Properties of BPFs for Approximating the Solution of Nonlinear Fredholm Integro Differential Equation

Transcription

1 Applied Mathematical Sciences, Vol. 6, 212, no. 32, Properties of BPFs for Approximating the Solution of Nonlinear Fredholm Integro Differential Equation Ahmad Shahsavaran 1 and Abar 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 Bloc Pulse FunctionsBPFs) 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; Bloc-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 wors have been focusing on the development of more advanced and efficient methods for integro differential equations such as Haar 1 Corresponding author. addresses: a.shahsavaran@iaub.ac.ir Ahmad Shahsavaran), shahsavaran@scitco.ir Abar Shahsavaran)

2 1564 Ahmad Shahsavaran and Abar 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 u x) = x, t)ψt, ut))dt + fx), x 1, 1) u) = u, where,, ψ L 2 [, 1) 2,f L 2 [, 1) are nown functions. The method consists of expanding the solution by BPFs with unnown coefficients. The properties of BPFs together with the collocation method are then utilized to evaluate the unnown coefficients and find an approximate solution to Eq. 1). 2. BPFs and function approximation We define a -set of BPFs as i 1 1, t< i, for all i =1, 2,..., B i t) =, elsewhere 2) the functions B r t) are disjoint and orthogonal. That is,, i j B j t)b i t) = 3) B i t), i = j, i j <B i t),b j t) > = 1, i = j 4) Now we approximate zt) as zt) z t) = z i B i t), 5) i=1 where, z i = <zt),b i t) >= zt)b it)dt. Also, Eq.5) can be restate in the matrix form z t) =z t Bt), 6) where, z =[z 1,z 2,...,z ] t, Bt) =[B 1 t),b 2 t),...,b t)] t and is a power of 2. Also, Kx, t) L 2 [, 1) 2 may be approximated in the matrix form as Kx, t) B t x)kbt), 7)

3 Fredholm integro differential equation 1565 where K =[K ij ] 1 i,j and K ij = 2 1 Kx, t)b ix)b j t)dxdt. Also the integration of BPFs is expandable into BPFs series: Bx)dx = PBt), 8) the -square matrix P is called the operational matrix of integration of the transform and is defined as follows: P = ) therefore, if we set A = Bt)B t t)dt, by using disjoint property of BPFs we obtain for i j, A ij = B it)b j t)dt =, for i = j, hence, A ij = = = i = 1, i 1 B 2 i t)dt B i t)dt 1dt A = 1 I, 1) where I is the identity matrix of order. Now we define zt) =ψt, ut)), 11)

4 1566 Ahmad Shahsavaran and Abar Shahsavaran since ut) = by using 1) and 11)-12) we obtain zt) =ψ t, = ψ t, = ψ t, ) u x)dx + u u x)dx + u, 12) ) ) x, t)zt)dt + fx) dx + u x, t)zt)dtdx + fx)dx + u ). 13) By approximating functions zt), Kx, t) and ft), as before, in the matrix form we have zt) z t) =B t t)z, 14) ft) B t t)f, 15) Kx, t) B t x)kbt), 16) by substituting the approximations 14)-16) into 13) we obtain ) B t t)z = ψ t, B t x)kbt)b t t)zdtdx + B t x)fdx + u ) ) ) = ψ t, B t x)k Bt)B t t)dt dxz + B t x)dx f + u. 17) Substituting 1) into 17) and using 8) implies B t t)z = ψ t, 1 ) ) ) B t x)dx Kz + B t x)dx f + u = ψ t, 1 ) Bt t)p t Kz + B t t)p t f + u, 18) collocating 18) at the points t j = j, j =1, 2,..., and using the fact that Bt j )=e j, where e j is the j-th column of the identity matrix of order, gives z j = ψ t j, 1 ) etj Pt Kz + e tj Pt f + u. 19)

5 Fredholm integro differential equation 1567 Equation 19) gives nonlinear equations which can be solved for the elements z j,j = 1, 2,..., using Newton s iterative method. So, zt) can be approximated by z t) using 5) and u x) may be evaluated as following u x) = x, t)ψt, ut))dt + fx) x, t)z t)dt + fx), therefore, we get desired approximation for ut) by ut) = 3. Numerical Examples u x)dx + u. Example 1. u x) = x2 + x2 t)ut)) 2 dt, u) =, with the exact solution ut) = t. Example 2. u x) =1 1 3 x3 + x3 ut)) 2 dt, u) =, with the exact solution ut) = t. Table 1: numerical results for example 1 t approximate for =16 approximate for =32 exact solution

6 1568 Ahmad Shahsavaran and Abar Shahsavaran Table 2: numerical results for example 2 t approximate for =16 approximate for =32 exact solution 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. References [1] A. Shahsavaran, Numerical solution of linear Volterra and Fredholm integro differential equations using Haar wavelets, Mathematics Scientific Journal, 12 21) [2] J. Biazar, H. Ghazvini, M. Eslami, He s homotopy perturbation method for systems of integro-differential equations, Chaos, Solitons and Fractals, 39 29) [3] M. T. Rashed, Lagrange interpolation to compute the numerical solutions of differential, integral and integro differential equations, Applied Mathematics and computation, ) [4] A. A. Dascioghlu, M. Sezer, Chebyshev polynomial solutions of systems of higher order linear Fredholm-Volterra integro differential equations, Journal of the Franlin Institute, ) [5] M. T. Kajani, M. Ghasemi, E. Babolian, Numerical solution of linear integro differential equation by using sine-cosine wavelets, Applied Mathematics and Computation, 18 26) [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 29)

7 Fredholm integro differential equation 1569 [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 ) Received: October, 211