A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse
functions and Bernoulli polynomials are presented. The operational matrices of
integration and product are given. These matrices are then utilized to reduce
the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic
equations. Illustrative examples are included to demonstrate the validity
and applicability of the technique.
1. Introduction
There is considerable literature that discusses approximating the solution of linear and nonlinear Hammerstein integral equations [1–6]. For Fredholm-Hammerstein integral equations, the classical method of successive approximations was introduced in [1]. A variation of the Nystrom method was presented in [2], and a collocation-type method was developed in [3]. In [4], Brunner applied a collocation-type method to nonlinear Volterra-Hammerstein integral equations and integrodifferential equations and discussed its connection with the iterated collocation method. Han [5] introduced and discussed the asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equations. The existence of solutions to nonlinear Hammerstein integral equations was discussed in [6]. Further, Reihani and Abadi [7] and Hsiao [8] applied rationalized Haar functions and hybrid of block-pulse functions and Legendre polynomials, respectively, for solving Fredholm and Volterra integral equations of the second kind. Several authors consider the nonlinear mixed Volterra-Fredholm integral equations of the form
(1)y(t)=f(t)+λ1∫0tκ1(t,s)g1(s,y(s))ds+λ2∫01κ2(t,s)g2(s,y(s))ds,0≤t,s≤1,
where λ1 and λ2 are constants and f(t) and the kernels κ1(t,s) and κ2(t,s) are given functions assumed to have nth derivatives on the interval 0≤x, t≤1. For the case g1(s,y(s))=yp(s) and g2(s,y(s))=yq(s), where p and q are nonnegative integers, Yalçinbaş [9], Bildik and Inc [10], and Hashemizadeh et al. [11] used Taylor series, modified decomposition method, and hybrid of block-pulse functions and Legendre polynomials, respectively, to find the solution. For the case g1(s,y(s))=F1(y(s)) and g2(s,y(s))=F2(y(s)), where F1(y(s)) and F2(y(s)) are given continuous functions which are nonlinear with respect to y(s), Yousefi and Razzaghi [12] applied Legendre wavelets to obtain the solution, and for the general case, where g1(s,y(s)) and g1(s,y(s)) are given continuous functions which are nonlinear with respect to s and y(s), Ordokhani [13] and Marzban et al. [14] applied the rationalized Haar functions and hybrid of block-pulse functions and Lagrange polynomials, respectively, to get the solution.
The available sets of orthogonal functions can be divided into three classes. The first class includes sets of piecewise constant basis functions (PCBF’s) (e.g., block-pulse, Haar, and Walsh). The second class consists of sets of orthogonal polynomials (e.g., Chebyshev, Laguerre, and Legendre). The third class is the set of sine-cosine functions in the Fourier series.
Orthogonal functions have been used when dealing with various problems of the dynamical systems. The approach is based on converting the underlying differential equation into an integral equation through integration, approximating various signals involved in the equation by truncated orthogonal functions, and using the operational matrix of integration P to eliminate the integral operations. The matrix P can be uniquely determined based on the particular orthogonal functions (see [15] and references therein). Among orthogonal polynomials, the shifted Legendre polynomial is computationally more effective [16]. The Bernoulli polynomials and Taylor series are not based on orthogonal functions; nevertheless, they possess the operational matrices of integration. However, since the integration of the cross product of two Taylor series vectors is given in terms of a Hilbert matrix [17], which is known to be ill-conditioned, the applications of Taylor series are limited.
Recently, different types of hybrid functions have been used for solving integral equations and proved to be a mathematical power tool [8, 11, 14].
In the present paper we introduce a new direct computational method to solve nonlinear mixed Volterra-Fredholm integral equations in (1). We approximate the solution not to the equation in its original form but rather to an equivalent equation z1(t)=g1(t,y(t)) and z2(t)=g2(t,y(t)), t∈[0,1]. The functions z1(t) and z2(t) are approximated by hybrid functions with unknown coefficients. These hybrid functions, which consist of block-pulse functions and Bernoulli polynomials together with their operational matrices of integration and product, are given. These matrices are then used to evaluate the coefficients of the hybrid functions for solution of nonlinear mixed Volterra-Fredholm integral equations.
The outline of this paper is as follows. In Section 2, we introduce properties of Bernoulli polynomials and hybrid functions. In Section 3, the numerical method is used to approximate the nonlinear mixed Volterra-Fredholm integral equations, and in Section 4, we report our numerical findings and demonstrate the accuracy of the proposed numerical scheme by considering five numerical examples.
2. Hybrid Functions2.1. Properties of Bernoulli Polynomials
The Bernoulli polynomials of order m are defined in [18] by
(2)βm(t)=∑k=0m(mk)αktm-k,
where αk, k=0,1,…,m are Bernoulli numbers. These numbers are a sequence of signed rational numbers, which arise in the series expansion of trigonometric functions [19] and can be defined by the identity
(3)tet-1=∑n=0∞αntnn!.
The first few Bernoulli numbers are
(4)α0=1,α1=-12,α2=16,α4=-130,
with α2k+1=0, k=1,2,3,….
The first few Bernoulli polynomials are
(5)β0(t)=1,β1(t)=t-12,β2(t)=t2-t+16,β3(t)=t3-32t2+12t.
According to [20], Bernoulli polynomials form a complete basis over the interval [0,1].
3. Hybrid of Block-Pulse Functions and Bernoulli Polynomials
Hybrid functions bnm(t), n=1,2,…,N, m=0,1,…,M are defined on the interval [0,tf] as [21]
(6)bnm(t)={βm(Ntft-n+1),t∈[n-1Ntf,nNtf],0,otherwise,
where n and m are the order of block-pulse functions and Bernoulli polynomials, respectively.
3.1. Function Approximation
Let H=L2[0,1], and assume that {b10(t),b20(t),…,bNM(t)}⊂H is the set of hybrid of block-pulse functions and Bernoulli polynomials, and
(7)Y=span{b10(t),b20(t),…,bN0(t),b11(t),hhhhhhhb21(t),…,bN1(t),…,b1M(t),hhhhhhhb2M(t),…,bNM(t)},
with f being an arbitrary element in H. Since Y is a finite dimensional vector space, f has the unique best approximation out of Y such as f0∈Y, that is,
(8)∀y∈Y,∥f-f0∥≤∥f-y∥.
Since f0∈Y, there exist unique coefficients c10,c20,…,cNM such that
(9)f≃f0=∑m=0M∑n=1Ncnmbnm(t)=CTB(t),
where
(10)BT(t)=[b10(t),b20(t),…,bN0(t),b11(t),hhhhb21(t),…,bN1(t),…,b1M(t),hhhhb2M(t),…,bNM(t)],(11)CT=[c10,c20,…,cN0,c11,c21,…,hhcN1,…,c1M,c2M,…,cNM].
Further, by using (9), we can obtain
(12)D=∫01B(t)BT(t)dt,
where D is a sparse invertible matrix [21].
3.2. Function of Two Variables Approximation
Let g(t,s) be a function of two independent variables defined for t∈[0,1] and s∈[0,1]. Then g can be expanded as
(13)g(t,s)=BT(t)G¯B(s)=∑j=0M∑i=1Nκij(t)bij(s).
Let the matrix G¯ be given by
(14)G¯=[φ1010φ2010…φNM10φ1020φ2020…φNM20⋮⋮⋮⋮φ10NMφ20NM…φNMNM].
From (13), we get
(15)[κ10(t),κ20(t),…,κNM(t)]D=[∫01g(t,s)b10(s)ds,∫01g(t,s)b20(s)ds,…,∫01g(t,s)bNM(s)ds].
Also, κij(t) can be expanded as
(16)κij(t)=∑q=0M∑p=1Nφpqijbpq(t),
where φpqij can be obtained similar to (15).
3.3. Operational Matrices of Integration and Product
The integration of the hybrid functions B(t) defined in (10) is given by
(17)∫0tB(t′)dt′≃PB(t),
where P is the N(M+1)×N(M+1) operational matrix of integration. The product of two hybrid functions with the vector C is given by
(18)B(t)BT(t)C≃C~B(t),
where C~ is the N(M+1)×N(M+1) product operational matrix. The matrices P and C~ are given in [22].
3.4. Approximation Errors
In this section we obtain bounds for the error of best approximation in terms of Sobolev norms. This norm is defined in the interval (a,b) for μ≥0 by
(19)∥f∥Hμ(a,b)=(∑k=0μ∫ab|f(k)(x)|2dx)1/2=(∑k=0μ∥f(k)∥L2(a,b)2)1/2,
where f(k) denotes the kth derivative of f. The symbol |f|Hμ;M(0,1) which is introduced in [23] is defined by
(20)|f|Hμ;M(0,1)=(∑k=min(μ,M+1)μ∥f(k)∥L2(0,1)2)1/2.
Theorem 1.
Suppose that f∈Hμ(0,1) with μ≥0. If PMf=∑m=0Mcmβm is the best approximation of f then
(21)∥f-PMf∥L2(0,1)≤cM-μ|f|Hμ;M(0,1)
and, for 1≤r≤μ,
(22)∥f-PMf∥Hr(0,1)≤cM2r-(1/2)-μ|f|Hμ;M(0,1),
where c depends on μ.
Proof.
Let f∈Hμ(0,1) with μ≥0 and let ∑m=0Mcm′pm be the best approximation of f, which is constructed by using shifted Legendre polynomials pm, m=0,…,M in the interval [0,1]. Then [23]
(23)∥f-∑m=0Mcm′pm∥L2(0,1)≤cM-μ|f|Hμ;M(0,1),
and, for 1≤r≤μ,
(24)∥f-∑m=0Mcm′pm∥Hr(0,1)≤cM2r-(1/2)-μ|f|Hμ;M(0,1).
Since the best approximation is unique [20], we have
(25)∥f-∑m=0Mcm′pm∥L2(0,1)=∥f-PMf∥L2(0,1),∥f-∑m=0Mcm′pm∥Hr(0,1)=∥f-PMf∥Hr(0,1),
and by using (23)–(25) we can obtain (21) and (22).
4. The Numerical Method
We approximate (1) as follows.
Define
(26)z1(t)=g1(t,y(t)),z2(t)=g2(t,y(t)),t∈[0,1].
By using (1) and (26), we have
(27)z1(t)=g1(t,f(t)+λ1∫0tκ1(t,s)z1(s)dshhhh+λ2∫01κ2(t,s)z2(s)ds),z2(t)=g2(t,f(t)+λ1∫0tκ1(t,s)z1(s)dshhhh+λ2∫01κ2(t,s)z2(s)ds).
By using (9) and (13) we get
(28)z1(t)=C1TB(t),z2(t)=C2TB(t),κ1(t,s)=BT(t)K1B(s),κ2(t,s)=BT(t)K2B(s).
By substituting (28) in (27) we have
(29)C1TB(t)=g1(t,f(t)+λ1∫0tBT(t)K1B(s)BT(s)C1dshhhhh+λ2∫01BT(t)K2B(s)BT(s)C2ds),C2TB(t)=g2(t,f(t)+λ1∫0tBT(t)K1B(s)BT(s)C1dshhhhh+λ2∫01BT(t)K2B(s)BT(s)C2ds).
By using (12), (17), and (18) we get
(30)C1TB(t)=g1(t,f(t)+λ1BT(t)K1C~1PB(t)hh+λ2BT(t)K2DC2),C2TB(t)=g2(t,f(t)+λ1BT(t)K1C~1PB(t)hhhh+λ2BT(t)K2DC2).
We collocate (30) at Newton-cotes nodes ti(31)ti=i+12N(M+1),i=0,1,…,2N(M+1)-2.
So we have
(32)C1TB(ti)=g1(ti,f(ti)+λ1BT(ti)K1C~1PB(ti)hhhhh+λ2BT(ti)K2DC2),C2TB(ti)=g2(ti,f(ti)+λ1BT(ti)K1C~1PB(ti)hhhh+λ2BT(ti)K2DC2),
for i=0,1,…,2N(M+1)-2.
Equation (32) can be solved for the unknown C1 and C2. The required approximations to the solution y(t) in (1) are given by
(33)y(t)=f(t)+λ1∫0tκ1(t,s)z1(s)ds+λ2∫01κ2(t,s)z2(s)ds,0≤t≤1.
5. Illustrative Example
In this section, five examples are given to demonstrate the applicability and accuracy of our method. Example 1 is a nonlinear mixed Volterra-Fredholm-Hammerstein integral equation which was considered in [13] by using rationalized Haar functions (RHF) and also solved in [24] by applying Chebyshev approximation. Examples 2 and 3 are the integral equations reformulation of the nonlinear two-point boundary value problems considered in [13, 25] by using RHF and Adomian method, respectively. Although the reformulated integral equations in Examples 2 and 3 are Fredholm-Hammerstein integral equations, the method described here can be used. Examples 1–3 were also solved in [14] by using hybrid of block-pulse functions and Lagrange polynomials. For Examples 1–3, we compare our findings with the numerical results in [14] which have been shown to be superior to those of [13, 24, 25]. Examples 4 and 5 are Fredholm and Volterra integral equations of the second kind, respectively, which were considered in [7] by using RHF which were also solved in [8] by using hybrid of block-pulse functions and Legendre polynomials. For Examples 4 and 5, we compare our findings with the numerical results in [8] which have been shown to be superior to those of [7].
For approximating an arbitrary time function, the advantages of Bernoulli polynomials βm(t), m=0,1,2,…,M where 0≤t≤1, over shifted Legendre polynomials pm(t), m=0,1,2,…,M, where 0≤t≤1, are given in [21] and over Lagrange polynomials Łm(t),m=0,1,2,…,M, where 0≤t≤1, are given below.
Advantages of Bernoulli Polynomials over Lagrange Polynomials. (a) The operational matrix of integration P in Bernoulli polynomials has less error than P for Lagrange polynomials. This is because, for P in β0(t),β1(t),…,βM(t), we ignore the term from the integration of βM(t), while for P in Lagrange polynomials in L0(t),L1(t),…,LM(t) we ignore the terms from the integration of each of Lm(t),m=0,1,…,M.
(b) Bernoulli polynomials have less terms than Lagrange polynomials. For example, β0(t),β1(t),β2(t),β3(t),β4(t),β5(t), and β6(t) have 1,2,3,3,4,4, and 5 terms, respectively, while L0(t),L1(t),…,L6(t) all have 7 terms, and this difference will increase by increasing m. Hence, for approximating an arbitrary function, we use less CPU time by applying Bernoulli polynomials as compared to Lagrange polynomials.
(c) The coefficients of individual terms in Bernoulli polynomials βm(t) are smaller than the coefficient of individual terms in the Lagrange polynomials Lm(t). Since the computational errors in the product are related to the coefficients of individual terms, the computational errors are less by using Bernoulli polynomials.
Example 1.
Consider the integral equation given in [13] by
(34)y(t)=2cost-2+3∫0tsin(t-s)cos2sds+67-6cos1∫01(1-s)cos2t(s+y(s))ds,hhhhhhhhhhhhhhhhhhhhhhhhhh0≤t<1.
The exact solution is y(t)=cost.
For this integral equation we choose N=1 and M=4.
Let
(35)z(t)=y(t)=WTB(t),(36)sin(t-s)cos2s=BT(t)Q1B(s),(1-s)cos2ts=BT(t)Q2B(s),(37)(1-s)cos2t=BT(t)Q3B(s),
where Q1, Q2, and Q3 are obtained similar to (13). By substituting (35)–(37) in (34) we have
(38)WTB(t)=2cos(t)-2+3BT(t)Q1PB(t)+67-6cos1(BT(t)Q2PB(1)+BT(t)Q3D1W),
where D1 can be calculated similar to (12). We collocate (38) at
(39)ti=i+110,i=0,1,…,8.
We get
(40)WTB(ti)=2cos(ti)-2+3BT(ti)Q1PB(ti)+67-6cos1×(BT(ti)Q2PB(1)+BT(ti)Q3D1W),hhhhhhhhhhhhhhhhhhhhhhi=0,1,…,8.
Solving (40) we obtain WT in (35). Table 1 shows the absolute errors of exact and approximate solutions in some points of the interval [0,1] obtained by the present method for N=1 and M=4 together with the method of [14]. In this Table M1 is order of Lagrange polynomials.
t
Method in [14] with N=2 and M1=4
Present method with N=1 and M=4
0.0
3.1021e-5
1.3322e-15
0.2
3.2341e-6
1.3322e-15
0.4
1.9092e-5
1.1102e-15
0.6
1.5029e-5
9.9920e-16
0.8
3.6499e-6
7.7715e-16
1.0
2.4290e-5
2.2204e-16
Example 2.
Consider
(41)y′′(t)-ey(t)=0,0≤t≤1,y(0)=y(1)=0,
which is of great interest in hydrodynamics [26]. This problem has the unique solution [3]
(42)ye(t)=-ln(2)+ln(λ(t)),
where
(43)λ(t)=(ccos((1/2)c)(t-(1/2))).
Here, c is the root of the equation
(44)(ccos(c/4))2=2.
Equation (41) can be reformulated as the integral equation
(45)y(t)=∫01k(t,s)ey(s)ds,0≤t≤1,
where
(46)k(t,s)={-s(1-t),s≤t,-t(1-s),t≤s.
Table 2 represents the computational results of the errors ∥y-ye∥L∞(0,1) for different values of N and M obtained by the present method together with different values of N and M1 in [14]. In Table 2, y and ye denote the approximate and exact solutions, respectively.
Methods
∥y-ye∥L∞(0,1)
Method of [14]
N=2, M1=5,
<10^{−7}
N=3, M1=5,
<10^{−8}
N=4, M1=5,
<10^{−9}
Present method
N=2, M=5,
<10^{−9}
N=3, M=5,
<10^{−11}
N=4, M=5,
<10^{−11}
Example 3.
In this example we consider the mathematical model for an adiabatic tubular chemical reactor discussed in [27, 28], which, in the case of steady state solutions, can be stated as the ordinary differential equation
(47)y′′(t)-λy′(t)+λμ(β-y(t))ey(t)=0,0≤t≤1,y′(0)=λy(0),y′(1)=0.
The problem can be converted into a Hammerstein integral equation of the form [28]
(48)y(t)=∫01k(t,s)G(s,y(s))ds,0≤t≤1,
where k(x,t) is defined by
(49)k(t,s)={1,s≤t,eλ(t-s),t≤s,G(s,y(s))=μ(β-y(s))ey(s).
The existence and uniqueness of the solution for this Hammerstein integral equation with respect to the value of parameters λ, μ, and β are given in [28]. In [25] the Adomian method is used to solve the integral equation (48) for the particular values of the parameters λ=10, μ=0.02, and β=3 which guarantee the existence and uniqueness of the solution for this integral equation [28].
Table 3 gives a comparison between the numerical results of y(t) in some points of the interval [0,1] obtained by the Adomian method given in [25], together with the method in [14] for N=4 and M1=7 and by the present method for N=4 and M=4.
t
Adomian method
Method in [14] with N=4 and M1=7
Present method with N=4 and M=4
0.0
0.006048
0.0060483739
0.0060483739
0.2
0.018192
0.0181929364
0.0181929364
0.4
0.030424
0.0304246702
0.0304246702
0.6
0.042669
0.0426691183
0.0426691183
0.8
0.054371
0.0543716533
0.0543716533
1.0
0.061458
0.0614587374
0.0614587374
Example 4.
Consider the Fredholm integral equation of the second kind [7]
(50)y(t)=e2t+(1/3)+∫01-13e2t-(5/3)sy(s)ds,
with the exact solution y(t)=e2t. Table 4 gives a comparison between the numerical results of y(t) in some points of the interval [0,1] obtained by the method given in [8] for N=2 and M3=11 and by the present method for N=2 and M=4. In this Table M3 is order of Legendre polynomials.
t
Method in [8] with N=2 and M3=11
Present method with N=2 and M=4
Exact solution
0.0
1.00000000000
1.00000000000
1.00000000000
0.0625
1.13314845282
1.13314845305
1.13314845306
0.1250
1.28402541665
1.28402541667
1.28402541668
0.1875
1.45499141433
1.45499141461
1.45499141461
0.2500
1.64872127062
1.64872127078
1.64872127070
0.3125
1.86824595710
1.86824595741
1.86824595743
0.3750
2.11700001648
2.11700001668
2.11700001661
0.4375
2.39887529357
2.39887529393
2.39887529396
0.5000
2.71828182826
2.71828182842
2.71828182845
0.5625
3.08021684845
3.08021684898
3.08021684891
0.6250
3.49034295717
3.49034295742
3.49034295746
0.6875
3.95507672235
3.95507672297
3.95507672292
0.7500
4.48168906993
4.48168907038
4.48168907033
0.8125
5.07841903648
5.07841903712
5.07841903718
0.8750
5.75460267546
5.75460267603
5.75460267600
0.9375
6.52081911946
6.52081912035
6.52081912033
1.0000
7.38905609819
7.38905609894
7.38905609893
Example 5.
Consider the Volterra linear integral equation of the second kind [7]
(51)y(t)=cos(t)-∫0t(t-s)cos(t-s)y(s)ds,
with the exact solution y(t)=(1/3)(2cos3t+1). Table 5 gives a comparison between the numerical results of y(t) in some points of the interval [0,1] obtained by the method given in [8] for N=2 and M3=11 and by the present method for N=2 and M=4.
t
Method in [8] with N=2 and M3=11
Present method with N=2 and M=4
Exact solution
0.0
1.0000000000
1.0000000000
1.0000000000
0.0625
0.9990222526
0.9960975654
0.9960975632
0.1250
0.9844055237
0.9844359392
0.9844359398
0.1875
0.9680159682
0.9651516526
0.9651516562
0.2500
0.9383506092
0.9384704781
0.9384704793
0.3125
0.9074518600
0.9047047767
0.9047047741
0.3750
0.8639866978
0.8642498478
0.8642498461
0.4375
0.8201591812
0.8175793124
0.8175793136
0.5000
0.7647877177
0.7652395632
0.7652395632
0.5625
0.7102158051
0.7078433572
0.7078433525
0.6250
0.6453877501
0.6460626343
0.6460626367
0.6875
0.5827577851
0.5806206938
0.5806207020
0.7500
0.5113646167
0.5122836956
0.5122836972
0.8125
0.4437393774
0.4418516716
0.4418516650
0.8750
0.3689792921
0.3701491788
0.3701491750
0.9375
6.2996549401
0.2980156676
0.2980156705
1.0000
0.2248834557
0.2262956409
0.2262956409
6. Conclusion
In the present work, the hybrid of block-pulse functions with Bernoulli polynomials is used to solve nonlinear mixed Volterra-Fredholm integral equations. The problem has been reduced to a problem of solving a system of algebraic equations. The matrices P and D in (17) and (12) have large numbers of zero elements and are sparse matrices, hence, the present method is very attractive and reduces the computer memory. Illustrative examples are given to demonstrate the validity and applicability of the proposed method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
TricomiF. G.LardyL. J.A variation of Nystroms method for Hammerstein integral equationsKumarS.SloanI. H.A new collocation-type method for Hammerstein integral equationsBrunnerH.Implicitly linear collocation methods for nonlinear Volterra equationsHanG.Asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equationsLiF.LiY.LiangZ.Existence of solutions to nonlinear Hammerstein integral equations and applicationsReihaniM. H.AbadiZ.Rationalized Haar functions method for solving Fredholm and Volterra integral equationsHsiaoC. H.Hybrid function method for solving Fredholm and Volterra integral equations of the second kindYalçinbaşS.Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equationsBildikN.IncM.Modified decomposition method for nonlinear Volterra-Fredholm integral equationsHashemizadehE.MaleknejadK.BasiratB.Hybrid functions approach for the nonlinear Volterra Fredholm integral equationsYousefiS.RazzaghiM.Legendre wavelets method for the nonlinear Volterra-Fredholm integral equationsOrdokhaniY.Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via rationalized Haar functionsMarzbanH. R.TabrizidoozH. R.RazzaghiM.A composite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsRazzaghiM.YousefiS.The Legendre wavelets operational matrix of integrationRazzaghiM.ElnagarG.Linear quadratic optimal control problems via shifted Legendre state parametrizationRazzaghiM.RazzaghiM.Instabilities in the solution of a heat conduction problem using taylor series and alternative approachesCostabileF.DellaccioF.GualtieriM. I.A new approach to Bernoulli polynomialsArfkenG.KreyszigE.MashayekhiS.OrdokhaniY.RazzaghiM.Hybrid functions approach for nonlinear constrained optimal control problemsMashayekhiS.OrdokhaniY.RazzaghiM.Hybrid functions approach for optimal control of systems described by integro-differential equationsCanutoC.HussainiM. Y.QuarteroniA.ZangT. A.BabolianE.FattahzadehF.RabokyE. G.A Chebyshev approximation for solving nonlinear integral equations of Hammerstein typeMadboulyN. M.McGheeD. F.RoachG. F.Adomian's method for Hammerstein integral equations arising from chemical reactor theoryBellmanR. E.KalabaR. E.PooreA. B.A tubular chemical reactor modelMadboulyN.