Solving a nonlinear system of second order boundary value problems

In this paper, a method is presented to obtain the analytical and approximate solutions of linear and nonlinear systems of second order boundary value problems. The analytical solution is represented in the form of series in the reproducing kernel space. In the mean time, the approximate solution un(x) is obtained by the n-term intercept of the analytical solution and is proved to converge to the analytical solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

Analytic solutions to a class of nonlinear infinite-delay-differential equations

The infinite-delay-differential equations (IDDEs) are studied and the analytic solution of a class of nonlinear IDDEs is presented based on the characteristics of the reproducing kernel space W₂[0,∞). Besides, the exact solution is represented in the form of series. It is proved that the n-term approximation un(x) converges to the exact solution u(x) of the IDDEs. Moreover, the approximate error of un(x) is monotone decreasing. The results of experiments showed that the proposed method in this paper is computationally efficient.     

Reproducing Kernel method for the solution of nonlinear hyperbolic telegraph equation with an integral condition

In this article, an iterative method is proposed for solving nonlinear hyperbolic telegraph equation with an integral condition. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n‐term approximation u_n(x, t) of the exact solution u(x, t) is obtained and is proved to converge to the exact solution. Moreover, the partial derivatives of u_n(x, t) are also convergent to the partial derivatives of u(x, t). Some numerical examples have been studied to demonstrate the accuracy of the present method. Results obtained by the method have been compared with the exact solution of each example and are found to be in good agreement with each other. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 867–886, 2011     

在再生核空间中求解非线性奇异两点边值问题

该文建立了一个迭代方法求解一类奇异两点边值问题(x_αu')'=f (x, u, u'), 其中x∈ (0,1),α< 2. 解的表达式是在再生核空间W₂[0,1]中以级数的形式给出的. 近似解一致收敛到准确解. 并且, 误差是单调下降的. 最后通过一些数值算例论述了所提方法的正确性与有效性.     

Solving singular two-point boundary value problem in reproducing kernel space

In this paper, we present a new method for solving singular two-point boundary value problem for certain ordinary differential equation having singular coefficients. Its exact solution is represented in the form of series in reproducing kernel space. In the mean time, the n  -term approximation _un(x)$u_n(x)$ to the exact solution u(x)$u(x)$ is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.     

Numerical method for Stokes' first problem for a heated generalized second grade fluid with fractional derivative

By using reproducing kernel theorem, a new method for solving a class of partial differential equation with fractional derivatives is proposed. We give the exact solution by a series form in the reproducing kernel space, and the n ‐term approximate solution is obtained from truncating the series. Moreover, the approximate solution, its integer order derivatives and fractional derivatives are uniformly convergent. The method is easy to implement and is verified by the numerical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1599–1609, 2011     

Constructive approximation of solution for fourth‐order nonlinear boundary value problems

In this paper, we obtain a sequence of approximate solution converging uniformly to the exact solution of a class of fourth‐order nonlinear boundary value problems. Its exact solution is represented in the form of series in the reproducing kernel space. The n‐term approximation u_n(x) is proved to converge to the exact solution u(x). Moreover, the derivatives of u_n(x) are also convergent to the derivatives of u(x). Copyright © 2008 John Wiley & Sons, Ltd.     

Solving integro-differential equations with Cauchy kernel

A simple method for solving the Fredholm singular integro-differential equations with Cauchy kernel is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. The advantage of the approach lies in the fact that, on the one hand, by improving the definition of traditional inner product, the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method; on the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives. Some examples are displayed to demonstrate the validity and applicability of the proposed method.     

The exact solution of a class of Volterra integral equation with weakly singular kernel

In this paper, the weakly singular Volterra integral equations with an infinite set of solutions are investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation. The aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory. Applying weighted integral, we provide a new definition for reproducing kernel space and obtain reproducing kernel function. Using the good properties of reproducing kernel function, the only smooth solution is exactly expressed in the form of series. The n-term approximate solution is obtained by truncating the series. Meanwhile, we prove that the derivative of approximation converges to the derivative of exact solution uniformly. The final numerical examples compared with other methods show that the method is efficient.     

A Reproducing Kernel Hilbert Space Method for Solving Integro-Differential Equations of Fractional Order

In this article, we implement a relatively new analytical technique, the reproducing kernel Hilbert space method (RKHSM), for solving integro-differential equations of fractional order. The solution obtained by using the method takes the form of a convergent series with easily computable components. Two numerical examples are studied to demonstrate the accuracy of the present method. The present work shows the validity and great potential of the reproducing kernel Hilbert space method for solving linear and nonlinear integro-differential equations of fractional order.     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

Meshless and analytical solutions to the time-dependent advection-diffusion-reaction equation with variable coefficients and boundary conditions

A variety of physical problems in science may be expressed using the advection-diffusion-reaction (ADR) equation that covers heat transfer and transport of mass and chemicals into a porous or a nonporous media. In this paper, the meshless generalised reproducing kernel particle method (RKPM) is utilised to numerically solve the time-dependent ADR problem in a general n-dimensional space with variable coefficients and boundary conditions. A time-dependent Robin boundary condition is formulated and precisely enforced in a novel approach. The accuracy and robustness of the meshless solution is verified against finite element simulations and a general one-dimensional analytical solution obtained in this study.     

One-point pseudospectral collocation for the one-dimensional Bratu equation

The one-dimensional planar Bratu problem is u_xx + λ exp(u) = 0 subject to u(±1) = 0. Because there is an analytical solution, this problem has been widely used to test numerical and perturbative schemes. We show that over the entire lower branch, and most of the upper branch, the solution is well approximated by a parabola, u(x) ≈ u₀ (1 − x²) where u₀ is determined by collocation at a single point x = ξ. The collocation equation can be solved explicitly in terms of the Lambert W-function as u(0) ≈ −W(−λ(1 − ξ²)/2)/(1 − ξ²) where both real-valued branches of the W-function yield good approximations to the two branches of the Bratu function. We carefully analyze the consequences of the choice of ξ. We also analyze the rate of convergence of a series of even Chebyshev polynomials which extends the one-point approximation to arbitrary accuracy. The Bratu function is so smooth that it is actually poor for comparing methods because even a bad, inefficient algorithm is successful. It is, however, a solution so smooth that a numerical scheme (the collocation or pseudospectral method) yields an explicit, analytical approximation. We also fill some gaps in theory of the Bratu equation. We prove that the general solution can be written in terms of a single, parameter-free β(x) without knowledge of the explicit solution. The analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly. We give three overlapping perturbative approximations to the eigenrelation, allowing the analytical solution to be easily evaluated throughout the entire parameter space.     

Application of reproducing kernel method to third order three-point boundary value problems

This paper investigates the analytical approximate solutions of third order three-point boundary value problems using reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions. This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

Norm expansion along a zero variety

The reproducing kernel function of a weighted Bergman space over domains in Cd is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d=2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z₁=z₂ as subvariety) and the ball (with z₂=0 as subvariety), as well as for a weighted Bargmann-Fock space over C² (with the diagonal z₁=z₂ as subvariety).     

A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems

This paper presents a new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between our method and other methods for solving an open fourth-order boundary value problem presented by Scott and Watts. The method is also applied to a nonlinear fourth-order boundary value problem. The numerical results demonstrate that the new method is quite accurate and efficient for fourth-order boundary value problems.     

Spaces of Dirichlet series with the complete Pick property

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form \(k\left( {s,u} \right) = \sum {{a_n}} {n^{ - s - \overline u }}\), and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space H _d ² in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of H _d ² . Thus, a family of multiplier algebras of Dirichlet series is exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic as a reproducing kernel Hilbert space to H _d ² and when its multiplier algebra is isometrically isomorphic to Mult(H _d ² ).     

A best approximation for the solution of one‐dimensional variable‐coefficient Burgers' equation

In this article, an iterative method for the approximate solution to one‐dimensional variable‐coefficient Burgers' equation is proposed in the reproducing kernel space W_(3,2). It is proved that the approximation w_n(x,t) converges to the exact solution u(x,t) for any initial function w₀(x,t) ε W_(3,2), and the approximate solution is the best approximation under a complete normal orthogonal system . Moreover the derivatives of w_n(x,t) are also uniformly convergent to the derivatives of u(x,t).© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Stability of the solutions of one-sided variational problems with applications to the theory of plasticity

In the paper one investigates the one-sided variational problem of the form $$/u - u_0 / = min, u \in M,$$ where ¦·¦ is the norm in some Hilbert space H₀, ? is a nonempty convex set, closed in the metric of H₀, and u₀ is a given element of this space. The fundamental results are: 1) The solution of the problem (1) is stable relative to small perturbations of the data of this problem: the element u₀, the norm ¦·¦, and the set ? the concept of small perturbation is precisely formulated. 2) Assume that the set ? is defined by the formula where g is an element of some Hilbert space containing the space H₀, ||| · ||| is some seminorm, and a is a positive constant. Let H⁽ⁿ⁾ be a subspace of the space H₀ on all of whose elements the seminorm ||| · ||| is finite. If u_n is the approximate solution of the problem (1), obtained as the solution of the problem ¦u?u₀¦=min, u_n ∈ M ∩ H⁽ⁿ⁾, then ¦u_*?u_n¦=0 (e_n(u_*)), where u_* is the exact solution of the problem (1), while e_n(u_*) is the best approximation of u_* by the elements of the subspace H⁽ⁿ⁾. The given results are used in a series of problems regarding the elastoplastic state according to the Saint-Venant-Mises theory; one assume that for these problems the Haar-Karman variational principle holds.     

Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method

This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.