This paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinear multipoint boundary value problems. The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth-kind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. The obtained numerical results are comparing favorably
with the analytical known solutions.
1. Introduction
Spectral methods are one of the principal methods of discretization for the numerical solution of differential equations. The main advantage of these methods lies in their accuracy for a given number of unknowns (see, e.g., [1–4]). For smooth problems in simple geometries, they offer exponential rates of convergence/spectral accuracy. In contrast, finite difference and finite-element methods yield only algebraic convergence rates. The three most widely used spectral versions are the Galerkin, collocation, and tau methods. Collocation methods [5, 6] have become increasingly popular for solving differential equations, also they are very useful in providing highly accurate solutions to nonlinear differential equations.
Many practical problems arising in numerous branches of science and engineering require solving high even-order and high odd-order boundary value problems. Legendre polynomials have been previously used for obtaining numerical spectral solutions for handling some of these kinds of problems (see, e.g., [7, 8]). In [9], the author has constructed some algorithms by selecting suitable combinations of Legendre polynomials for solving the differentiated forms of high-odd-order boundary value problems with the aid of Petrov-Galerkin method, while in the two papers [10, 11], the authors handled third- and fifth-order differential equations using Jacobi tau and Jacobi collocation methods.
Multipoint boundary value problems (BVPs) arise in a variety of applied mathematics and physics. For instance, the vibrations of a guy wire of uniform cross-section composed of N parts of different densities can be set up as a multipoint BVP, as in [12]; also, many problems in the theory of elastic stability can be handled by the method of multipoint problems [13]. The existence and multiplicity of solutions of multipoint boundary value problems have been studied by many authors; see [14–17] and the references therein. For two-point BVPs, there are many solution methods such as orthonormalization, invariant imbedding algorithms, finite difference, and collocation methods (see, [18–20]). However, there seems to be little discussion about numerical solutions of multipoint boundary value problems.
Second-order multipoint boundary value problems (BVP) arise in the mathematical modeling of deflection of cantilever beams under concentrated load [21, 22], deformation of beams and plate deflection theory [23], obstacle problems [24], Troesch's problem relating to the confinement of a plasma column by radiation pressure [25, 26], temperature distribution of the radiation fin of trapezoidal profile [21, 27], and a number of other engineering applications. Many authors have used numerical and approximate methods to solve second-order BVPs. The details about the related numerical methods can be found in a large number of papers (see, for instance, [21, 23, 24, 28]). The Walsh wavelets and the semiorthogonal B-spline wavelets are used in [23, 29] to construct some numerical algorithms for the solution of second-order BVPs with Dirichlet and Neumann boundary conditions. Na [21] has found the numerical solution of second-, third-, and fourth-order BVPs by converting them into initial value problems and then applying a class of methods like nonlinear shooting, method of reduced physical parameters, method of invariant imbedding, and so forth. The presented approach in this paper can be applied to both BVPs and IVPs with a slight modification, but without the transformation of BVPs into IVPs or vice versa.
Wavelets theory is a relatively new and an emerging area in mathematical research. It has been applied to a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for wave form representation and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms, (see [30, 31]).
The application of Legendre wavelets for solving differential and integral equations is thoroughly considered by many authors (see, for instance, [32, 33]). Also, Chebyshev wavelets are used for solving some fractional and integral equations (see, [34, 35]).
Chebyshev polynomials have become increasingly crucial in numerical analysis, from both theoretical and practical points of view. It is well known that there are four kinds of Chebyshev polynomials, and all of them are special cases of the more widest class of Jacobi polynomials. The first and second kinds are special cases of the symmetric Jacobi polynomials (i.e., ultraspherical polynomials), while the third and fourth kinds are special cases of the nonsymmetric Jacobi polynomials. In the literature, there is a great concentration on the first and second kinds of Chebyshev polynomials Tn(x) and Un(x) and their various uses in numerous applications, (see, for instance, [36]). However, there are few articles that concentrate on the other two types of Chebyshev polynomials, namely, third and fourth kinds Vn(x) and Wn(x), either from theoretical or practical point of view and their uses in various applications (see, e.g., [37]). This motivates our interest in such polynomials. We therefore intend in this work to use them in a marvelous application of multipoint BVPs arising in physics.
There are several advantages of using Chebyshev wavelets approximations based on collocation spectral method. First, unlike most numerical techniques, it is now well established that they are characterized by exponentially decaying errors. Second, approximation by wavelets handles singularities in the problem. The effect of any such singularities will appear in some form in any scheme of the numerical solution, and it is well known that other numerical methods do not perform well near singularities. Finally, due to their rapid convergence, Chebyshev wavelets collocation method does not suffer from the common instability problems associated with other numerical methods.
The main aim of this paper is to develop two new spectral algorithms for solving second-order multipoint BVPs based on shifted third- and fourth-kind Chebyshev wavelets. The method reduces the differential equation with its boundary conditions to a system of algebraic equations in the unknown expansion coefficients. Large systems of algebraic equations may lead to greater computational complexity and large storage requirements. However the third- and fourth-kind Chebyshev wavelets collocation method reduces drastically the computational complexity of solving the resulting algebraic system.
The structure of the paper is as follows. In Section 2, we give some relevant properties of Chebyshev polynomials of third and fourth kinds and their shifted ones. In Section 3, the third- and fourth-kind Chebyshev wavelets are constructed. Also, in this section, we ascertain the convergence of the Chebyshev wavelets series expansion. Two new shifted Chebyshev wavelets collocation methods for solving second-order linear and nonlinear multipoint boundary value problems are implemented and presented in Section 4. In Section 5, some numerical examples are presented to show the efficiency and the applicability of the presented algorithms. Some concluding remarks are given in Section 6.
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The Chebyshev polynomials Vk(x) and Wk(x) of third and fourth kinds are polynomials of degree k in x defined, respectively, by (see [38])
(1)Vk(x)=cos(k+(1/2))θcos(θ/2),Wk(x)=sin(k+(1/2))θsin(θ/2),
where x=cosθ; also they can be obtained explicitly as two particular cases of Jacobi polynomials Pk(α,β)(x) for the two nonsymmetric cases correspond to β=-α=±1/2. Explicitly, we have
(2)Vk(x)=(2kk!)2(2k)!Pk(-1/2,1/2)(x),Wk(x)=(2kk!)2(2k)!Pk(1/2,-1/2)(x).
It is readily seen that
(3)Wk(x)=(-1)kVk(-x).
Hence, it is sufficient to establish properties and relations for Vn(x) and then deduce their corresponding properties and relations for Wn(x) (by replacing x by -x).
The polynomials Vn(x) and Wn(x) are orthogonal on (-1,1); that is,
(4)∫-11ω1(x)Vk(x)Vj(x)dx=∫-11ω2(x)Wk(x)Wj(x)dx={π,k=j,0,k≠j,
where
(5)ω1(x)=1+x1-x,ω2(x)=1-x1+x,
and they may be generated by using the two recurrence relations
(6)Vk(x)=2xVk-1(x)-Vk-2(x),k=2,3,…,
with the initial values
(7)V0(x)=1,V1(x)=2x-1,(8)Wk(x)=2xWk-1(x)-Wk-2(x),k=2,3,…,
with the initial values
(9)W0(x)=1,W1(x)=2x+1.
The shifted Chebyshev polynomials of third and fourth kinds are defined on [0,1], respectively, as
(10)Vn*(x)=Vn(2x-1),Wn*(x)=Wn(2x-1).
All results of Chebyshev polynomials of third and fourth kinds can be easily transformed to give the corresponding results for their shifted ones.
The orthogonality relations of Vn*(t) and Wn*(t) on [0,1] are given by
(11)∫01w1*Vm*(t)Vn*(t)dt=∫01w2*Wm*(t)Wn*(t)dt={π2,m=n,0,m≠n,
where
(12)ω1*=t1-t,ω2*=1-tt.
3. Shifted Third- and Fourth-Kind Chebyshev Wavelets
Wavelets constitute of a family of functions constructed from dilation and translation of single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, then we have the following family of continuous wavelets:
(13)ψa,b(t)=|a|-1/2ψ(t-ba),a,b∈ℝ,a≠0.
Each of the third- and fourth-kind Chebyshev wavelets ψnm(t)=ψ(k,n,m,t) has four arguments: k,n∈ℕ, m is the order of the polynomial Vm*(t) or Wm*(t), and t is the normalized time. They are defined explicitly on the interval [0,1] as
(14)ψnm(t)={2(k+1)/2πVm*(2kt-n),resp.,2(k+1)/2πWm*(2kt-n),t∈[n2k,n+12k],0⩽m⩽M,0000000000000⩽n⩽2k-1,0,otherwise.
3.1. Function Approximation
A function f(t) defined over [0,1] may be expanded in terms of Chebyshev wavelets as
(15)f(t)=∑n=0∞∑m=0∞cnmψnm(t),
where
(16)cnm=(f(t),ψnm(t))ωi*=∫01ωi*f(t)ψnm(t)dt,
and the weights wi*, i=1,2, are given in (12).
Assume that f(t) can be approximated in terms of Chebyshev wavelets as
(17)f(t)≃∑n=02k-1∑m=0Mcnmψnm(t).
3.2. Convergence Analysis
In this section, we state and prove a theorem to ascertain that the third- and fourth-kind Chebyshev wavelets expansion of a function f(t), with bounded second derivative, converges uniformly to f(t).
Theorem 1.
Assume that a function f(t)∈Lω1*2[0,1], ω1*=t/(1-t) with |f′′(t)|⩽L, can be expanded as an infinite series of third-kind Chebyshev wavelets; then this series converges uniformly to f(t). Explicitly, the expansion coefficients in (16) satisfy the following inequality:
(18)|cnm|<22πLm2(n+1)5/2(m4-1),∀n⩾0,m>1.
Proof.
From (16), it follows that
(19)cnm=2(k+1)/2π∫n/2k(n+1)/2kf(t)Vm*(2kt-n)ω1*(2kt-n)dt.
If we make use of the substitution 2kt-n=cosθ in (19), then we get
(20)cnm=2(-k+1)/2π∫0πf(cosθ+n2k)×cos(m+(1/2))θcos(θ/2)1+cosθ1-cosθsinθdθ=2(-k+3)/2π∫0πf(cosθ+n2k)cos(m+12)θcos(θ2)dθ=2(-k+1)/2π∫0πf(cosθ+n2k)×[cos(m+1)θ+cosmθ]dθ,
which in turn, and after performing integration by parts two times, yields
(21)cnm=125k/22π∫0πf′′(cosθ+n2k)γm(θ)dθ,
where
(22)γm(θ)=1m+1(sinmθm-sin(m+2)θm+2)+1m(sin(m-1)θm-1-sin(m+1)θm+1).
Now, we have
(23)|cnm|=|125k/22π∫0πf′′(cosθ+n2k)γm(θ)dθ|=125k/22π|∫0πf′′(cosθ+n2k)γm(θ)dθ|⩽L25k/22π∫0π|γm(θ)|dθ⩽Lπ2(5k+1)/2[1m+1(1m+1m+2)+1m(1m-1+1m+1)]=L2π25k/2[1m2+2m+1m2-1]<2L2π25k/2(m2m4-1).
Finally, since n⩽2k-1, we have
(24)|cnm|<22πLm2(n+1)5/2(m4-1).
Remark 2.
The estimation in (18) is also valid for the coefficients of fourth-kind Chebyshev wavelets expansion. The proof is similar to the proof of Theorem 1.
4. Solution of Multipoint BVPs
In this section, we present two Chebyshev wavelets collocation methods, namely, third-kind Chebyshev wavelets collocation method (3CWCM) and fourth-kind Chebyshev wavelets collocation method (4CWCM), to numerically solve the following multipoint boundary value problem (BVP):
(25)a(x)y′′(x)+b(x)y′(x)+c(x)y(x)+f(x,y′)+g(x,y)=0,0⩽x⩽1,(26)α0y(0)+α1y′(0)=∑i=1m0λiy(ξi)+δ0,β0y(1)+β1y′(1)=∑i=1m1μiy(ηi)+δ1,
where a(x), b(x), and c(x) are piecewise continuous on [0,1]; a(0)a(1) may equal zero; 0<ξi; ηi<1; αi, βi, λi, μi, and δi are constants such that (α02+α12)(β02+β12)≠0; f is a nonlinear function of y′, and g is a nonlinear function in y.
Consider an approximate solution to (25) and (26) which is given in terms of Chebyshev wavelets as
(27)yk,M(x)=∑n=02k-1∑m=0Mcnmψnm(x);
then the substitution of (27) into (25) enables one to write the residual of (25) in the form
(28)R(x)=∑n=02k-1∑m=2Mcnma(x)ψnm′′(x)+∑n=02k-1∑m=1Mcnmb(x)ψnm′(x)+∑n=02k-1∑m=0Mcnmc(x)ψnm(x)+f(x,∑n=02k-1∑m=1Mcnmψnm′(x))+g(x,∑n=02k-1∑m=0Mcnmψnm(x)).
Now, the application of the typical collocation method (see, e.g., [5]) gives
(29)R(xi)=0,i=1,2,…,2k(M+1)-2,
where xi are the first (2k(M+1)-2) roots of V2k(M+1)*(x) or W2k(M+1)*(x). Moreover, the use of the boundary conditions (26) gives
(30)α0∑n=02k-1∑m=0Mcnmψnm(0)+α1∑n=02k-1∑m=1Mcnmψnm′(0)=∑i=1m0∑n=02k-1∑m=0Mλicnmψnm(ξi)+δ0,β0∑n=02k-1∑m=0Mcnmψnm(1)+β1∑n=02k-1∑m=1Mcnmψnm′(1)=∑i=1m1∑n=02k-1∑m=0Mμicnmψnm(ηi)+δ1.
Equations (29) and (30) generate 2k(M+1) equations in the unknown expansion coefficients, cnm, which can be solved with the aid of the well-known Newton's iterative method. Consequently, we get the desired approximate solution yk,M(x) given by (27).
5. Numerical Examples
In this section, the presented algorithms in Section 4 are applied to solve both of linear and nonlinear multipoint BVPs. Some examples are considered to illustrate the efficiency and applicability of the two proposed algorithms.
Example 1.
Consider the second-order nonlinear BVP (see [6, 28]):
(31)y′′+38y+21089(y′)2+1=0,0<x<1,y(0)=0,y(13)=y(1).
The two proposed methods are applied to the problem for the case corresponding to k=0 and M=8. The numerical solutions are shown in Table 1. Due to nonavailability of the exact solution, we compare our results with Haar wavelets method [6], ADM solution [28] and ODEs Solver from Mathematica which is carried out by using Runge-Kutta method. This comparison is also shown in Table 1.
Comparison between different solutions for Example 1.
x
3CWCM
4CWCM
Haar method [6]
ADM method [28]
ODEs solver from Mathematica
0.1
0.06560
0.06560
0.06561
0.0656
0.06560
0.3
0.16587
0.16587
0.16588
0.1658
0.16587
0.5
0.22369
0.22369
0.22369
0.2236
0.22369
0.7
0.23820
0.23820
0.23821
0.2382
0.23820
0.9
0.20920
0.20920
0.20910
0.2092
0.20920
Example 2.
Consider the second-order linear BVP (see, [39, 40]):
(32)y′′=sinhx-2,0<x<1,y′(0)=0,y(1)=3y(35).
The exact solution of problem (32) is given by
(33)y(x)=12(sinh1-3sinh35)+sinhx-x2-x+1125.
In Table 2, the maximum absolute error E is listed for k=1 and various values of M, while in Table 3, we give a comparison between the best errors resulted from the application of various methods for Example 2, while in Figure 1, we give a comparison between the exact solution of (32) with three approximate solutions.
The maximum absolute error E for Example 2.
k
M
10
11
12
13
14
15
0
3CWCM
5.173·10-9
7.184·10-11
4.418·10-12
5.018·10-14
2.442·10-15
2.220·10-16
4CWCM
3.324·10-10
2.533·10-11
2.811·10-12
1.665·10-14
1.110·10-15
2.220·10-16
M
4
5
6
7
8
9
1
3CWCM
4.644·10-6
1.001·10-8
1.840·10-9
2.547·10-10
6.247·10-11
5.681·10-12
4CWCM
2.15·10-6
5.247·10-9
7.548·10-10
1.004·10-10
2.154·10-11
4.257·10-12
The best errors for Example 2.
Method in [39]
Method in [40]
3CWCM
4CWCM
9.00·10-6
4.00·10-5
2.22·10-16
2.22·10-16
Different solutions of Example 2.
Example 3.
Consider the second-order singular nonlinear BVP (see [40, 41]):
(34)x(1-x)y′′+6y′+2y+y2=6coshx+(2+x-x2+sinhx)sinhx,0000000000000000000000000000<x<1,y(0)+y(23)=sinh(23),y(1)+12y(45)=12sinh(45)+sinh1,
with the exact solution y(x)=sinhx. In Table 4, the maximum absolute error E is listed for k=0 and various values of M, while in Table 5 we give a comparison between the best errors resulted from the application of various methods for Example 3. This table shows that our two algorithms are more accurate if compared with the two methods developed in [40, 41].
The maximum absolute error E for Example 3.
M
8
9
10
11
12
13
3CWCM
4.444·10-10
2.759·10-11
4.389·10-13
1.643·10-14
4.441·10-16
2.220·10-16
4CWCM
4.478·10-9
2.179·10-10
1.527·10-12
2.975·10-14
4.441·10-16
2.220·10-16
The best errors for Example 3.
Method in [40]
Method in [41]
3CWCM
4CWCM
3.00·10-8
6.60·10-7
2.22·10-16
2.22·10-16
Example 4.
Consider the second-order nonlinear BVP (see [42]):
(35)y′′+(1+x+x3)y2=f(x),0<x<1,y(0)=16y(29)+13y(79)-0.0286634,y(1)=15y(29)+12y(79)-0.0401287,
where
(36)f(x)=19[(-3(1-2x)2+(1+x+x3)sin(x-x2))-6cos(x-x2)+sin(x-x2)×(-3(1-2x)2+(1+x+x3)sin(x-x2))].
The exact solution of (35) is given by y(x)=(1/3)sin(x-x2). In Table 6, the maximum absolute error E is listed for k=2 and various values of M, and in Table 7 we give a comparison between best errors resulted from the application of various methods for Example 4. This table shows that our two algorithms are more accurate if compared with the method developed in [42].
The maximum absolute error E for Example 4.
M
4
5
6
7
8
9
3CWCM
2.881·10-3
3.441·10-3
1.212·10-5
1.933·10-5
1.990·10-6
3.010·10-8
4CWCM
1.241·10-3
8.542·10-4
7.526·10-6
1.002·10-6
6.321·10-7
2.354·10-9
The best errors for Example 4.
Method in [42]
3CWCM
4CWCM
8.00·10-6
3.010·10-8
2.35·10-9
Example 5.
Consider the second-order singular linear BVP:
(37)y′′+f(x)y=g(x),0<x<1,y(0)+16y(14)=3e4,y(1)+16y(34)=3e34,
where
(38)f(x)={3x,0⩽x⩽12,2x,12<x⩽1,
and g(x) is chosen such that the exact solution of (37) is y(x)=x(1-x)ex. In Table 8, the maximum absolute error E is listed for k=1 and various values of M, while in Figure 2, we give a comparison between the exact solution of (37) with three approximate solutions.
The maximum absolute error E for Example 5.
M
7
8
9
10
11
12
13
14
3CWCM
7.1·10-6
3.2·10-7
1.3·10-8
7.4·10-10
1.3·10-11
3.5·10-13
9.3·10-15
8.1·10-16
4CWCM
2.8·10-5
1.5·10-6
6.3·10-8
2.4·10-9
7.8·10-11
2.3·10-12
6.3·10-14
1.7·10-15
Different solutions of Example 5.
Example 6.
Consider the following nonlinear second-order BVP:
(39)y′′+(y′)2-64y=32,0<x<1,(40)y(0)+y(14)=1,(41)4y(12)-y(1)=0,
with the exact solution y(x)=16x2. We solve (39) using 3CWCM for the case corresponding to k=0 and M=3, to obtain an approximate solution of y(x). If we make use of (27), then the approximate solution y0,3(x) can be expanded in terms of third-kind Chebyshev wavelets as
(42)y0,3(x)=c0,02π+c0,12π(4x-3)+c0,22π(16x2-20x+5).
If we set
(43)vi=2πc0,i,i=0,1,2,
then (42) reduces to the form
(44)y0,3(x)=v0+v1(4x-3)+v2(16x2-20x+5).
If we substitute (44) into (39), then the residual of (39) is given by
(45)R(x)=2v2+[v1+v2(8x-5)]2-4[(16x2-20x+5)v0+v1(4x-3)+v2(16x2-20x+5)]-2.
We enforce the residual to vanish at the first root of V3*(x)=64x3-112x2+56x-7, namely, at x1=0.18825509907063323, to get
(46)6.10388v22+0.5v12-3.49396v2v1-2.60388v2+4.49396v1-2v0=1.
Furthermore, the use of the boundary conditions (40) and (41) yields
(47)2v0-5v1+6v2=1,3v0-5v1-5v2=0.
The solution of the nonlinear system of (46) and (47) gives
(48)v0=10,v1=5,v2=1,
and consequently
(49)y0,3(x)=10+5(4x-3)+(16x2-20x+5)=16x2,
which is the exact solution.
Remark 3.
It is worth noting here that the obtained numerical results in the previous solved six examples are very accurate, although the number of retained modes in the spectral expansion is very few, and again the numerical results are comparing favorably with the known analytical solutions.
6. Concluding Remarks
In this paper, two algorithms for obtaining numerical spectral wavelets solutions for second-order multipoint linear and nonlinear boundary value problems are analyzed and discussed. Chebyshev polynomials of third and fourth kinds are used. One of the advantages of the developed algorithms is their availability for application on singular boundary value problems. Another advantage is that high accurate approximate solutions are achieved using a few number of terms of the approximate expansion. The obtained numerical results are comparing favorably with the analytical ones.
Acknowledgment
The authors would like to thank the referee for his valuable comments and suggestions which improved the paper in its present form.
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