Abstract
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.
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1 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].
In this paper, we will consider the boundary value problem
with \(0\le q\le 2r\) fixed; \(\alpha _i\), \(i=0,\dots , r-1,\) and \(\beta _i\), \(i=0,\dots , r,\) finite real constants. The function f is defined and continuous in \([0,1]\times D\), \(D\subset \mathrm{I\!R}^{q+1}\).
The existence and uniqueness of solution of high-order BVPs are discussed in [1], but there numerical methods and examples are only mentioned. Problem (1.1) does not appear in [1], for the particularity of the boundary conditions. The boundary conditions in (1.1) have physical meanings. For example, for \(r = 1\), they represent the position and acceleration at the end point and the velocity at the starting point of the system. However, to the best of authors’ knowledge, there are no concret physical problems at least in the mathematical literature.
We call the BVP (1.1) Lidstone–Euler second-type boundary value problem in contrast to Lidstone–Euler (first-type) boundary value problem in [10].
The interpolatory theory and qualitative as well as quantitative study of BVPs are directly connected [1, 7, 10, 16]. The boundary conditions in (1.1), that is
represent the Birkhoff-type interpolatory problem with the following incidence matrix, in Schoenberg notation: [28]:
The corresponding interpolation series has been considered in [24, 25, 27, 28], and in [36] for analogous problems.
Our study on the BVP (1.1) starts from the interpolatory problem (1.2).
The paper is organized as follows: in Sect. 2, we consider the Lidstone–Euler interpolation problem (1.2) [8, 11] using the Lidstone–Euler second-type (or even) polynomials, and we give some new results concerning the bounds of error and the convergence. In Sect. 3, we consider the existence and uniqueness of the solution of problem (1.1). In Sect. 4, we discuss some computational aspects and we give two algorithms for computing a numerical solution of the problem. In Sect. 5, we present some numerical examples to illustrate the applicability of the proposed methods. The results clearly show that the described procedures are able to produce good results in terms of accuracy. Finally, some conclusions are given.
2 Lidstone–Euler Second-Type Polynomials and Related Interpolation Problem
Lidstone–Euler second-type polynomials \({\mathscr {S}}_k(x)\), also called Lidstone–Euler even polynomials, have been introduced in [8, 11, 12]. They satisfy the BVP of second order
Polynomials \({\mathscr {S}}_k(x)\) are connected to Euler polynomials by the identity
where \(E_k(x)\) is the classic Euler polynomial of degree k [8]. Moreover, \({\mathscr {S}}_k(x)\) satisfies
\(\varepsilon _k (x)\) being the Lidstone–Euler first-type polynomial [8, 10, 11].
The first polynomials \({\mathscr {S}}_k(x)\) are
Relations (2.1) and (2.2) justify the name Lidstone–Euler type given to the sequence \(\left\{ {\mathscr {S}}_k\right\} _k\). This sequence is important, because it acts as a fundamental polynomial sequence for the Birkhoff interpolation problem given by the incidence matrix (1.3).
Theorem 1
For the Lidstone–Euler second-type polynomials, the following identity holds:
where
Proof. The proof follows by induction. For \(k=1\), the thesis is trivial. For \(k\ge 2\), we observe that the polynomial \({\mathscr {S}}_{k+1}(x)\) is the solution of the boundary value problem
From the inductive hypothesis, we have
From the theory of differential equations, the solution of (2.7) is
Thus, the thesis follows.
Remark 1
From (2.4), \(g_1(x,t)\le 0\), \(0\le x,t\le 1\). Thus, from (2.5),
Hence, in view of (2.3), we get \( (-1)^k{\mathscr {S}}_k(x)\ge 0\), \( 0\le x\le 1\).
Theorem 2
The Lidstone–Euler second-type polynomials can be written as
where
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.
Remark 2
Since \(g_1(x,t)\le 0\), from (2.10) it holds \((-1)^{k+1} G_k(x,t)\ge 0\).
Proposition 1
Let \(G_k(x,t)\) be the function defined in (2.9)–(2.10). Then
-
(i)
\(\displaystyle G_{k}(1,t)=0,\quad k\ge 1 ;\)
-
(ii)
\(\displaystyle \left. \frac{\partial }{\partial x} G_{k}(x,t)\right| _{x=0} =0,\quad k\ge 0;\)
-
(iii)
\(\displaystyle \frac{\partial ^{2s}}{\partial x^{2s}}G_{k}(x,t) = G_{k-s}(x,t),\qquad s=0,\dots , k-1,\quad k\ge 1;\)
-
(iv)
\(\displaystyle \frac{\partial ^{2s+1}}{\partial x^{2s+1}}G_{k}(x,t) =\frac{\partial }{\partial x}G_{k-s}(x,t),\qquad s=0,\dots , k-1,\quad k\ge 1.\)
Proof. The first two identities follow from the definition of \(G_k\left( x,t\right) \) 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 \(k\ge 1\), from (2.4), (2.10), and the inductive hypothesis, we get
To prove property (2.13), observe that
By differentiating, we obtain
Hence
From (2.12), it follows:
The following two results have been proved in [11].
Theorem 3
The polynomial \(Q_{\ell }\) defined by
is the unique polynomial of degree \(2\ell \) that satisfies the interpolatory conditions
with \({{\tilde{\alpha }}}_i\), \(i=0,\dots , \ell -1\) and \({{\tilde{\beta }}}_i\), \(i=0,\dots , \ell \), given real numbers.
Hence, we call \(Q_\ell (x)\) the Lidstone–Euler type interpolant polynomial of second kind and the conditions (2.15) the Lidstone–Euler type interpolation conditions of second kind.
Corollary 1
Let f be a \(2\ell \)-differentiable function in [0, 1]. The polynomial
is the unique polynomial of degree \(2\ell \), such that
The polynomial \(Q_\ell [f](x)\) 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
with
and
Proof
The proof follows after easy calculations tacking into account the inequality \( \Bigl | E_{n}(x)\Bigr | \le \frac{2}{3 \pi ^{n-1}}\) [21, p. 303]. \(\square \)
For any \(x\in [0,1]\), we can define the remainder as
Theorem 4
If \(f\in C^{2\ell +1} [0,1]\), the following identity holds:
with \(G_\ell (x,t)\) defined in (2.10).
Proof
From definition, \(\displaystyle T_\ell ^{(2\ell +1)}[f](x)=f^{(2\ell +1)}(x)\). By integrating in \(\left[ x,1\right] \), we get
with \(G_0(x,t)\) defined as in (2.9).
For the second derivative of \(T_\ell ^{(2\ell -2)}[f](x)\), we have
Moreover, from (2.17) and (2.20),
hence
By repeating s times the same procedure, we get (2.21). \(\square \)
Theorem 5
(Cauchy representation) If \(f\in C^{2\ell +1} [0,1]\), there exists \(\xi \in (0,1)\), such that
Proof
The result follows from the mean value theorem for integrals, since \(f\in C^{2\ell +1} [0,1]\), and Remark 2. \(\square \)
We can derive a different representation of the remainder.
Theorem 6
(Peano’s representation of the remainder) [11] If \(f\in C^{2\ell +1} [0,1]\), the following identity holds
where
\(\left( \cdot \right) _+\) being the known truncated power function [18].
Corollary 2
[11] For the Peano kernel \(K_\ell (x,t)\), we get
that is
Proposition 4
For any \(\ell \ge 1\), we get
Proof
The thesis follows from (2.21), (2.22) and the uniqueness of Peano’s kernel. \(\square \)
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. \(\square \)
Theorem 8
With the previous hypothesys and notations, the following bounds hold:
where \(\displaystyle \gamma _{\ell ,2i}=\frac{1}{2^{\ell -i}},\ \gamma _{\ell ,2i+1}=\frac{1}{2^{\ell -i+1}}\). It also holds
with
Proof
From Proposition 1 and Theorem 4, for \(i=0,\ldots ,\ell \)
and for \(i=0,\ldots ,\ell -1\)
From the last two inequalities and Proposition 2, we have
Relations (2.27) and (2.28) can be written as (2.25).\(\square \)
Remark 3
We explicitly note that \(\gamma _{\ell ,0}= \frac{1}{2^{\ell }}\), \(\gamma _{\ell ,1}= \gamma _{\ell ,2}= \frac{1}{2^{\ell -1}}.\)
From the previous inequalities, the following theorem can be proved.
Theorem 9
Let be \(f\in C^\infty \left[ 0,1\right] \). Then, for a fixed k
absolutely and uniformly in \(\left[ 0,1\right] \), providing that there exists a positive constant \(\lambda \), with \(|\lambda |<2\), and an integer m, such that \(f^{(2\ell +1)}=O\left( \lambda ^{\ell -\zeta +1}\right) \), for all \(\ell \ge m\) and \(\zeta \) as in (2.26).
Remark 4
We observe that the functions \(\sin x\) and \(\cos x\) satisfy Theorem 9.
3 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).
If \(f\equiv 0\), problem (1.1) has a unique solution \(y\left( x\right) = Q_r \left( x\right) \), where \(Q_r \left( x\right) \) is defined in (2.14).
The following theorem provides sufficient conditions for the existence and uniqueness of the solution of problem (1.1).
Theorem 10
Suppose that
-
(i)
\(k_s>0\), \(0\le s\le q\), are real given numbers and let M be the maximum of \(\Bigl | f\left( x,y_0,\dots , y_{q}\right) \Bigr |\) on the compact set \([0,1]\times \Omega \), where
$$\begin{aligned} \Omega =\left\{ \left( y_0,\dots , y_{q}\right) \Big |\, |y_s|\le 2 k_s,\; s=0,\dots , q\right\} ; \end{aligned}$$ -
(ii)
\(C_{2s}<k_{2s},\; \; C_{2s+1}<k_{2s+1}\), where \(C_{2s}\) and \(C_{2s+1}\) are defined in (2.18) and (2.19), respectively;
-
(iii)
\(\displaystyle \frac{M}{2^{r-s}}<k_{2s},\ \ s=0,\dots , \Bigl \lfloor \frac{q}{2}\Bigr \rfloor ; \quad \frac{M}{2^{r-s-1}}<k_{2s+1},\ \ s=0,\dots , \Bigl \lfloor \frac{q-1}{2}\Bigr \rfloor ; \)
-
(iv)
the function f satisfies a uniform Lipschitz condition in \(\left( y(x), y'(x),\dots ,\right. \left. y^{(q)}(x)\right) \), that is, there exists a nonnegative constant L, such that the inequality
$$\begin{aligned} \bigl |f\left( x,y_0,\dots , y_q\right) -f\left( x,{\overline{y}}_0,\dots , {\overline{y}}_q\right) \bigr | \le L\sum _{k=0}^q \bigl |y_k-{\overline{y}}_k\bigr | \end{aligned}$$holds whenever \(\left( y_0,\dots , y_q\right) \) and \(\left( {\overline{y}}_0,\dots , {\overline{y}}_q\right) \) belong to \(\Omega \);
-
(v)
\(\displaystyle (q+1)DL<1\), where \(\displaystyle D= \max _{0\le s\le q} \left\{ \max _{0\le x,t\le 1} \left| \frac{\partial ^s}{\partial x^s} K_r(x,t)\right| \right\} .\)
Then, the boundary value problem (1.1) has a unique solution on \(\Omega \).
Proof
It is known [16] that problem (1.1) is equivalent to the Fredholm integral equation
where \(Q_r[y]\left( x\right) \) is the polynomial defined in (2.16) and \(K_r(x,t)\) is the Peano kernel as in (2.24).
We define the operator \(T:C^{q}[0,1]\rightarrow C^{2r+1}[0,1]\) as follows:
Obviously, any fixed point of T is a solution of the boundary value problem (1.1).
For all \(y\in C^{q} [0,1]\), we introduce the norm \(\displaystyle \left\| y\right\| =\max _{0\le s\le q} \left\{ \max _{0\le t\le 1} \left| y^{(s)} (t)\right| \right\} \), 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 \(y(x)\in B\). Then
From hypotheses (i) and (iii), and Propositions 1, 2, and 4 , we have that \( T B \subseteq B\). In fact
and
From inequalities (3.2)–(3.3), we get that the sets \( \left\{ \bigl ( T[y]\bigr )^{(s)}(x) \; \Big | y(x)\in \right. \left. B \right\} \) are uniformly bounded and equicontinuous in [0, 1], for all \( 0\le s\le q\). From the Ascoli–Arzela theorem, this implies that \(\overline{T B}\) is compact. Hence, from the Shauder fixed point theorem, there exists a fixed point of T in \(\Omega \).
Now, we will prove the uniqueness. Suppose that there exist two distinct solutions y(x), z(x) of problem (1.1). It results
\(s=0,\dots , q\). Hence
so that
From hypothesis (v), the uniqueness of the solution follows.\(\square \)
4 Computational Aspects
For the numerical solution of high odd-order boundary value problems, many approaches have been proposed such as: spline interpolation [9, 30], non-polynomial 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.
4.1 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 \(g\left( x\right) \) defined in \(\left[ 0,1\right] \), the \(n-th\) Bernstein polynomial for g is given by
Then, it is known the following
Theorem 11
[18] Let g be a bounded real function in \(I=[0,1]\). Then
at any point \(x\in I\) at which g is continuous, and, if we pose \({{\overline{R}}}_n\left[ g\right] \left( x\right) =B_n\left[ g\right] \left( x\right) -g(x)\),
where \(\omega \) is the modulus of continuity of g on I. If \(g\in C(I)\), the convergence is uniform in I.
Moreover, if g is twice differentiable in I, then
Theorem 12
[5] If \(g\in C^{2s}(I)\), \(s\ge 1\), then the Bernestein polynomial for g has the following asymptotic expansion:
where \(\displaystyle h=\frac{1}{n}\), the functions \(S_i[g]\left( x\right) \), \(i=0,\dots , s\), do not depend on h and \(E_h[g](x)\rightarrow 0 \) for \(h\rightarrow 0\).
From Theorems 11 and 12, we can prove the following theorem.
Theorem 13
Let y(x) be the solution of problem (1.1). Then
where \(\displaystyle x_k=\frac{k}{n}\)
Moreover, if \(y\in C^{2r+3}(I)\), 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 \(g\equiv y^{(2r+1)}\). \(\square \)
The proposed method for the numerical solution of problem (1.1) is based on the results of the previous theorems.
\(\forall n \in \mathrm{I\!N}\), let us set
Corollary 3
For the solution y(x) of problem (1.1), we get
uniformly in \(x\in I\). Moreover
We call \(\phi _n\left( x\right) \) 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.
Theorem 14
Let n, m be two positive integers and \(\displaystyle h=\frac{1}{n}\). Moreover, let \(y(x)\in C^{2(r+m+1)}(I)\) be the solution of problem (1.1). Then
where the functions \(\displaystyle {\overline{S}}_i\left[ y\right] \left( x\right) \) do not depend on \(\displaystyle h\), and \({\overline{E}}_h\left[ y\right] \left( x\right) \rightarrow 0\) for \(h\rightarrow 0\).
Proof
The proof follows after easy calculations by applying Theorem 12 at relation (3.1). \(\square \)
The expansion (4.4) in Theorem 14 suggests to apply the extrapolation procedure [5, 32] described in the following Theorem.
Theorem 15
Let \(y\in C^{2(r+m)}[0,1]\), with m a fixed positive integer. Let \(\left\{ n_k\right\} _k\) be an increasing sequence of positive integers and \(\displaystyle h_k=n_k^{-1}\). We define a sequence of polynomials of degree \(n_{i+k}\) as follows:
Then, for \(i=0,\dots , m-k\)
Moreover, the following representations of the error and of \(T_k^{(i)}\) hold:
From Theorem 15, for any \(z\in [0,1]\), y(z) is approximated by \(T_{m}^{(0)}[y](z)\), \(n_m\) being the last element of the considered numerical sequence \(\left\{ n_i\right\} _i\).
5 Algorithm for Practical Calculations
To calculate the first-order approximation \(\phi _n(x)\) by formula (4.2), we need the values \(\displaystyle y_i^{(s)}\approx y^{(s)}\left( x_i\right) \), \(s=0,\dots , q, \ \ i=0,\ldots ,n,\) with \(s\ne 2j+1,\; j=0,\dots , \Bigl \lfloor \frac{q-1}{2}\Bigr \rfloor ,\) when \(i=0\) and \(s\ne 2j,\; j=0,\dots , \Bigl \lfloor \frac{q}{2}\Bigr \rfloor ,\) when \(i=n\).
To this aim, we consider the algebraic system of dimension \(n(q+1)\)
and \(y_0^{(2j+1)}=y^{(2j+1)}(0)=\alpha _j\), \(j=0,\dots , \Bigl \lfloor \frac{q-1}{2}\Bigr \rfloor ,\) \(y_n^{(2j)}=y^{(2j)}(1)=\beta _j\), \(j=0,\dots , \Bigl \lfloor \frac{q}{2}\Bigr \rfloor \).
Let us put \(\displaystyle Y_n^q=({{\overline{Y}}}_0, \dots , \overline{Y}_{q} )^T\), with \(\displaystyle {{\overline{Y}}}_{2j}= \left( y^{(2j)}_0,\dots , y_{n-1}^{(2j)}\right) \), \(j=0,\dots , \Bigl \lfloor \frac{q}{2}\Bigr \rfloor \), \(\displaystyle \overline{Y}_{2j+1}= \left( y^{(2j+1)}_1,\dots , y_{n}^{(2j+1)}\right) \), \(j=0,\dots , \Bigl \lfloor \frac{q-1}{2}\Bigr \rfloor \)
with \(\displaystyle A_j\in \mathrm{I\!R}^{n\times (n+1)}\)
Moreover
with
and
with
Thus, system (5.1) can be written in the form
or
Proposition 5
For the matrix \(A_n^q\), the following relation holds:
Proof. \(\displaystyle \Vert A_n^q\Vert _\infty = \max \nolimits _{0\le s\le q} \Vert A_s\Vert _\infty .\) 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 \(\int _0^1 b_{n,k}(t) \mathrm{d}t=\frac{1}{n+1}\), we have that
-
if \(s=2j\), \(j=0,\dots , \Bigl \lfloor \frac{q}{2}\Bigr \rfloor ,\) then
$$\begin{aligned} \begin{aligned} \left| p_{n,k}^{(2j) }\left( x_{i-1}\right) \right|&\le \int _0^1 \left| \frac{\partial ^{2j}}{\partial x^{2j}}K_{r}(x,t) \right| _{x=x_{i-1}} b_{n,k}(t) \mathrm{d}t\\&=\int _0^1 \left| K_{r-j}(x_{i-1}, t) \right| b_{n,k}(t) \mathrm{d}t \le \frac{1}{n+1} \left| K_{r-j}(x_{i-1},{{\overline{t}}}) \right| , \quad {{\overline{t}}}\in (0,1); \end{aligned} \end{aligned}$$hence
$$\begin{aligned} \Vert A_s\Vert _\infty= & {} \max _{0\le i\le n} \sum _{k=0}^n | p_{n,k}^{(2j) }\left( x_{i-1}\right) |\le \frac{1}{n+1} \sum _{k=0}^n \max _{0\le i\le n} \left| K_{r-j}(x_{i-1},{{\overline{t}}}) \right| \\= & {} \max _{0\le i\le n} \left| K_{r-j}(x_{i-1},{{\overline{t}}}) \right| ; \end{aligned}$$ -
if \(s=2j+1\), \(j=0,\dots , \Bigl \lfloor \frac{q-1}{2}\Bigr \rfloor ,\) then
$$\begin{aligned} \begin{aligned}&\qquad \left| p_{n,k}^{(2j+ 1) }\left( x_{i}\right) \right| \le \int _0^1 \left| \frac{\partial ^{2j+1}}{\partial x^{2j+1}}K_{r}(x,t) \right| _{x=x_{i-1}} b_{n,k}(t) \mathrm{d}t\\&\qquad =\int _0^1 \left| \frac{\partial }{\partial x}K_{r-j}(x,t) \right| _{x=x_{i-1}} b_{n,k}(t) \mathrm{d}t \le \frac{1}{n+1} \left| \frac{\partial }{\partial x} K_{r-j}(x_{i},{{\tilde{t}}}) \right| , \quad {{\tilde{t}}}\in (0,1); \end{aligned} \end{aligned}$$hence
$$\begin{aligned} \Vert A_s\Vert _\infty= & {} \max _{0\le i\le n} \sum _{k=0}^n | p_{n,k}^{(2j+1) }\left( x_{i}\right) |\le \frac{1}{n+1} \sum _{k=0}^n \max _{0\le i\le n} \left| \frac{\partial }{\partial x} K_{r-j}(x_{i},{{\tilde{t}}}) \right| \\= & {} \max _{0\le i\le n} \left| \frac{\partial }{\partial x} K_{r-j}(x_{i},{{\tilde{t}}}) \right| . \end{aligned}$$
From definition of \(K_l(x,t)\), there exist \(M_1, M_2\in \mathrm{I\!R}\), such that \(\left| K_l(x,t)\right| \le M_1\) and \(\displaystyle \left| \frac{\partial }{\partial x} K_l(x,t)\right| \le M_2\), \(l\ge 0\), for all \(0\le x,t\le 1\). From this, the result follows. \(\square \)
Lemma 1
With the previous notations and hypothesis, system (5.2) has a unique solution if \(T=L\left\| A\right\| _\infty <1\). The solution can be calculated by the iterations
with a fixed \(\left( Y_n\right) _{0} \in \mathrm{I\!R}^{n(q+1)}\). 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. \(\square \)
The previous Lemma allows us to consider the first-order approximating function
Proposition 6
Let y(x) be the solution of problem (1.1) and \(\overline{\phi }_n\left( x\right) \) the first-order approximation. Then
if \(LZ<1\), where \(\displaystyle Z=(q+1) D\) with D as in Theorem 10.
Proof
For all \(j=0,\dots , q\) and for all \(x \in [0,1]\), from (4.2), we get
and by differentiating the (5.3)
\(\square \)
From (5.4), (5.5) and (4.1), we obtain
Hence, from the Lipschitz property of f
Finally, the last inequality yields
hence, the thesis follows, according to Corollary 3.\(\square \)
The Bernstein extrapolation method for BVP (1.1) can be summarized as follows:
-
1.
Data input: problem (1.1);
-
2.
for a fixed \(n\in \mathrm{I\!N}\), solve the algebraic system (5.1);
-
3.
for a fixed \(m\in \mathrm{I\!N}\), choose a sequence \(n_i\), \(i=0,\dots , m\) and calculate \({{\overline{\phi }}}_{n_i} (x)\);
-
4.
for \(k=1,\dots , m-1\), \(i=0,\dots , m-k\), calculate \(T_k^{(i)}\).
5.1 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.
Let y(x) be the solution of problem (1.1). For any \(n\in \mathrm{I\!N}\), if \(y(x)\in C^{2r+n+2}[0,1]\), we can approximate \(y^{(2r+1)}\left( x\right) \) in \(x\in [0,1]\) by the well-known Lagrange interpolation polynomial
where
\(l_i\left( x\right) \) being the fundamental Lagrange polynomials and \(\omega _n(x)=\prod _{i=0}^n (x-x_i)\), with \(x_i\), for \(i=0\dots ,n\), \(n+1\) distinct points in [0, 1] and \(\xi _x\) a point in the smallest interval containing x and all \(x_i\), \(i=0,\dots , n\).
Then, by substituting (5.6) in (3.1), we have
where
and the remainder term \(T_{r,n}[y](y,x)\) is given by
Relation (5.7) suggests to consider the implicitly defined polynomial
Theorem 16
[16] The polynomial \(y_{n}(x)\) of degree \(2r+n+1\) defined in (5.8) is a collocation polynomial for (1.1) at nodes \(x_i\).
An algorithm for practical calculation is similar to that used in [16].
6 Numerical Examples
As we said, there are no specific methods for the numerical solution of problem (1.1). Neither specific numerical examples we have found in the literature, to compare the numerical results. Anyhow, in the following, we report some problems to validate the theoretical results previously given. Since the analytical solutions of the considered examples are known, we compute the true errors \(\forall x\in I\) fixed
Example 1
Consider the following problem:
The analytical solution is \(\displaystyle { y\left( x\right) = \log (1+x)}\).
The first approximating polynomials \({{\overline{\phi }}}_n(x)\) are
Figure 1 shows the graphs of the error functions \(e_{B,n_k}(x)\) with \(n_k=4+2k\), \(k=0,\dots , 3\) (Fig. 1a) and the graph of \(E_{B,3}(x)\) (Fig. 1b). Figure 1a shows the low convergence of the approximating polynomial sequence \(\left\{ {{\overline{\phi }}}_n\right\} \).
The absolute errors in \(x=\frac{1}{2}\) using extrapolation for different sequences \(n_k\), \(k=0,\dots , m\), are displayed in Table 1.
Figure 2 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 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:
The analytical solution is \(\displaystyle { y\left( x\right) = e^x}\).
Figure 3 shows the graphs of the error functions \(e_{B,n_k}(x)\) with \(n_k=2+2k\), \(k=0,\dots , 3\) (Fig. 3a) and the graph of \(E_{B,2}(x)\) (Fig. 3b).
The errors in \(x=\frac{1}{2}\) 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.
7 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 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.
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Dedicated to Professor Francesco Altomare on his 70th birthday.
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Costabile, F.A., Gualtieri, M.I. & Napoli, A. Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools. Mediterr. J. Math. 18, 180 (2021). https://doi.org/10.1007/s00009-021-01822-5
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DOI: https://doi.org/10.1007/s00009-021-01822-5