0
0
(13)
0
By approximating functions z(t), K(x, t) and f (t), as before, in the matrix form we have z(t) zk (t) = Bt (t)z, f (t) Bt (t)f, K(x, t) Bt (x)KB(t),
(14) (15) (16)
by substituting the approximations (14)-(16) into (13) we obtain t 1
t t t t B (x)KB(t)B (t)zdtdx + B (x)fdx + u0 B (t)z = ψ t, 0 0 0 t 1
t
t t t = ψ t, B (x)K B(t)B (t)dt dxz + B (x)dx f + u0 . t
0
0
0
(17) Substituting (10) into (17) and using (8) implies
t
t 1 t t B (x)dx Kz + B (x)dx f + u0 B (t)z = ψ t, k 0 0
1 t t t t = ψ t, B (t)P Kz + B (t)P f + u0 , k t
(18)
collocating (18) at the points tj = kj , j = 1, 2, ..., k and using the fact that B(tj ) = ej , where ej is the j-th column of the identity matrix of order k, gives
1 t t t t zj = ψ tj , ej P Kz + ej P f + u0 . k
(19)
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Fredholm integro differential equation
Equation (19) gives k nonlinear equations which can be solved for the elements zj , j = 1, 2, ..., k using Newton’s iterative method. So, z(t) can be approximated by zk (t) using (5) and u (x) may be evaluated as following 1 u (x) = k(x, t)ψ(t, u(t))dt + f (x) 0 1 k(x, t)zk (t)dt + f (x), 0
therefore, we get desired approximation for u(t) by t u(t) = u (x)dx + u0 . 0
3. Numerical Examples Example 1. 1 u (x) = 54 − 13 x2 + 0 (x2 − t)(u(t))2 dt, u(0) = 0, with the exact solution u(t) = t. Example 2.
u(x) = 1 − 13 x3 + u(0) = 0,
1 0
x3 (u(t))2 dt,
with the exact solution u(t) = t. t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Table 1: numerical results for example 1 approximate for k=16 approximate for k=32 exact .0 .0 .08810 .08885 .17654 .17802 .26568 .26784 .35585 .35861 .44741 .45067 .54070 .54433 .63606 .63991 .73384 .73773 .83439 .83811
solution .0 .1 .2 .3 .4 .5 .6 .7 .8 .9
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Ahmad Shahsavaran and Akbar Shahsavaran
t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Table 2: numerical results for example 2 approximate for k=16 approximate for k=32 exact .0 .0 .10001 .10000 .20016 .20004 .30084 .30021 .40267 .40067 .50653 .50165 .61354 .60344 .72509 .70637 .84280 .81087 .96856 .91741
solution .0 .1 .2 .3 .4 .5 .6 .7 .8 .9
Conclusion In present paper, BPFs together with the collocation points are applied to solve the nonlinear Fredholm integro differential equations. The benefit of this method are low cost of setting up the equations due to properties of BPFs mentioned in section 2 and very cheap as computational cost. In addition, the nonlinear system of algebraic equations (19) is sparse. Moreover, the method may be more accurate by using larger k. References [1] A. Shahsavaran, Numerical solution of linear Volterra and Fredholm integro differential equations using Haar wavelets, Mathematics Scientific Journal, 12 (2010) 85-96. [2] J. Biazar, H. Ghazvini, M. Eslami, He’s homotopy perturbation method for systems of integro-differential equations, Chaos , Solitons and Fractals, 39 (2009) 1253-1258. [3] M. T. Rashed, Lagrange interpolation to compute the numerical solutions of differential, integral and integro differential equations, Applied Mathematics and computation, 151 (2004) 869-878. [4] A. A. Dascioghlu, M. Sezer, Chebyshev polynomial solutions of systems of higher order linear Fredholm-Volterra integro differential equations, Journal of the Franklin Institute, 342 (2005) 688-701. [5] M. T. Kajani, M. Ghasemi, E. Babolian, Numerical solution of linear integro differential equation by using sine-cosine wavelets, Applied Mathematics and Computation, 180 (2006) 569-574. [6] A. R. Vahidi, E. Babolian, G. A. Cordshooli, Z. Azimzadeh, Numerical solution of Fredholm integro differential equation by Adomian decomposition method, Int. Journal of Math. Analysis, 36 (2009) 1769-1773.
Fredholm integro differential equation
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[7] S. Yalcinbas, M. Sezer, The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000) 291-308. Received: October, 2011
