Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions

Page 1 of 20 RESEARCH Open Access Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions Andrei Rontó1, Miklos Rontó2, Gabriela Holubová3 and Petr Nečesal3* * Correspondence: pnecesal@kma. zcu.cz 3 Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic Full list of author information is available at the end of the article Abstract The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numerical-analytic 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, numerical-analytic method, Chebyshev interpolation polynomials 1 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 functional-differential equations. We refer, e.g., to the books [3-5], the articles [6-12], 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 integro-differential 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. © 2011 Rontó et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Page 2 of 20 In this article, we introduce the Chebyshev polynomial version of the known numerical-analytic 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 quasi-linear one containing the p-Laplace 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 ⎧ dx1 ⎪ ⎪ = f1 (t, x1 , x2 ), t ∈ (0, T), ⎪ ⎨ dt dx2 = f2 (t, x1 , x2 ), ⎪ ⎪ ⎪ ⎩ dt x1 (0) = x1 (T) = 0. In the vector form, we have ⎧ ⎨ dx = f (t, x), t ∈ (0, T), dt ⎩ Ax(0) + C1 x(T) = 0, (1) (2) where x = col(x1, x2), f(t, x) = col(f1(t, x), f2(t, x)) and     10 00 . A= , C1 = 00 10 Let the function f(t, x) be defined and continuous in the domain [0, T] × D, D = [−a1 , a1 ] × [a2 , b2 ] ⊂ R2 . (3) To avoid dealing with singular matrix C1 in (2), which does not enable us to express explicitly x(T), it is useful to carry out the following parametrization Page 3 of 20 x2 (T) = λ, (4) where λ ∈  ⊆ [a2 , b2 ]. (5) Thus, instead of (2) we use the equivalent problem with two-point parameterized boundary conditions ⎧ ⎨ dx = f (t, x), t ∈ (0, T), (6) dt ⎩ Ax(0) + Cx(T) = d(λ), x2 (T) = λ, where  C=  01 , 10 d(λ) = col(λ, 0). The two-point parameterized boundary conditions in (6) allow us to write x(T) = C−1 d(λ) − C−1 Ax(0), which will be used in the sequel for the construction of the iterative scheme. Throughout the text, C([0, T], ℝ2) is the Banach space of vector functions with continuous components and L1([0, T], ℝ2) 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  1 (7) δD (f ); = max f (t, u) − min f (t, u) , 2 (t,u)∈[0,T]×D (t,u)∈[0,T]×D for which the following estimate is true (cf. [5,16]) δD (f ) ≤ max |f (t, u)| . (t,u)∈[0,T]×D (8) For z Î ℝ2 of the form z = co1(0, z2 ), z2 ∈ [a2 , b2 ] ⊆ [a2 , b2 ] (9) and l Î Λ we define the vector γ : D ×  → R2+ γ = γ (z, λ) := T δD (f ) + |C−1 d(λ) − (C−1 A + I2 )z|, 2 (10) 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 ® ℝ2 is continuous. (A2) The function f satisfies the following Lipschitz condition: there exists a nonnegative constant square matrix K of order 2 such that ∀t ∈ [0, T]∀u, v ∈ D : |f (t (...truncated)