Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

General nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.


Introduction
Boundary value problems (BVPs) with higher order differential equations play an important role in a variety of different branches of applied mathematics, engineering, and many other fields of advanced physical sciences. For examples, third-order BVPs arise in several physical problems, such as the deflection of a curved beam, the motion of rockets, thin-film flows, electromagnetic waves, or gravity-driven flows (see [3,4] and references therein). Fifth-order differential equations are used in mathematical modelling of viscoelastic flows [17]. Seventh-order BVPs arise in modelling induction motor with two rotor circuits [26]. Nineth-order BVPs are known to arise in hydrodynamic, hydromagnetic stability, and mathematical modelling of AFTI-F16 fighters [2,23]. For more details on accuracy of high-order BVPs, see [6,[13][14][15]19,20,22,33,34, and references therein].

Theorem 1.
For the Lidstone-Euler second-type polynomials, the following identity holds: where (2.4) g k (x, t) = From the theory of differential equations, the solution of (2.7) is Thus, the thesis follows.

Theorem 2. The Lidstone-Euler second-type polynomials can be written as
with g 1 defined as in (2.4).
Proof. Relation (2.8) can be proved by induction. For k = 1, it is trivially true. For k > 1, from (2.8) and the inductive hypothesis, the boundary value problem (2.6) can be written as The solution of (2.11) is and this completes the proof.
Proof. The first two identities follow from the definition of G k (x, t) and the boundary conditions in (2.1). Property (iii) is obtained from the first of (2.1) and Theorem 2. Relation (iv) follows from (iii).

Proposition 2.
For the function G k (x, t), the following inequalities hold: Proof. The proof of (2.12) follows by induction. For k = 0, the thesis is trivial. For k ≥ 1, from (2.4), (2.10), and the inductive hypothesis, we get To prove property (2.13), observe that From (2.12), it follows: The following two results have been proved in [11].
is the unique polynomial of degree 2 that satisfies the interpolatory conditions Hence, we call Q (x) the Lidstone-Euler type interpolant polynomial of second kind and the conditions (2.15) the Lidstone-Euler type interpolation conditions of second kind.
is the unique polynomial of degree 2 , such that (2.17) The polynomial Q [f ](x) is called the Lidstone-Euler interpolant polynomial of second kind for the function f .

Proposition 3.
For the derivatives of the Lidstone-Euler type interpolant polynomial of second kind, there exist constants C 2s and C 2s+1 , such that Proof. The proof follows after easy calculations tacking into account the inequality E n (x) ≤ 2 3π n−1 [21, p. 303]. For any x ∈ [0, 1], we can define the remainder as Theorem 4. If f ∈ C 2 +1 [0, 1], the following identity holds: with G (x, t) defined in (2.10).
We can derive a different representation of the remainder.
If we set the following theorems provide bounds for the remainder and its derivatives.

Theorem 7.
With the previous hypotheses and notations, the following bound holds: Proof. The thesis follows from Cauchy representation and Proposition 2.

Theorem 8.
With the previous hypothesys and notations, the following bounds hold: Proof. From Proposition 1 and Theorem 4, for i = 0, . . . , From the last two inequalities and Proposition 2, we have Relations (2.27) and (2.28) can be written as (2.25).
Remark 3. We explicitly note that From the previous inequalities, the following theorem can be proved.
absolutely and uniformly in [0, 1], providing that there exists a positive constant λ, with |λ| < 2, and an integer m, such that f ( Remark 4. We observe that the functions sin x and cos x satisfy Theorem 9.

The Second-Type Lidstone-Euler Boundary Value Problem
In this section, we will investigate the existence and uniqueness of the solution of the second-type Lidstone-Euler BVP (1.1). As we said, to the best of the authors' knowledge, the existence and uniqueness of the solution of (1.1) have not previously been investigated. Similar BVPs with different boundary conditions have been much studied (see, for example, [1] and references therein).
Proof. It is known [16] that problem (1.1) is equivalent to the Fredholm integral equation We define the operator T : C q [0, 1] → C 2r+1 [0, 1] as follows: Obviously, any fixed point of T is a solution of the boundary value problem (1.1).
For all y ∈ C q [0, 1], we introduce the norm y = max 0≤s≤q max 0≤t≤1 y (s) (t) , so that C q [0, 1] becomes a Banach space. Moreover, we consider the set The operator T maps B into itself. To show this, let y(x) ∈ B. Then From hypotheses (i) and (iii), and Propositions 1, 2, and 4 , we have that T B ⊆ B. In fact and From hypothesis (v), the uniqueness of the solution follows.

Computational Aspects
For the numerical solution of high odd-order boundary value problems, many approaches have been proposed such as: spline interpolation [9,30], nonpolynomial spline techniques [29], Galerkin methods [17], variational iterative techniques [31,33], and modified decomposition method [35]. In the following, we present two numerical approaches: Bernstein extrapolation and collocation methods.

Bernstein Extrapolation Methods
The first numerical approach for the numerical solution of the BVP (1.1) is based on extrapolated Bernstein polynomials [5]. We recall that, given a real function g (x) defined in [0, 1], the n − th Bernstein polynomial for g is given by Then, it is known the following where ω is the modulus of continuity of g on I. If g ∈ C(I), the convergence is uniform in I. Moreover, if g is twice differentiable in I, then .

Theorem 12. [5]
If g ∈ C 2s (I), s ≥ 1, then the Bernestein polynomial for g has the following asymptotic expansion: where h = 1 n , the functions S i [g] (x), i = 0, . . . , s, do not depend on h and From Theorems 11 and 12, we can prove the following theorem.

Theorem 13. Let y(x) be the solution of problem (1.1).
Then (4.1) Moreover, if y ∈ C 2r+3 (I), then and the convergence is uniform. Proof. If y(x) is the solution of problem (1.1), we get relation (3.1). The thesis follows after easy calculations from Theorem 12 if g ≡ y (2r+1) .
The proposed method for the numerical solution of problem (1.1) is based on the results of the previous theorems.
∀n ∈ IN, let us set We call φ n (x) the approximating solution of first order, and the error is bounded by (4.3).
To have approximating functions of higher order, we use the following asymptotic expansion.  Proof. The proof follows after easy calculations by applying Theorem 12 at relation (3.1).
The expansion (4.4) in Theorem 14 suggests to apply the extrapolation procedure [5,32] described in the following Theorem.

Algorithm for Practical Calculations
To calculate the first-order approximation φ n (x) by formula (4.2), we need the values y , when i = 0 and s = 2j, j = 0, . . . , q 2 , when i = n. To this aim, we consider the algebraic system of dimension n(q + 1) Moreover Thus, system (5.1) can be written in the form Proposition 5. For the matrix A q n , the following relation holds: Since all Bernstein basis functions b n,k (t) of the same order have the same definite integral over the interval [0, 1], that is Kr−j(xi−1, t) , t ∈ (0, 1); hence As ∞ = max with a fixed (Y n ) 0 ∈ IR n(q+1) . Moreover, at the jth iteration, the error is Proof. The proof follows by standard technique by applying the well-known contraction principle and Proposition 5.
The previous Lemma allows us to consider the first-order approximating function Proof. For all j = 0, . . . , q and for all x ∈ [0, 1], from (4.2), we get  Hence, from the Lipschitz property of f Finally, the last inequality yields hence, the thesis follows, according to Corollary 3.
The Bernstein extrapolation method for BVP (1.1) can be summarized as follows: 1

Collocation-Birkhoff-Lagrange Approach
The collocation-Birkhoff-Lagrange approach to BVPs has been proposed in [16]. Here, we use this approach as comparisons with the method described above.
An algorithm for practical calculation is similar to that used in [16].

Numerical Examples
As we said, there are no specific methods for the numerical solution of prob  The analytical solution is y (x) = log(1 + x). The first approximating polynomials φ n (x) are Figure 1 shows the graphs of the error functions e B,n k (x) with n k = 4 + 2k, k = 0, . . . , 3 (Fig. 1a) and the graph of E B,3 (x) (Fig. 1b). Figure 1a shows the low convergence of the approximating polynomial sequence φ n . The absolute errors in x = 1 2 using extrapolation for different sequences n k , k = 0, . . . , m, are displayed in Table 1.  Table 2 lists the comparison between the approximation by extrapolated Bernstein polynomials (using one of the sequences considered in Table 1) and collocation-Birkhoff-Lagrange polynomials of degree 10 and 12, respectively. Example 2. Consider the following problem: y (0) = y (0) = 1.
The errors in x = 1 2 using extrapolation for different sequences n k are displayed in Table 3. Figure 4 shows the graphs of the error functions e L,n (x) in the case of collocation on equidistant nodes, for several values of the degree n of the approximating polynomials.    Table 4 lists the comparison between the approximation by extrapolated Bernstein polynomials and collocation-Birkhoff-Lagrange polynomials.

Conclusions
In this paper, we considered general nonlinear high odd-order differential equations with Lidstone-Euler boundary conditions of second type. First, we studied the associated interpolation problem and we obtained new properties, such as the integral Cauchy and Peano representation of the error, bounds for the error, and its derivatives, and we deducted the interesting convergence properties of the interpolatory polynomial sequences. Then, we    Table 4. Comparison between extrapolated Bernstein polynomials and collocation-Birkhoff-Lagrange polynomials-Problem (6.2) ||E B,m || ||e L,n || n k = 2 + 2k m = 4 m = 5 n = 7 n = 9 3.9448e−09 9.9367e−12 6.6291e−11 2.2598e−11 considered the associated Lidstone-Euler second-type boundary value problem, from both a theoretical and a computational point of view. Particularly, we gave a theorem of existence and uniqueness of the solution of the problem and we proposed two different numerical approaches for the approximate solution: one of them is based on extrapolated Bernstein polynomials, the other one on Lagrange interpolation and collocation principle. We note the convergence properties of the Bernstein extrapolation method and, in some cases, a greater computational accuracy of the Lagrange-collocation methods. From the numerical examples, we can observe that the order of the error is similar with the two different methods. Further developments are possible and desirable as well theoretical as well computational.