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A new generalized Jacobi Galerkin operational matrix of derivatives: two algorithms for solving fourthorder boundary value problems
Advances in Difference Equations volumeÂ 2016, ArticleÂ number:Â 22 (2016)
Abstract
This paper reports a novel Galerkin operational matrix of derivatives of some generalized Jacobi polynomials. This matrix is utilized for solving fourthorder linear and nonlinear boundary value problems. Two algorithms based on applying Galerkin and collocation spectral methods are developed for obtaining new approximate solutions of linear and nonlinear fourthorder two point boundary value problems. In fact, the key idea for the two proposed algorithms is to convert the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable numerical solvers. The convergence analysis of the suggested generalized Jacobi expansion is carefully discussed. Some illustrative examples are given for the sake of indicating the high accuracy and effectiveness of the two proposed algorithms. The resulting approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by other existing techniques in the literature.
1 Introduction
Spectral methods are global methods. The main idea behind spectral methods is to approximate solutions of differential equations by means of truncated series of orthogonal polynomials. The spectral methods play prominent roles in various applications such as fluid dynamics. The three most used versions of spectral methods are: tau, collocation, and Galerkin methods (see for example [1â€“8]). The choice of the suitable used spectral method suggested for solving the given equation depends certainly on the type of the differential equation and also on the type of the boundary conditions governed by it.
In the collocation approach, the test functions are the Dirac delta functions centered at special collocation points. This approach requires the differential equation to be satisfied exactly at the collocation points. The taumethod is a synonym for expanding the residual function as a series of orthogonal polynomials and then applying the boundary conditions as constraints. The tau approach has an advantage that it can be applied to problems with complicated boundary conditions. In the Galerkin method, the test functions are chosen in a way such that each member of them satisfies the underlying boundary conditions of the given differential equation.
There is extensive work in the literature on the numerical solutions of highorder boundary value problems (BVPs). The great interest in such problems is due to their importance in various fields of applied science. For example, a large number of problems in physics and fluid dynamics are described by problems of this kind. In this respect, there is a huge number of articles handling both high odd and high evenorder BVPs. For example, in the sequence of papers, [5, 9â€“11], the authors have obtained numerical solutions for evenorder BVPs by applying the Galerkin method. The main idea for obtaining these solutions is to construct suitable basis functions satisfying the underlying boundary conditions on the given differential equation, then applying Galerkin method to convert each equation to a system of algebraic equations. The suggested algorithms in these articles are suitable for handling one and two dimensional linear evenorder BVPs. The Galerkin and PetrovGalerkin methods have the advantage that their applications on linear problems enable one to investigate carefully the resulting systems, especially their complexities and condition numbers.
There are many algorithms in the literature which are applied for handling fourthorder boundary value problems. For example, Bernardi et al. in [12] suggested some spectral approximations for handling two dimensional fourthorder problems. In the two leading articles of Shen [13, 14], the author developed direct solutions of fourthorder two point boundary value problems. The suggested algorithms in these articles are based on constructing compact combinations of Legendre and Chebyshev polynomials together with the application of the Galerkin method. Many other techniques were used for solving fourthorder BVPs, for example, variational iteration method is applied in [15], nonpolynomial sextic spline method in [16], quintic nonpolynomial spline method in [17], and the Galerkin method (see [18, 19]). Theorems which list the conditions for the existence and uniqueness of solution of such problems are thoroughly discussed in the important book of Agarwal [20].
The approach of employing operational matrices of differentiation and integration is considered an important technique for solving various kinds of differential and integral equations. The main advantage of this approach is its simplicity in application and its capability for handling linear differential equations as well as nonlinear differential equations. There are a large number of articles in the literature in this direction. For example, the authors in [6], employed the tau operational matrices of derivatives of Chebyshev polynomials of the second kind for handling the singular LaneEmden type equations. Some other studies in [21, 22] employ tau operational matrices of derivatives for solving the same type of equations. The operational matrices of shifted Chebyshev, shifted Jacobi, and generalized Laguerre polynomials and other kinds of polynomials are employed for solving some fractional problems (see for example, [23â€“27]). In addition, recently in the two papers of AbdElhameed [28, 29] one introduced and used two Galerkin operational matrices for solving, respectively, the sixthorder two point BVPs and LaneEmden equations.
In this paper, our main aim is fourfold:

Establishing a novel Galerkin operational matrix of derivatives of some generalized Jacobi polynomials.

Investigating the convergence analysis of the suggested generalized Jacobi expansion.

Employing the introduced operational matrix of derivatives to numerically solve linear fourthorder BVPs based on the application of Galerkin method.

Employing the introduced operational matrix of derivatives for solving the nonlinear fourthorder BVPs based on the application of collocation method.
The contents of the paper is organized as follows. SectionÂ 2 is devoted to presenting an overview on classical Jacobi and generalized Jacobi polynomials. SectionÂ 3 is concerned with deriving the Galerkin operational matrix of derivatives of some generalized Jacobi polynomials. In SectionÂ 4, we implement and present two numerical algorithms for the sake of handling linear and nonlinear fourthorder BVPs based on the application of generalized JacobiGalerkin operational matrix method (GJGOMM) for linear problems and generalized Jacobi collocation operational matrix method (GJCOMM) for nonlinear problems. Convergence analysis of the generalized Jacobi expansion is discussed in detail in SectionÂ 5. Numerical examples including some discussions and comparisons are given in SectionÂ 6 for the sake of testing the efficiency, accuracy, and applicability of the suggested algorithms. Finally, conclusions are reported in SectionÂ 7.
2 An overview on classical Jacobi and generalized Jacobi polynomials
The classical Jacobi polynomials \(P_{n}^{(\alpha,\beta)}(x)\) associated with the real parameters (\(\alpha>1\), \(\beta>1\)) (see [30] and [31]) are a sequence of polynomials defined on \([1,1]\). Define the normalized orthogonal polynomials \(R_{n}^{(\alpha,\beta)}(x)\) (see [32])
and define the shifted normalized Jacobi polynomials on \([a,b]\) as
The polynomials \(\tilde{R}^{(\alpha ,\beta )}_{n}(x)\) are orthogonal on \([a,b]\) with respect to the weight function \((bx)^{\alpha } (xa)^{\beta }\), in the sense that
where
It should be noted here that the Legendre polynomials are particular polynomials of Jacobi polynomials. In fact, \(R_{n}^{(0,0)}(x)=L_{n}(x)\), where \(L_{n}(x)\) is the standard Legendre polynomial of degree n.
Let \(w^{\alpha,\beta}(x)=(bx)^{\alpha}(xa)^{\beta}\). We denote by \(L^{2}_{w^{\alpha,\beta}}(a,b)\) the weighted \(L^{2}\) space with inner product:
and the associated norm \(\u\_{w^{\alpha,\beta}}=(u,u)^{\frac{1}{2}}_{w^{\alpha,\beta}}\). Now, the definition of the shifted Jacobi polynomials will be extended to include the cases in which Î± and/or \(\beta\le1\). Now assume that \(\ell,m\in\mathbb{Z}\), and define
It is worthy to note here that in the case of \([a,b]=[1,1]\), the polynomials defined in (4) are the socalled generalized Jacobi polynomials \((J_{i}^{(\ell,m)}(x))\), which are defined by Guo et al. in [33]. Now, the symmetric generalized Jacobi polynomials \(J^{(n,n)}_{i}(x)\) can be expressed explicitly in terms of the Legendre polynomials, while the symmetric shifted generalized Jacobi polynomials \(\tilde{J}^{(n,n)}_{i}(x)\) can be expressed in terms of the shifted Legendre polynomials. These results are given in the following two lemmas.
Lemma 1
For every nonnegative integer n, and for all \(i\ge2n\), one has
and in particular,
Proof
For the proof of Lemma 1, see [9].â€ƒâ–¡
Now, Lemma 2 is a direct consequence of Lemma 1.
Lemma 2
For every nonnegative integer n, for all \(i\ge2n\), one has
and in particular,
The following lemma is also of interest in the sequel.
Lemma 3
The following integral formula holds:
Proof
Lemma 3 follows if we integrate equation (8) (for the case \([a,b]=[1,1]\)) together with the aid of the following integral formula:
â€ƒâ–¡
3 Generalized Jacobi Galerkin operational matrix of derivatives
In this section, a novel operational matrix of derivatives will be developed. For this purpose, we choose the following set of basis functions:
It is easy to see that, the set of polynomials \(\{\phi_{i}(x): i=0,1,2,\ldots\}\) is a linearly independent set. Moreover, they are orthogonal on \([a,b]\) with respect to the weight function \(w(x)=\frac{1}{(xa)^{2} (bx)^{2}}\), in the sense that
Let us denote \(H_{w}^{r}(I)\) (\(r=0,1,2,\ldots\)), as the weighted Sobolev spaces, whose inner products and norms are denoted by \((\cdot,\cdot)_{r,w}\) and \(\\cdot\_{r,w}\), respectively (see [4]). To account for homogeneous boundary conditions, we define
where \(I=(a,b)\).
Define the following subspace of \(H_{0,w}^{2}(I)\):
Any function \(f(x)\in H_{0,w}^{2}(I)\) can be expanded as
where
Assume that \(f(x)\) in equation (12) can be approximated as
where
Now, we are going to state and prove the main theorem, from which a novel Galerkin operational matrix of derivatives will be introduced.
Theorem 1
If the polynomials \(\phi_{i}(x)\) are selected as in (11), then for all \(i\ge1\), one has
where \(\eta_{i}(x)\) is given by
Proof
The key idea is to prove Theorem 1 on \([1,1]\), and hence the proof on the general interval \([a,b]\) can easily be transported. Now, we intend to prove the relation
where
and
To prove (18), it is sufficient to prove that the following identity holds, up to a constant:
where
Indeed
If we make use of Lemma 3, then the latter equationafter performing some manipulationsis turned into the relation
where
After performing some rather lengthy manipulations on the right hand side of (21), equation (19) is obtained.
Now, if x in (18) is replaced by \(\frac{2xab}{ba}\), then after performing some manipulations, we get
where \(\eta_{i}(x)\) is given by
and this completes the proof of Theorem 1.â€ƒâ–¡
Now, with the aid of Theorem 1, the first derivative of the vector \(\boldsymbol {\Phi}(x)\) defined in (15) can be expressed in matrix form:
where \(\boldsymbol {\eta}(x)= (\eta_{0}(x),\eta_{1}(x),\dots,\eta _{N}(x) )^{T}\), and \(H= (h_{ij} )_{0\leqslant i,j\leqslant N}\) is an \((N+1)\times(N+1)\) matrix whose nonzero elements can be given explicitly from equation (16) by
For example, for \(N=5\), the operational matrix M is the following \((6\times6)\) matrix:
Corollary 1
The second, third and fourthorder derivatives of the vector \(\boldsymbol {\Phi}(x)\) are given, respectively, by
4 Two algorithms for fourthorder two point BVPs
In this section, we are interested in developing two numerical algorithms for solving both of the linear and nonlinear fourthorder two point BVPs. The Galerkin operational matrix of derivatives that introduced in SectionÂ 3 is employed for this purpose. The linear equations are handled by the application of the Galerkin method, while the nonlinear equations are handled by the application of the typical collocation method.
4.1 Linear fourthorder BVPs
Consider the linear fourthorder boundary value problem
subject to the homogeneous boundary conditions
If \(u(x)\) is approximated as
then making use of equations (23)(26), the following approximations for \(y^{(\ell)}(x)\), \(1\le\ell\le4\), are obtained:
where
If we substitute equations (29)(31) into equation (27), then the residual, \(r(x)\), of this equation can be written
The application of the Galerkin method (see [4]) yields the following \((N+1)\) linear equations in the unknown expansion coefficients, \(c_{i}\), namely
Thus equation (33) generates a set of \((N+1)\) linear equations which can be solved for the unknown components of the vector C, and hence the approximate spectral solution \(u_{N}(x)\) given in (29) can be obtained.
Remark 1
It should be noted that the problem (27), governed by the nonhomogeneous boundary conditions
can easily be transformed to a problem similar to (27)(28) (see [10]).
4.2 Solution of nonlinear fourthorder two point BVPs
Consider the following nonlinear fourthorder boundary value problem:
governed by the homogeneous boundary conditions
If \(u^{(\ell)}(x)\), \(0\le\ell\le4\), are approximated as in (29)(31), then the following nonlinear equations in the unknown vector C can be obtained:
An approximate solution \(u_{N}(x)\) can be obtained by employing the typical collocation method. For this purpose, equation (37) is collocated at \((N+1)\) points. These points may be taken to be the zeros of the polynomial \(\tilde{R}^{(2,2)}_{N+1}(x)\), or by any other choice. Hence, a set of \((N+1)\) nonlinear equations is generated in the expansion coefficients, \(c_{i}\). This nonlinear system can be solved with the aid of a suitable solver, such as the wellknown Newton iterative method. Therefore, the corresponding approximate solution \(u_{N}(x)\) can be obtained.
5 Convergence analysis of the approximate expansion
In this section, the convergence analysis of the suggested generalized Jacobi approximate solution will be investigated. We will state and prove a theorem in which the expansion in (12) of a function \(f(x)=(xa)^{2} (bx)^{2} G(x)\in H_{0,w}^{2}(I)\), where \(G(x)\) is of bounded fourth derivative, converges uniformly to \(f(x)\).
Theorem 2
A function \(f(x)=(xa)^{2} (bx)^{2} G(x)\in H_{0,w}^{2}(I)\), \(w(x) =\frac{1}{(xa)^{2} (bx)^{2}}\) with \(G^{(4)}(x)\leqslant M\), can be expanded as an infinite sum of the basis given in (12). This series converges uniformly to \(f(x)\), and the coefficients in (12) satisfy the inequality
Proof
From equation (13), one has
and with the aid of equation (11), the coefficients \(c_{i}\) may be written alternatively in the form
Making use of Lemma 2, the polynomials \(\tilde{J}^{(2,2)}_{i}(x)\) can be expanded in terms of the shifted Legendre polynomials, and so the coefficients \(c_{i}\) take the form
If the last relation is integrated by parts four times, then the repeated application of equation (10) yields
where \(I^{(4)}(x)\) is given by
which can be written as
and then the coefficients \(c_{i}\) take the form
Now, making use of the substitution \(\frac{2xab}{ba}=\cos \theta\) enables one to put the coefficients \(c_{i}\) in the form
Taking into consideration the assumption \(\vert G^{(4)}(x)\vert \le M\), then we have
From a Bernstein type inequality (see [34]), it is easy to see that
and hence (44) together with the last inequality leads to the estimation
Finally it is easy show that for all \(i\ge6\),
This completes the proof of the theorem.â€ƒâ–¡
6 Numerical results and discussions
In this section, the two proposed algorithms in SectionÂ 4 are applied to solve linear and nonlinear fourthorder two point boundary value problems. The numerical results ensure that the two algorithms are very efficient and accurate.
Example 1
Consider the fourthorder linear boundary value problem (see [35]):
The exact solution of (46) is
TableÂ 1 lists the maximum absolute errors E, which resulted from the application of GJGOMM, for various values of N, while in TableÂ 2 we display a comparison between the relative errors obtained by the application of the two methods namely, the firstorder method (1OM) and the secondorder methods (2OMs) developed in [35] with the relative errors resulting from the application of GJGOMM.
Example 2
Consider the following fourthorder nonlinear boundary value problem (see [36, 37]):
The exact solution of the above problem is
In TableÂ 3, we list the maximum absolute errors using GJCOMM for various values ofÂ N. Let \(E_{1},E_{2},E_{3}\), and \(E_{4}\) denote the maximum absolute errors if the selected collocation points are respectively, the zeros of the shifted Legendre polynomial \(L^{*}_{N+1}(x)\), the shifted Chebyshev polynomials of the first and second kinds \(T^{*}_{N+1}(x)\), \(U^{*}_{N+1}(x)\), and the shifted symmetric Jacobi polynomial \({\tilde{R}^{(2,2)}}_{N+1}(x)\), while FiguresÂ 1 and 2 display a comparison between the maximum absolute errors resulting from the application of GJCOMM for \(N=4\) and 6, respectively. TableÂ 3 and FiguresÂ 1 and 2 show that the best choice among the previous choices for the collocation points are obtained if the selected collocation points are the zeros of the polynomial \(\tilde{R}^{(2,2)}_{N+1}(x)\). TableÂ 4 displays a comparison between the errors obtained by the application of GJCOMM for \(N=4\), with the errors resulting from the application of the three methods developed in [36, 37]. This comparison ascertains that our results are more accurate than those obtained in [36, 37].
Example 3
Consider the following fourthorder nonlinear boundary value problem (see [37]):
where \(g(x)=x^{10}+4 x^{9}4 x^{8}4 x^{7}+8 x^{6}4 x^{4}+120 x48\). The exact solution of the above problem is
In TableÂ 5, we list the maximum absolute errors using GJCOMM for various values of N. Let E denote the maximum pointwise errors if the selected collocation points are the zeros of the polynomial \(\tilde{R}^{(2,2)}_{N+1}(x)\). Moreover, TableÂ 6 displays a comparison between the errors obtained by the application of GJCOMM with the method developed in [37] for the case \(N=2\). The comparison ascertains that our results are more accurate than those obtained by [37].
Example 4
Consider the following nonlinear fourthorder boundary value problem (see [38]):
with the exact solution \(y(x) =\sinh(x)+1\).
In TableÂ 7, the absolute errors are listed for various values of N. In order to compare the absolute errors obtained by applying GJCOMM with those obtained by applying RHKSM in [38], we list the absolute errors obtained by the application of RHKSM in the last column of this table. This table shows that the approximate solution of problem (49) obtained by using GJCOMM is of high efficiency and more accurate than the approximate solution obtained by RHKSM [38].
7 Concluding remarks
In this article, a novel operational matrix of derivatives of certain generalized Jacobi polynomials is derived and used for introducing spectral solutions of linear and nonlinear fourthorder two point boundary value problems. The two spectral methods, namely the Galerkin and collocation methods are employed for this purpose. The main advantages of the introduced algorithms are their simplicity in application, and also their high accuracy, since highly accurate approximate solutions can be achieved by using a small number of terms of the suggested expansion. The numerical results are convincing and the resulting approximate solutions are very close to the exact ones.
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AbdElhameed, W.M., Ahmed, H.M. & Youssri, Y.H. A new generalized Jacobi Galerkin operational matrix of derivatives: two algorithms for solving fourthorder boundary value problems. Adv Differ Equ 2016, 22 (2016). https://doi.org/10.1186/s1366201607532
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DOI: https://doi.org/10.1186/s1366201607532