Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 134272, 8 pages http://dx.doi.org/10.1155/2014/134272
Research Article Bernstein Collocation Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations in the Most General Form AyGegül Akyüz-DaGcJoLlu,1 NeGe EGler Acar,2 and CoGkun Güler3 1
Department of Mathematics, Faculty of Arts & Sciences, Pamukkale University, 20070 Denizli, Turkey Department of Mathematics, Faculty of Arts & Sciences, Mehmet Akif Ersoy University, 15030 Burdur, Turkey 3 Department of Mathematical Engineering, Faculty of Chemistry and Metallurgy, Yildiz Technical University, 34210 Istanbul, Turkey 2
Correspondence should be addressed to Cos¸kun G¨uler;
[email protected] Received 14 May 2014; Accepted 17 July 2014; Published 10 August 2014 Academic Editor: Naseer Shahzad Copyright © 2014 Ays¸eg¨ul Aky¨uz-Das¸cıo˘glu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A collocation method based on the Bernstein polynomials defined on the interval [𝑎, 𝑏] is developed for approximate solutions of the Fredholm-Volterra integrodifferential equation (FVIDE) in the most general form. This method is reduced to linear FVIDE via the collocation points and quasilinearization technique. Some numerical examples are also given to demonstrate the applicability, accuracy, and efficiency of the proposed method.
1. Introduction
under the initial
The quasilinearization method was introduced by Bellman and Kalaba [1] to solve nonlinear ordinary or partial differential equations as a generalization of the Newton-Raphson method. The origin of this method lies in the theory of dynamic programming. In this method, the nonlinear equations are expressed as a sequence of linear equations and these equations are solved recursively. The main advantage of this method is that it converges monotonically and quadratically to the exact solution of the original equations [2]. Therefore, the quasilinearization method is an effective approach for obtaining approximate solutions of nonlinear equations such as differential equations [3–7], functional equations [8, 9], integral equations [10–12], and integrodifferential equations [13–15]. In this paper, we consider the nonlinear FVIDE in the general form 𝑔 (𝑥, 𝑦 (𝑥) , 𝑦 (𝑥) , . . . , 𝑦(𝑚) (𝑥)) 𝑏
= 𝜆 1 ∫ 𝑓 (𝑥, 𝑡, 𝑦 (𝑡) , 𝑦 (𝑡) , . . . , 𝑦(𝑚) (𝑡)) 𝑑𝑡 𝑎
𝑥
(𝑚)
+ 𝜆 2 ∫ V (𝑥, 𝑡, 𝑦 (𝑡) , 𝑦 (𝑡) , . . . , 𝑦 𝑎
(𝑡)) 𝑑𝑡,
(1)
𝑚−1
∑ 𝜏𝑗𝑘 𝑦(𝑘) (𝑐) = 𝜇𝑗 ;
𝑗 = 0, 1, . . . , 𝑚 − 1, 𝑐 ∈ [𝑎, 𝑏] ,
(2)
𝑘=0
or boundary conditions 𝑚−1
∑ [𝛼𝑗𝑘 𝑦(𝑘) (𝑎) + 𝛽𝑗𝑘 𝑦(𝑘) (𝑏)] = 𝛾𝑗 ;
𝑗 = 0, 1, . . . , 𝑚 − 1.
𝑘=0
(3) Here 𝑔 : [𝑎, 𝑏] × R𝑚+1 → R, 𝑓 : [𝑎, 𝑏] × [𝑎, 𝑏] × R𝑚+1 → R, and V : [𝑎, 𝑏] × [𝑎, 𝑏] × R𝑚+1 → R are known functions, 𝛼𝑗𝑘 , 𝛽𝑗𝑘 , 𝜏𝑗𝑘 , 𝜇𝑗 , 𝛾𝑗 , 𝜆 1 , and 𝜆 2 are known constants, and 𝑦(𝑥) is an unknown function. Besides, we approximate to the nonlinear FVIDE (1) by the generalized Bernstein polynomials defined on the interval [𝑎, 𝑏] as 𝑛
𝑦 (𝑥) ≅ 𝐵𝑛 (𝑦; 𝑥) = ∑𝑦 (𝑎 + 𝑖=0
(𝑏 − 𝑎) 𝑖 ) 𝑝𝑖,𝑛 (𝑥) , 𝑛
(4)
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where 𝑝𝑖,𝑛 (𝑥) denotes the generalized Bernstein basis polynomials of the form 𝑝𝑖,𝑛 (𝑥) =
1 𝑛 ( ) (𝑥 − 𝑎)𝑖 (𝑏 − 𝑥)𝑛−𝑖 ; (𝑏 − 𝑎)𝑛 𝑖
𝑖 = 0, 1, . . . , 𝑛. (5)
For convenience, we set 𝑝𝑖,𝑛 (𝑥) = 0, if 𝑖 < 0 or 𝑖 > 𝑛. Bernstein polynomials have many useful properties such as the positivity, continuity, differentiability, integrability, recursion’s relation, symmetry, and unity partition of the basis set over the interval [𝑎, 𝑏]. For more information about the Bernstein polynomials, see [16, 17]. Recently, these polynomials have been used for the numerical solutions of differential equations [4, 18, 19], integral equations [20–24], and integrodifferential equations [25–27]. Now, we give two main theorems for the generalized Berntein polynomials and their basis forms that were proved by Akyuz Dascioglu and Isler [4] as follows.
terms of the generalized Bernstein polynomials. For this, we firstly express this nonlinear equation as a sequence of linear FVIDEs via the quasilinearization technique iteratively. After that, using the collocation points yields the system of linear algebraic equations. This system represents a matrix equation given by the following theorem. Finally, solving this system with the conditions we get the desired approximate solution. Theorem 3. Let 𝑥𝑠 be collocation points defined on the interval [𝑎, 𝑏], and let the functions 𝑔, 𝑓, and V be able to expand by Taylor series with respect to 𝑦(𝑘) ; 𝑘 = 0, 1, . . . , 𝑚. Suppose that nonlinear FVIDE (1) has the generalized Bernstein polynomial solution. Then, the following matrix relation holds: 𝑚
[ ∑ (G𝑟,𝑘 P − 𝜆 1 F𝑟,𝑘 − 𝜆 2 V𝑟,𝑘 ) N𝑘 ] Y𝑟+1 = H𝑟 ; 𝑘=0
(10)
𝑟 = 0, 1, . . . ,
𝑘
Theorem 1. If 𝑦 ∈ 𝐶 [𝑎, 𝑏], for some integer 𝑚 ≥ 0, then lim 𝐵𝑛(𝑘) (𝑦; 𝑥) = 𝑦(𝑘) (𝑥) ;
𝑛→∞
𝑘 = 0, 1, . . . , 𝑚
(6)
converges uniformly. Proof. The above theorem can be easily proved by applying transformation 𝑡 = (𝑥 − 𝑎)/(𝑏 − 𝑎) to the theorem given on the interval [0, 1] by Phillips [28]. Theorem 2 (see [4]). There is a relation between generalized Bernstein basis polynomials matrix and their derivatives in the form (𝑘)
P
𝑘
(𝑥) = P (𝑥) N ;
𝑘 = 1, . . . , 𝑚
(7)
where N is defined in Theorem 2, P = [𝑝𝑖,𝑛 (𝑥𝑠 )], G𝑟,𝑘 = 𝑟,𝑘 diag[𝐺𝑟,𝑘 (𝑥𝑠 )], F𝑟,𝑘 = [𝐹𝑠,𝑖 ], and V𝑟,𝑘 = [𝑉𝑠,𝑖𝑟,𝑘 ] are (𝑛 + 1) × (𝑛 + 1) matrices, and Y𝑟+1 = [𝑦𝑟+1 (𝑎 + (𝑏 − 𝑎)𝑖/𝑛)] and H𝑟 = [ℎ𝑟 (𝑥𝑠 )] are (𝑛 + 1) × 1 matrices for 𝑖, 𝑠 = 0, 1, . . . , 𝑛, such that 𝐺𝑟,𝑘 (𝑥𝑠 ) = 𝑔𝑦𝑟(𝑘) (𝑥𝑠 , 𝑦𝑟 (𝑥𝑠 ) , 𝑦𝑟 (𝑥𝑠 ) , . . . , 𝑦𝑟(𝑚) (𝑥𝑠 )) , 𝑏
𝑟,𝑘 𝐹𝑠,𝑖 = ∫ 𝑓𝑦𝑟(𝑘) (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) 𝑝𝑖,𝑛 (𝑡) 𝑑𝑡, 𝑎
𝑥𝑠
𝑉𝑠,𝑖𝑟,𝑘 = ∫ V𝑦𝑟(𝑘) (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) 𝑝𝑖,𝑛 (𝑡) 𝑑𝑡. 𝑎
(11)
such that P (𝑥) = [𝑝0,𝑛 (𝑥) 𝑝1,𝑛 (𝑥) ⋅ ⋅ ⋅ 𝑝𝑛,𝑛 (𝑥)] .
(8)
Here the elements of (𝑛 + 1) × (𝑛 + 1) matrix N = (𝑑𝑖𝑗 ), 𝑖, 𝑗 = 0, 1, . . . , 𝑛, are defined by 𝑛 − 𝑖; 𝑖𝑓 𝑗 = 𝑖 + 1 { { {2𝑖 − 𝑛; 𝑖𝑓 𝑗 = 𝑖 1 { 𝑑𝑖𝑗 = { −𝑖; 𝑖𝑓 𝑗 = 𝑖 − 1 𝑏−𝑎{ { { 0; 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. {
(9)
In highlight of these theorems, a collocation method based on the generalized Bernstein polynomials, given in Section 2, is developed for the approximate solutions of the nonlinear FVIDE in the most general form (1) via the quasilinearization technique iteratively. In Section 3, some numerical examples are presented for exhibiting the accuracy and applicability of the proposed method. Finally, the paper ends with the conclusions in Section 4.
Here 𝑟 is iteration index, 𝑝𝑖,𝑛 (𝑥) is generalized Bernstein basis polynomials, and ℎ𝑟 (𝑥) is given in the following proof. Proof. Firstly, by applying the quasilinearization method to the nonlinear FVIDE (1), we obtain a sequence of linear FVIDEs: 𝑔 (𝑥, 𝑦𝑟 (𝑥) , 𝑦𝑟 (𝑥) , . . . , 𝑦𝑟(𝑚) (𝑥)) 𝑚
(𝑘) + ∑ (𝑦𝑟+1 (𝑥) − 𝑦𝑟(𝑘) (𝑥)) 𝑔𝑦𝑟(𝑘) 𝑘=0
𝑚
𝑏
𝑘=0
𝑎
(𝑘) = ∑ [𝜆 1 ∫ (𝑦𝑟+1 (𝑡) − 𝑦𝑟(𝑘) (𝑡)) 𝑓𝑦𝑟(𝑘) 𝑑𝑡 𝑥
+𝜆 2 ∫
𝑎
(12) (𝑘) (𝑦𝑟+1
(𝑡) −
𝑦𝑟(𝑘)
(𝑡)) V𝑦𝑟(𝑘) 𝑑𝑡]
𝑏
2. Method of the Solution Our aim is to obtain a numerical solution of the nonlinear FVIDE in the general form (1) under conditions (2) or (3) in
+ 𝜆 1 ∫ 𝑓 (𝑥, 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) 𝑑𝑡 𝑎
𝑥
+ 𝜆 2 ∫ V (𝑥, 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) 𝑑𝑡, 𝑎
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3
where the expressions 𝑔𝑦𝑟(𝑘) , 𝑓𝑦𝑟(𝑘) , and V𝑦𝑟(𝑘) represent partial differentiation of the functions 𝑔, 𝑓, and V with respect to 𝑦𝑟(𝑘) , and these are defined, respectively, as
𝑓𝑦𝑟(𝑘) = V𝑦𝑟(𝑘) =
(𝑥) , . . . , 𝑦𝑟(𝑚)
𝜕𝑓
(𝑡) , . . . , 𝑦𝑟(𝑚)
𝜕𝑦𝑟(𝑘)
𝑥𝑠
(𝑥)) ,
V𝑟,𝑘 (𝑥𝑠 ) = ∫ V𝑦𝑟(𝑘) (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) P (𝑡) 𝑑𝑡. 𝑎
(𝑥, 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑘) 𝜕𝑦𝑟 𝜕V
𝑏
F𝑟,𝑘 (𝑥𝑠 ) = ∫ 𝑓𝑦𝑟(𝑘) (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) P (𝑡) 𝑑𝑡, 𝑎
𝜕𝑔
(𝑥, 𝑦𝑟 (𝑥) , 𝑦𝑟 𝜕𝑦𝑟(𝑘)
𝑔𝑦𝑟(𝑘) =
Here F𝑟,𝑘 (𝑥𝑠 ) and V𝑟,𝑘 (𝑥𝑠 ) are denoted by
(𝑡)) ,
(13)
(𝑥, 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) .
Here 𝑦0 (𝑥) is a reasonable initial approximation of the function 𝑦(𝑥), and 𝑦𝑟 (𝑥) is always considered known and is obtained from previous iteration. The recurrence relation (12) can now be written compactly in the form 𝑚
P(𝑡) and 𝐺𝑟,𝑘 (𝑥) are defined, respectively, in Theorems 2 and 3. For 𝑠 = 0, 1, . . . , 𝑛, the system (18) can be written compactly in the matrix form W𝑟 Y𝑟+1 = H𝑟 ;
−𝜆 2 ∫
𝑥
𝑎
(𝑘) V𝑦𝑟(𝑘) 𝑦𝑟+1
𝑚
where matrices are clearly
(𝑡) 𝑑𝑡] = ℎ𝑟 (𝑥)
denoting ℎ𝑟 (𝑥):
G𝑟,𝑘
ℎ𝑟 (𝑥) = −𝑔 (𝑥, 𝑦𝑟 , 𝑦𝑟 , . . . , 𝑦𝑟(𝑚) ) 𝑏
𝑘=0
𝑎
+ ∑ [𝑔𝑦𝑟(𝑘) 𝑦𝑟(𝑘) (𝑥) − 𝜆 1 ∫ 𝑓𝑦𝑟(𝑘) 𝑦𝑟(𝑘) (𝑡) 𝑑𝑡 𝑥
−𝜆 2 ∫ V𝑦𝑟(𝑘) 𝑦𝑟(𝑘) (𝑡) 𝑑𝑡]
(15)
𝑎
P (𝑥0 ) [P (𝑥1 )] ] [ P = [ . ], [ .. ] [P (𝑥𝑛 )]
V𝑟,𝑘
𝑎
𝑥
+ 𝜆 2 ∫ V (𝑥, 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) 𝑑𝑡. 𝑎
Notice that (14) is a linear FVIDE with variable coefficients, since 𝑦𝑟 (𝑥) is known function of 𝑥. 𝑦𝑟+1 (𝑥) is an unknown function that has the Bernstein polynomial solution; also this function and its derivatives can be expressed by 𝑟 = 0, 1, . . . . (16)
By utilizing Theorem 2 and collocation points, above relation becomes (𝑘) 𝑦𝑟+1
𝑘
(𝑥𝑠 ) = P (𝑥𝑠 ) N Y𝑟+1 ;
𝑘 = 0, 1, . . . , 𝑚.
(17)
Substituting the collocation points and the relation (17) into (14), we obtain a linear algebraic system: 𝑚
∑ [𝐺𝑟,𝑘 (𝑥𝑠 ) P (𝑥𝑠 ) − 𝜆 1 F𝑟,𝑘 (𝑥𝑠 ) − 𝜆 2 V𝑟,𝑘 (𝑥𝑠 )] N𝑘 Y𝑟+1
𝑘=0
0 𝐺𝑟,𝑘 (𝑥0 ) 0 𝐺𝑟,𝑘 (𝑥1 ) .. .. . . 0 0 [
... ...
F𝑟,𝑘
V𝑟,𝑘 (𝑥0 ) [V𝑟,𝑘 (𝑥1 )] ] [ =[ ], .. ] [ . [V𝑟,𝑘 (𝑥𝑛 )]
0 0 .. .
] ] ], ] d . . . 𝐺𝑟,𝑘 (𝑥𝑛 )]
[ [ =[ [
𝑏
+ 𝜆 1 ∫ 𝑓 (𝑥, 𝑡, 𝑦𝑟 (𝑡) , 𝑦𝑟 (𝑡) , . . . , 𝑦𝑟(𝑚) (𝑡)) 𝑑𝑡
(𝑘) 𝑦𝑟+1 (𝑥) ≃ 𝐵𝑛(𝑘) (𝑦𝑟+1 ; 𝑥) = P(𝑘) (𝑥) Y𝑟+1 ;
(21)
𝑘=0
(14)
𝑚
(20)
W𝑟 = ∑ (G𝑟,𝑘 P − 𝜆 1 F𝑟,𝑘 − 𝜆 2 V𝑟,𝑘 ) N𝑘 ,
(𝑘) (𝑘) ∑ [𝑔𝑦𝑟(𝑘) 𝑦𝑟+1 (𝑥) − 𝜆 1 ∫ 𝑓𝑦𝑟(𝑘) 𝑦𝑟+1 (𝑡) 𝑑𝑡 𝑎
𝑟 = 0, 1, . . .
so that
𝑏
𝑘=0
(19)
F𝑟,𝑘 (𝑥0 ) [F𝑟,𝑘 (𝑥1 )] ] [ =[ ], .. ] [ . (𝑥 ) F [ 𝑟,𝑘 𝑛 ] ℎ𝑟 (𝑥0 ) [ℎ𝑟 (𝑥1 )] ] [ H𝑟 = [ . ] . [ .. ] [ℎ𝑟 (𝑥𝑛 )]
Hence, the proof is completed. Corollary 4. For 𝜆 1 = 𝜆 2 = 0, the nonlinear FVIDE (1) is reduced to the 𝑚th order nonlinear differential equation 𝑔 (𝑥, 𝑦 (𝑥) , 𝑦 (𝑥) , . . . , 𝑦(𝑚) (𝑥)) = 0,
(18)
(23)
and by utilizing Theorem 3, this equation can be written as 𝑚
̂𝑟; ∑ G𝑟,𝑘 PN𝑘 Y𝑟+1 = H
𝑟 = 0, 1, . . . .
(24)
𝑘=0
Here the matrices P, N, and G𝑟,𝑘 are defined as above, and ̂ 𝑟 = [̂ℎ𝑟 (𝑥𝑠 )] are denoted by elements of the matrix H 𝑚
̂ℎ (𝑥 ) = ∑ 𝐺 (𝑥 ) 𝑦(𝑘) (𝑥 ) 𝑟 𝑠 𝑟,𝑘 𝑠 𝑠 𝑟 𝑘=0
= ℎ𝑟 (𝑥𝑠 ) .
(22)
−
𝑔 (𝑥𝑠 , 𝑦𝑟 (𝑥𝑠 ) , 𝑦𝑟
(25) (𝑥𝑠 ) , . . . , 𝑦𝑟(𝑚)
(𝑥𝑠 )) .
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Corollary 5. For 𝑚 = 0, the nonlinear FVIDE (1) is reduced to the nonlinear Fredholm-Volterra integral equation in the form 𝑏
𝑥
𝑎
𝑎
𝑔 (𝑥, 𝑦 (𝑥)) = 𝜆 1 ∫ 𝑓 (𝑥, 𝑡, 𝑦 (𝑡)) 𝑑𝑡 + 𝜆 2 ∫ V (𝑥, 𝑡, 𝑦 (𝑡)) 𝑑𝑡 (26)
such that 𝑦(0) (𝑡) = 𝑦(𝑡). From Theorem 3, this equation has the iteration matrix form as [G𝑟 − 𝜆 1 F𝑟 − 𝜆 2 V𝑟 ] Y𝑟+1 = H𝑟 ;
𝑟 = 0, 1, . . . .
(27)
Here G𝑟 = diag[𝐺𝑟 (𝑥𝑠 )], F𝑟 = [𝐹𝑟,𝑠,𝑖 ], V𝑟 = [𝑉𝑟,𝑠,𝑖 ], H𝑟 = [ℎ𝑟 (𝑥𝑠 )], and elements of these matrices are denoted as follows: 𝐺𝑟 (𝑥𝑠 ) = 𝑔𝑦𝑟 (𝑥𝑠 , 𝑦𝑟 (𝑥𝑠 )) ,
where the matrices are 𝑚−1
I𝑗 = ∑ 𝜏𝑗𝑘 P (𝑐) N𝑘 , 𝑘=0
(30)
𝑚−1
𝑘
𝑘
B𝑗 = ∑ [𝛼𝑗𝑘 P (𝑎) N + 𝛽𝑗𝑘 P (𝑏) N ] . 𝑘=0
Besides, (29) can be denoted by the augmented matrices [I𝑗 ; 𝜇𝑗 ] and [B𝑗 ; 𝛾𝑗 ]. Step 4. To obtain the solution of nonlinear FVIDE (1) under the given conditions, we insert the elements of the row matrices [I𝑗 ; 𝜇𝑗 ] or [B𝑗 ; 𝛾𝑗 ] to the end of the augmented matrix [W𝑟 ; H𝑟 ]. In this way, we have the new augmented matrix ̃ 𝑟 ], that is, (𝑛 + 𝑚 + 1) × (𝑛 + 1) rectangular matrix. ̃ 𝑟; H [W ̃ 𝑟 ) = rank[W ̃ 𝑟; H ̃ 𝑟 ] = 𝑛 + 1, then unknown Step 5. If rank(W coefficients 𝑦𝑟+1 are uniquely determined for each iteration 𝑟. This kind of systems can be solved by the Gauss Elimination, Generalized Inverse, and QR factorization methods.
𝑏
𝐹𝑟,𝑠,𝑖 = ∫ 𝑓𝑦𝑟 (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡)) 𝑝𝑖,𝑛 (𝑡) 𝑑𝑡, 𝑎
𝑥𝑠
𝑉𝑟,𝑠,𝑖 = ∫ V𝑦𝑟 (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡)) 𝑝𝑖,𝑛 (𝑡) 𝑑𝑡, 𝑎
ℎ𝑟 (𝑥𝑠 )
3. Numerical Results
= 𝑔𝑦𝑟 (𝑥𝑠 , 𝑦𝑟 (𝑥𝑠 )) 𝑦𝑟 (𝑥𝑠 ) − 𝑔 (𝑥𝑠 , 𝑦𝑟 (𝑥𝑠 ))
Four numerical examples are given to illustrate the applicability, accuracy, and efficiency of the proposed method. All results are computed by using an algorithm written in Matlab 7.1. Besides, in the tables, the absolute and L2 -norm errors are computed numerically on the collocation points 𝑥𝑠 = 𝑎 + (𝑏 − 𝑎)𝑠/𝑛; 𝑠 = 0, 1, . . . , 𝑛, by the folowing formulas:
𝑏
− 𝜆 1 ∫ [𝑓 (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡)) − 𝑓𝑦𝑟 (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡)) 𝑦𝑟 (𝑡)] 𝑑𝑡 𝑎
𝑥𝑠
− 𝜆 2 ∫ [V (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡)) − V𝑦𝑟 (𝑥𝑠 , 𝑡, 𝑦𝑟 (𝑡)) 𝑦𝑟 (𝑡)] 𝑑𝑡. 𝑎
(28)
𝐸abs = 𝑦 (𝑥𝑠 ) − 𝐵𝑛 (𝑦𝑟 ; 𝑥𝑠 ) , 𝑥𝑠
(31)
Now we can solve the nonlinear FVDIE (1) under the initial (2) or boundary (3) conditions as follows.
𝐸2 = √ ∫ (𝑦 (𝑥) − 𝐵𝑛 (𝑦𝑟 ; 𝑥)) 𝑑𝑥
Step 1. Firstly, we use Theorem 3 for the nonlinear FVDIE (1) and determine the matrices in (10). This matrix equation is a system of linear algebraic equations with 𝑛-unknown coefficients 𝑦𝑟+1 (𝑎 + (𝑏 − 𝑎)𝑖/𝑛). Let the augmented matrix corresponding equation (10) be denoted by [W𝑟 ; H𝑟 ].
such that 𝐵𝑛 (𝑦𝑟 ; 𝑥) is a Bernstein approximation for the 𝑟th iteration function 𝑦𝑟 and 𝑦(𝑥) is an exact solution.
Step 2. We need to choose the first iteration function 𝑦0 (𝑥) for calculating the W𝑟 and H𝑟 . Notice that this function can be obtained in a variety of ways. For instance, it can be obtained from the physical situation for engineering problems. However, a very rough choice for the first iteration function such as initial value is enough for the procedure to converge. We can also consider that the first iteration function as the highest degree polynomial satisfied the given conditions (2) or (3). Step 3. From expression (17), initial (2) and boundary (3) conditions can be written in the matrix forms, respectively, I𝑗 Y𝑟+1 = 𝜇𝑗 , B𝑗 Y𝑟+1 = 𝛾𝑗 ,
0
Example 1. Consider the nonlinear Volterra integrodifferential equation 𝑦 (𝑥) = 2𝑥 −
𝑥 1 sin (𝑥4 ) + ∫ 𝑥2 𝑡 cos (𝑥2 𝑦 (𝑡)) 𝑑𝑡; 2 0
0 < 𝑥,
(32)
𝑡
