Abstract
The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numericalanalytic method based upon successive approximations constructed in analytic form. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function. We provide a justification of the polynomial version of the method with several illustrating examples.
2000 Mathematics Subject Classification: 34B15; 65L10.
Keywords:
nonlinear boundary value problem; numericalanalytic method; Chebyshev interpolation polynomials1 Introduction
In studies of solutions of various types of nonlinear boundary value problems for ordinary differential equations side by side with numerical methods, it is often used an appropriate technique based upon some types of successive approximations constructed in analytic form. This class of methods includes, in particular, the approach suggested at first in [1,2] for investigation of periodic solutions. Later, appropriate versions of this method were developed for handling more general types of nonlinear boundary value problems for ordinary and functionaldifferential equations. We refer, e.g., to the books [35], the articles [612], and the series of survey articles [13] for the related references.
According to the basic idea, the given boundary value problem is replaced by the Cauchy problem for a suitably modified system of integrodifferential equations containing some artificially introduced parameters. The solution of the perturbed problem is searched in analytic form by successive iterations. The perturbation term, which depends on the original differential equation, on the introduced parameters and on the boundary conditions, yields a system of algebraic or transcendental determining equations. These equations enable us to determine the values of the introduced parameters for which the original and the perturbed problems coincide. Moreover, studying solvability of the approximate determining systems, we can establish existence results for the original boundary value problem.
In this article, we introduce the Chebyshev polynomial version of the known numericalanalytic method based on successive iterations. At the beginning, we follow the ideas presented by Rontó and Rontó [14] and by Rontó and Shchobak [15], which contains existence results for a system of two nonlinear differential equations with separated boundary conditions. In order to avoid some technical difficulties, we deal in this article, for simplicity, with nonlinear differential equations with homogeneous Dirichlet boundary conditions. On the other hand, our basic recurrence relation has the same general form as it is presented in [15]. In Section 2, we state the studied problem and the corresponding setting. Sections 3 and 4 contain the construction of the sequence of successive approximations, its convergence analysis, the properties of the limit function, and its correspondence to the solution of the original boundary value problem. The existence questions are discussed as well. Main results of the article are in Section 5, which contains a justification of the Chebyshev polynomial version of the introduced method with corresponding convergence analysis and error estimates. Results in Section 5 allow us to construct the Chebyshev polynomial approximations of the solution of the nonlinear boundary value problem, which essentially simplify the computations of successive approximations in analytic form and simplify also the form of the determining equation. In Section 6, we illustrate the applicability of our approach to three Dirichlet boundary value problems: the linear one, the semilinear one, and the quasilinear one containing the pLaplace operator. Finally, let us note that presented polynomial version of the numericalanalytic method in this article can be extended to more general nonlinear boundary value problems studied in [14].
2 Problem setting and preliminaries
We consider the following system of two nonlinear differential equations with Dirichlet boundary conditions
In the vector form, we have
where x = col(x_{1}, x_{2}), f(t, x) = col(f_{1}(t, x), f_{2}(t, x)) and
Let the function f(t, x) be defined and continuous in the domain
To avoid dealing with singular matrix C_{1 }in (2), which does not enable us to express explicitly x(T), it is useful to carry out the following parametrization
where
Thus, instead of (2) we use the equivalent problem with twopoint parameterized boundary conditions
where
The twopoint parameterized boundary conditions in (6) allow us to write
which will be used in the sequel for the construction of the iterative scheme.
Throughout the text, C([0, T], ℝ^{2}) is the Banach space of vector functions with continuous components and L_{1}([0, T], ℝ^{2}) is the usual Banach space of vector functions with Lebesgue integrable components.
Moreover, the signs ·, ≤, ≥, max, and min operations will be everywhere understood componentwise. Let us define the vector
for which the following estimate is true (cf. [5,16])
For z ∈ ℝ^{2 }of the form
and λ ∈ Λ we define the vector
where I_{2 }is the unit matrix of order 2. In the sequel, we use the following assumptions.
(A1) The function f : [0, T] × D → ℝ^{2 }is continuous.
(A2) The function f satisfies the following Lipschitz condition: there exists a nonnegative constant square matrix K of order 2 such that
(A3) The subset
is nonempty, where B(z, γ(z, λ)) := {u ∈ ℝ^{2 }: u  z ≤ γ(z, λ)}.
(A4) The greatest eigenvalue r(Q) of the nonnegative matrix
satisfies the inequality r(Q) < 1.
Remark 1. The history and possible improvements of the constant in the definition of Q can be found in [5,17,18].
We will use the auxiliary sequence {α_{m}} of continuous functions α_{m }= α_{m}(t), t ∈ [0, T], defined by
It is obvious that, in particular,
According to [[16], Lemma 4] or [[5], Lemma 2.4], we have the following estimates
3 Successive approximations and convergence analysis
To investigate the solution of the parameterized boundary value problem (6) let us introduce the sequence of functions defined by the recurrence relation
where d(λ) = col(λ, 0) and x_{0}(t, z, λ) = z, z ∈ D_{γ}. Let us note that for m = 0, 1, 2, ..., we have x_{m}(t, z, λ) = col (x_{m,1}(t, z, λ), x_{m,2}(t, z, λ)). Moreover, all the functions x_{m }= x_{m}(t, z, λ) are continuously differentiable and satisfy the initial condition x_{m}(0, z, λ) = z as well as the boundary conditions in (6).
Let us establish the uniform convergence of the sequence (14) and the relation of the limit function to the solution of some additively modified boundary value problem.
Theorem 2. Let the assumptions (A1)(A4) be satisfied. Then for all z ∈ D_{γ }and λ ∈ Λ, the following statements hold
1. The sequence {x_{m}} converges uniformly in t ∈ [0, T] to the limit function
which satisfies the initial condition x*(0, z, λ) = z and the boundary conditions in (6).
2. For all t ∈ [0, T], the limit function x* satisfies the identity
Moreover, x* is continuously differentiable and it is a unique solution of the Cauchy problem for the additively modified differential equation
3. The following error estimate holds
Remark 3. We emphasize that the first component of the vector z is fixed and coincide with the value of x_{1}(0) in the first boundary condition in (1), while its second component z_{2 }is considered as free parameter. Thus, the expression "for all z", actually means "for all z_{2}".
Proof (of Theorem 2). First, we show that for all (t, z, λ) ∈ [0, T] × D_{γ }× Λ and m ∈ ℕ, all functions x_{m }= x_{m}(t, z, λ) belong to D. Indeed, using the estimate in [[19], Lemma 2], an arbitrary continuous function u : [0, T] → ℝ satisfies
Thus, we have
Therefore, we conclude that x_{1}(t, z, λ) ∈ D, whenever (t, z, λ) ∈ [0, T] × D_{γ }× Λ. By induction, we obtain that for all m ∈ ℕ, we have
i.e., all functions x_{m }are also contained in D.
For m = 0, 1, 2, ..., let us define
Due to the assumption (A2), we have
Relation (19) yields
and thus using (20), we obtain
By induction, we obtain for m = 1, 2, ... that
Using (12) and (13), we have
and thus, for all (t, z, λ) ∈ [0, T] × D_{γ }× Λ and j, m ∈ ℕ, we obtain
Due to the assumption (A4), the sequence {Q^{m}} converges to the zero matrix for m → +∞. Hence, (21) implies that {x_{m}} is a Cauchy sequence in the Banach space C([0, T], ℝ^{2}) and therefore, the limit function x* = x*(t, z, λ) exists. Passing to the limit for j → +∞ in (21), we obtain the final error estimate (17).
The limit function x* satisfies the initial condition x*(0, z, λ) = z as well as the boundary conditions in (6), since these conditions are satisfied by all functions x_{m }= x_{m}(t, z, λ) of the sequence {x_{m}}. Passing to the limit in the recurrence relation (14) for x_{m}, we show that the limit function x* satisfies the identity (15). If we differentiate this identity, we obtain that x* is a unique solution of the Cauchy problem (16).
Let us find a relation of the limit function x* = x*(t, z, λ) of the sequence {x_{m}} and the solution of the parameterized boundary value problem (6). For this purpose, let us define the function Δ : ℝ^{2 }→ ℝ^{2}
Theorem 4. Let the assumptions (A1)(A4) be satisfied. The limit function x* of the sequence {x_{m}} is a solution of the boundary value problem (6) if and only if the value of the vector parameters z ∈ D_{γ }and λ ∈ Λ are such that
Proof. It is sufficient to apply Theorem 2 and notice that the equation in (16) coincides with the original equation in (6) if and only if the relation Δ(z, λ) = 0 holds.
Remark 5. The function Δ = Δ(z, λ) is called the determining function and the equation Δ(z, λ) = 0 is called the determining equation, because it determines the values of the unknown parameters z ∈ D_{γ }and λ ∈ Λ involved in the recurrence relation (14).
4 Properties of the limit function and the existence theorem
Let us investigate some properties of the limit function x* of the sequence {x_{m}} and the determining function Δ.
Lemma 6. Under the assumptions (A1)(A4), the limit function x* satisfies the following Lipschitz condition for all t ∈ [0, T], all z, y ∈ D_{γ }and λ ∈ Λ
Proof. Using the assumption (A2), we obtain
Similarly, using the above estimate, we have
and by induction, we obtain
Using the estimates in (13), we get
and passing to the limit for m → +∞, due to the assumption (A4), we obtain the final inequality
Lemma 7. Under the assumptions (A1)(A4), the determining function Δ is well defined and bounded in D_{γ }× Λ. Furthermore, it satisfies the following Lipschitz condition for all z, y ∈ D_{γ }and λ ∈ Λ
Proof. It follows from Theorem 2 that the limit function x* of the sequence {x_{m}} exists and is continuously differentiable in D_{γ }× Λ. Therefore, Δ is bounded and the assumption (A2) implies
Using Lemma 6 and taking into account that , we get
Let us define the mth approximate determining function
which has the following property.
Lemma 8. Let the assumptions (A1)(A4) be satisfied. Then for any z ∈ D_{γ}, λ ∈ Λ and m ∈ ℕ,
Proof. Using the assumption (A2) and (17), we have
Let us introduce the relation f_{1 }⊳_{Ω }f_{2 }of two vector functions f_{1 }= f_{1}(x) and f_{2 }= f_{2}(x), which means that for all x ∈ Ω, at least one component of f_{1}(x) is greater then the corresponding component of f_{2}(x).
Definition 9. Let Ω be an arbitrary nonempty set. For any pair of functions
we write f_{1 }⊳_{Ω }f_{2 }if and only if there exists a function k : Ω → {1, 2} such that for all x ∈ Ω
The following statement provides sufficient conditions for the solvability of the boundary value problem (6).
Theorem 10. Let the assumptions (A1)(A4) be satisfied. Moreover, let there exist m ∈ ℕ and nonempty set Ω ⊂ D_{γ }× Λ such that the approximate determining function Δ_{m }satisfies
and the Brouwer degree of Δ_{m }over Ω with respect to 0 satisfies
Then, there exists a pair (z*, λ*) ∈ Ω such that
and the corresponding limit function x* = x*(t, z*, λ*) of the sequence {x_{m}} solves the boundary value problem (6).
Proof. Let us introduce the mapping P : [0, 1] × Ω → ℝ^{2}
Since the mappings (z, λ) ↦ Δ(z, λ) and (z, λ) ↦ Δ_{m }(z, λ) are continuous due to the continuity of x_{m}, x* and f, the mapping P is continuous as well. Moreover, using Lemma 8 and (23), we have
for all Θ ∈ [0, 1] and (z, λ) ∈ ∂Ω. Thus, the mapping P is the admissible homotopy connecting Δ_{m }and Δ and the Brouwer degrees deg(Δ, Ω, 0) and deg (Δ_{m}, Ω, 0) are well defined. The invariance property of the Brouwer degree under homotopy implies that
The assumption (24) then guarantees the existence of (z*, λ*) ∈ Ω such that
Applying Theorem 4, we obtain that the limit function x* = x* (t, z*, λ*) of the sequence {x_{m}} is the solution of the boundary value problem (6).
5 Polynomial successive approximations
In order to make the computations of x_{m }possible or more easier, we give a justification of a polynomial version of the iterative scheme (14). At first, we recall some results of the theory of approximations in [20].
We denote by H_{q }a set of all polynomials of degree not higher than q and by E_{q}(f, P_{q}) the deviation of the function f from the polynomial P_{q }∈ H_{q}
There exists a unique polynomial for which
This polynomial is said to be a polynomial of the best uniform approximation of f in H_{q }and the number E_{q}(f) is called the error of the best uniform approximation. It is known that
Definition 11. Let f : [0, T] → ℝ be a uniformly continuous function and δ be a positive real number. Then we define the modulus of continuity of f as
where the supremum is taken over all t, s ∈ [0, T] for which t  s ≤ δ.
Let us note that the modulus of continuity ω(f, δ) is a continuous nondecreasing function of the variable δ, such that
Definition 12. We say that the function f : [0, T] → ℝ satisfies the Dini condition if
Let us note that, e.g., all αHölder continuous functions on [0, T] with 0 < α ≤ 1 satisfy the Dini condition.
For a given function f, let us denote by f^{q }the interpolation Chebyshev polynomial of degree q, q ∈ ℕ, which satisfies
where t_{i }are the Chebyshev interpolation nodes in the interval [0, T]
Lemma 13 (see [20]). If the function f satisfies the Dini condition, then the sequence {f^{q}} of the corresponding interpolation Chebyshev polynomials converges uniformly on [0, T] to f and the following estimate holds
for all t ∈ [0, T].
Let us introduce the sufficiency for the Dini condition for a composite function F(t) = f(t, x(t)).
Lemma 14. Let the function f = f(t, x) satisfy the Dini condition with respect to t ∈ [0, T] and the Lipschitz condition with respect to x ∈ D. Then for any continuously differentiable function x = x(t), t ∈ [0, T], with the values in D, the composite function F(t) = f(t, x(t)) satisfies the Dini condition in the interval [0, T].
Proof. Taking into account the Lipschitz condition, we obtain
Since f and x both satisfy the Dini condition, we conclude that F satisfies the Dini condition as well.
For a given function f = f(t, z), let us define
Using Lemmas 13 and 14, the function F(t) = f(t, x(t)) and its interpolation Chebyshev polynomial F^{q}(t) = f^{q}(t, x(t)) satisfy
Let us note that and the sequence {F^{q}} uniformly converges to the function F on the interval [0, T].
Remark 15. In the case of vector functions f, the error of the best uniform approximation E_{q}(f), the modulus of continuity ω(f, δ) and L_{q }are vectors as well. The Dini condition and the construction of the corresponding Chebyshev polynomials are understood componentwise.
Let us return again to the boundary value problem (6) considered in the domain [0, T] × D × Λ. To investigate the solution of the parameterized boundary value problem (6), instead of (14), we introduce the sequence of vector polynomials of degree (q + 1)
where is the vector of interpolation Chebyshev polynomial of degree q corresponding to f . Let us point out that the coefficients of the interpolation polynomials depend on the parameters z and λ. Moreover, all the functions are continuously differentiable and satisfy the initial condition as well as the boundary conditions in (6).
Let us define the domain
where
Theorem 16. Let the assumptions (A1)(A4) be satisfied with instead of D_{γ}. Then for all , λ ∈ Λ, the following statements hold
1. The sequence converges uniformly in t ∈ [0, T] to the limit function
which satisfies the initial condition x*(0, z, λ) = z and the boundary conditions in (6).
2. The following error estimate holds
Proof. We show that for all and m ∈ ℕ, all functions belong to D. Similarly as in the proof of Theorem 2, we have
Therefore, we conclude that , whenever . By induction, we obtain that for all m ∈ ℕ, we have
i.e., all functions are also contained in D.
For j = 1, 2, ..., m and for all , we estimate
Taking into account that
and using the assumption (A2) and the estimate (26), we get
In particular, for j = 1 and j = 2 we have
and by induction
Using (12) and (13), we get
and due to the assumption (A4), we obtain
By Theorem 2, we can write
Recall that the sequence {Q^{m}} converges to the zero matrix for m → +∞ and L_{q }tends to the zero vector for q → +∞, which implies immediately that the sequence converges uniformly to x* on [0, T].
Let us define the mth approximate polynomial determining function
Lemma 17. Let the assumptions (A1)(A4) be satisfied with instead of D_{γ}. Then, for all , λ ∈ Λ and m ∈ ℕ,
Proof. Due to the assumption (A2), (26) and the error estimate (28), we get
Theorem 18. Let the assumptions (A1)(A4) be satisfied with instead of D_{γ}. Moreover, let there exist m ∈ ℕ and nonempty set Ω^{q }⊂ D_{γ }× Λ such that the approximate polynomial determining function satisfies
and the Brouwer degree of over Ω^{q }with respect to 0 satisfies
Then there exists a pair (z*, λ*) ∈ Ω^{q }such that
and the corresponding limit function x* = x*(t, z*, λ*) of the sequence solves the boundary value problem (6).
Proof. Using the same steps as in the proof of Theorem 10, we construct the admissible homotopy P_{q }: [0, 1] × Ω^{q }→ ℝ^{2}
and we get
The assumption (31) then guarantees the existence of (z*, λ*) ∈ Ω^{q }such that
Applying Theorem 4, we obtain that the limit function x* = x* (t, z*, λ*) of the sequence is the solution of the boundary value problem (6).
6 Examples
In this section, we introduce three particular boundary value problems in the form of the system (1). The first problem is a linear one and enables us to build the sequence {x_{m}} directly by the recurrence relation (14). The second problem is nonlinear and it is impossible to integrate in (14) in a closed form. Thus, we use the Chebyshev interpolation of the integrand to construct the sequence of successive approximations also in this case. In the last example, we use again the polynomial version of presented method in order to approximate a solution of the nonlinear Dirichlet problem containing pLaplacian.
Example 1. Let us consider the following linear problem with the Dirichlet boundary conditions
which has a unique solution given in a closed form. Let us denote this solution by .
All the assumptions (A1)(A4) are satisfied. Let us take x_{0}(t, z, λ) = col(0, z_{2}) and construct the successive approximations X_{m }of the exact solution x° for m = 0, 1, 2, ... in the following way:
1. using the recurrence relation (14), evaluate x_{m+1}(t, z, λ),
2. solve the (m+1)th approximate determining equation Δ_{m+1}(z, λ) = 0 (system of two linear equations) and denote its solution by (z_{m+1}, λ_{m+1}),
3. define X_{m+1}(t) := x_{m+1}(t, z_{m+1}, λ_{m+1}).
Figure 1 contains both components X_{m,1 }and X_{m,2 }of approximations X_{m }for m = 1, 2, 11 and also their differences from the exact solution x°. Let us point out that the maxima of both components of X_{11}(t)  x°(t) for t ∈ 〈0, 1〉 are both less then 10^{9}.
Figure 1. The approximations X_{1}, X_{2 }and X_{11 }of the exact solution x° of (32).
Example 2. Let us investigate the nonlinear Dirichlet boundary value problem
which has a solution in the form . In this case, it is not possible to construct the sequence {X_{m}} of approximations using the iterative scheme as in the previous Example 1. Thus, we use the following polynomial version of the iterative scheme. Choose and for m = 0, 1, 2, ..., proceed the steps:
1. define and realize the Chebyshev interpolation,
2. define
3. define and realize the Chebyshev interpolation with parameters z and λ
4. solve the (m + 1)th approximate polynomial determining equation
and denote its solution by (z_{m+1}, λ_{m+1}),
In Figure 2, it is possible to compare polynomial approximations for q = 15, m = 1, 2, 11 and the corresponding differences from the exact solution x°. Let us note that the maximum of for t ∈ 〈0, 1〉 is less then 10^{7 }for j = 1 and less then 2·10^{7 }for j = 2.
Figure 2. The polynomial approximations of the exact solution x° of (33) for q = 15.
Example 3. Let us consider the following nonlinear problem with the Dirichlet boundary conditions
where p > 1, sin_{p }is the generalized sine function (see [21] for the definition),
Let us recall that sin_{p }is 2π_{p}periodic function on ℝ, which coincide with the sin function for p = 2. Moreover, the pair (sin_{p}(π_{p}t), (π_{p }cos_{p}(π_{p}t))^{p1}) is a solution of the following initial value problem
For ε = 0, the problem (34) reads as the Dirichlet boundary value problem with pLaplacian
For the problem (34), all the assumptions (A1)(A4) are satisfied in the linear case p = 2. If p ≠ 2, then there exist bounded domains D = 〈a_{1}, a_{2}〉 × 〈b_{1}, b_{2}〉 for which the second assumption (A2) concerning the Lipschitz condition of f is not satisfied. Thus, we have to take into account the following additional assumptions on D in order to satisfy the assumption (A2):
1. for 1 < p < 2, we have to ensure that ε < a_{1 }< a_{2 }or a_{1 }< a_{2 }< ε,
2. for 2 < p, we have to ensure that 0 < b_{1 }< b_{2 }or b_{1 }< b_{2 }< 0.
Let us note that for p > 2, the function , which appears in the first component of f, is not Lipschitz continuous on any interval containing zero. On the other hand, in the case of 1 < p < 2, the function g_{ε }in the second component of f is not Lipschitz continuous on any interval containing ε.
Thus, all the assumptions (A1)(A4) are satisfied if we take, e.g., , and . The polynomial version of the iterative scheme from the previous Example 2 is applicable in this case. Figure 3 shows for q = 15, m = 1, 2, 11. Their differences from the exact solution of the problem (34) are also available. Let us note that the maximum of for t ∈ 〈0, 1〉 is less then 7·10^{4 }for j = 1 and less then 6·10^{4 }for j = 2.
Figure 3. The approximations of the exact solution x° of (34) for , and q = 15.
Last Figure 4 shows for , ε = 0 and q = 15, m = 1, 2, 11. In spite of the fact that the assumption (A2) is not satisfied in this case, we obtain the polynomial approximation , which differs from the exact solution x° less than 4·10^{3 }in both components.
Figure 4. The approximations of the exact solution x° of (34) for , ε = 0 and q = 15.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed to each part of this study equally, and also read and approved the final manuscript.
Acknowledgements
The authors were partially supported by the Ministry of Education, Youth and Sports of the Czech Republic, grant no. ME09109 (Program KONTAKT) and by MSM 4977751301 (G. Holubová, P. Nečesal), and by the Hungarian Scientific Research Fund OTKA throught grant no. 68311 (M. Rontó). This research was carried out as part of the TAMOP4.2.1.B10/2/KONV20100001 project with support by the European Union, cofinanced by the European Social Fund. Finally, the research was supported in part by the Academy of Sciences of the Czech Republic through the Institutional Research Plan No.~AV0Z10190503 (A.~Ront\'o).
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