Hilbert Space Method Research Articles

A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems

In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments are discussed and the approximate solutions are obtained to be more correct compared to the other obtained results in the literature.

Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense

Abstract This article is concerned with design and comprehensive study of a numerical approach for solving Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense. The proposed technique is using the reproducing kernel Hilbert space to approximate the solution to those class of fractional differential equations in the form of uniformly convergent series with respect to space variables. An accurate computational algorithm is presented to confirm the gained analysis. The relevant theorems and characterizations related to Riccati and Bernoulli equations are included among the reproducing kernel theory. Supplementary problems at the end of the article serve as a complete review of the utilized method and theories. The numerical consequences of the proposed approach are practically effective and impressive, whilst the utilized theories lay the foundation stone for addressing such issues. In light of the summary, conclusions and future recommendations are also provided.

Reproducing kernel Hilbert space method for the solutions of generalized Kuramoto–Sivashinsky equation

Reproducing kernel Hilbert space method is given for the solution of generalized Kuramoto–Sivashinsky equation. Reproducing kernel functions are obtained to get the solution of the generalized Kuramoto–Sivashinsky equation. Two examples have been introduced to prove the accuracy of the method. The obtained results show that the reproducing kernel Hilbert space method gives approximate analytical solutions which are very close to the exact solution of the generalized Kuramoto–Sivashinsky equation, which demonstrates the power of the proposed technique. We prove the efficiency of the reproducing kernel Hilbert space method in this paper.

Solving optimal control problems of Fredholm constraint optimality via the reproducing kernel Hilbert space method with error estimates and convergence analysis

Modeling of dynamic systems of optimal control problems (OCPs) is very important issue in applied sciences and engineering. In this analysis, by developed the reproducing kernel Hilbert space (RKHS) method within the calculus of variations, the OCP is solved with respect to initial conditions and Fredholm operator optimality. The solution methodology involves the use of two generalized Hilbert spaces (HSs) for both range and domain spaces. Numerical algorithm and procedure of solution are assembled compatibility with the optimal formulation of the problem. The convergence analysis and error rating of the utilized method are considered under some presumptions, which provide the theoretical structure behind the technique. The optimal profiles show the performance of the numerical solutions and the effect of the Fredholm operator in the optimal results. In this approach, computational simulations are introduced to delineate the suitability, straightforwardness, and relevance of the calculations created.

Multidimensional balanced credibility model with time effect and two level random common effects

This paper extends the multidimensional credibility model under balanced loss function to account for not only certain conditional dependence over time for claim amounts but also dependence across individual risks and over portfolio risks. By means of orthogonal projection method in Hilbert space of random vectors, the inhomogeneous and homogeneous multidimensional credibility estimators are derived, which generalize some well known existing results in credibility theory. Moreover, the unbiased estimators of structural parameters are investigated. Finally, we present a numerical example to show the existence of the multidimensional credibility estimators and their difference from the existing ones.

A reproducing kernel Hilbert space method for solving the nonlinear three‐point boundary value problems

AbstractThe main objective of this paper is to present a reproducing kernel Hilbert space method for computing solutions of nonlinear third‐order ordinary differential equations under multipoint boundary conditions. Some other aspects of the reproducing kernel Hilbert space method, such as convergence analysis and error estimate, are also discussed. The proposed method is a well‐performance technique for calculating the best approximate solution of nonlinear boundary value problems. Our numerical experiments validate our theoretical findings and demonstrate the desirable performance of the algorithm in terms of simplicity, accuracy, and efficiency.

Approximation scheme for handling coupled systems of differential equations within reproducing kernel method

This paper proposes an efficient numerical method to obtain analytical-numerical solutions for a class of system of boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method. The analytical solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential equations.

New styles of periodic solutions of the classical six-body problem

New styles of periodic solutions (weaving styles and chasing styles) of the classical equal mass six-body problem are shown to exist using a variational method and computational algorithm. These solutions with axial symmetry and collision-free of the bodies. We first prove existence of collision-free minimizers of the Lagrangian action functional on suitable class of symmetric loops, which yields collision-free periodic solutions of the equal mass six-body problem. Then we derive approximate analytical solutions for the new families of the weaving and chasing periodic solutions in the form of uniformly convergent series with easily computable components in the reproducing kernel Hilbert space, the derivation was performed by using the multi-step reproducing kernel Hilbert space method. Moreover, the new styles of periodic solutions are numerically illustrated.

Application of reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations

In this article, we propose a new method that determines an efficient numerical procedure for solving second-order fuzzy Volterra integro-differential equations in a Hilbert space. This method illustrates the ability of the reproducing kernel concept of the Hilbert space to approximate the solutions of second-order fuzzy Volterra integro-differential equations. Additionally, we discuss and derive the exact and approximate solutions in the form of Fourier series with effortlessly computable terms in the reproducing kernel Hilbert space W_{2}^{3} [ a,b ] oplus W_{2}^{.3} [ a,b ]. The convergence of the method is proven and its exactness is illustrated by three numerical examples.

The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations

The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find. The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity. From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs). The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization.

New Numerical Method for Solving Tenth Order Boundary Value Problems

In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables.

Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlevé equations in Hilbert space

In this analysis, by developed the reproducing kernel Hilbert space method within the Atangana–Baleanu fractional approach, the Bagley–Torvik and Painlevé equations are solved with respect to initial conditions of necessity. The solution methodology involves the use of two Hilbert spaces for both range and domain space. Numerical algorithm and procedure of solution are assembled compatibility with the cogent formulation of the problem. The method of solution of the utilized problems is studied under some hypotheses, which provides the theoretical structure behind the technique. The solutions profiles show the performance of the numerical solutions and the effect of the Atangana–Baleanu fractional approach in the obtained results. In this approach, computational simulations are introduced to delineate suitability, straightforwardness, and relevance of the calculations created.

Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator

In this article, by popularization of the reproducing kernel Hilbert space method in the sense of the Atangana–Baleanu fractional operator; set of first-order integrodifferential equations are solved with respect to Fredholm operator and initial conditions of optimality. The solvability approach based on use of the generalized Mittag–Leffler function in order to avoid nonsingular and nonlocal kernel functions appears in the classical fractional operator's. The procedure of solution is studied and described in details under some hypotheses, which provides the theoretical structure behind the utilized numerical method. Indeed, error analysis and convergence of numerical solution for the identification of the method is introduced in Hilbert space. In this analysis, some computational results and graphical representations are presented to demonstrated the suitability and portability of the utilized new fractional operator. Finally, the gained results reach to that; the utilized method is simple, direct, and powerful tool in finding numerical solutions for considered fractional equations.

Research on Fault Evolution Feature Extraction and Identification of Sun Gear Cracks in Planetary Gearbox Based on Volterra High-Order Kernel Generalized Frequency Response Graphic Analysis

In this paper, relying on the Volterra series nonlinear system model and the high-order kernel Hilbert’s reconstructed kernel fast solved algorithm, a fault feature frequency domain identification method based on Volterra high-order kernel generalized frequency response graph analysis is proposed. Firstly, the method uses the system input and output vibration signals to determine the Volterra model. Then, the Volterra high-order kernel function is solved quickly by reproducing kernel Hilbert space method, and the generalized frequency response function is used to identify the model. Finally, multidimensional high-order spectral pattern analysis is used to separate and extract the fault and degree characteristic information implied by frequency and phase coupling in the third-order kernel function. Following the theoretical approach, in the experimental part, this paper uses the planetary gearbox fault loading test rig to complete the data collection and establishes the Volterra experimental model through the measured data. The generalized frequency responses of each order kernel function are compared and analyzed and the capability of distinguishing and the adaptability of different order kernel functions for the degree of crack failure are discussed. The effects of changing the memory length of the Volterra model and the order of the kernel function on the recognition result are verified. The final experimental results show that the use of reproducing kernel Hilbert space can effectively avoid the dimension disaster problem that occurs in the high-order kernel solution process. Moreover, the third-order kernel can describe more intuitively the nonlinear system model under multifactor coupling than the second-order kernel. Finally, Volterra series model the third-order kernel’s generalized frequency response can effectively distinguish between nondefective and faulty gears, and its resolution is enough to distinguish the degree of failure of gear cracks.

Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates

Numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science, which means increased attention to fractional calculus. This paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in Hilbert space within appropriate kernel functions. The algorithm relies on the reproducing kernel Hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the Fourier coefficients of orthogonalization process. The Caputo fractional derivatives are introduced to address these issues. Moreover, the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions. The superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples. The outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method.

GENERALIZED JACOBI REPRODUCING KERNEL METHOD IN HILBERT SPACES FOR SOLVING THE BLACK-SCHOLES OPTION PRICING PROBLEM ARISING IN FINANCIAL MODELLING

Based on the reproducing kernel Hilbert space method, a new approach is proposed to approximate the solution of the Black-Scholes equation with Dirichlet boundary conditions and introduce the reproducing kernel properties in which the initial conditions of the problem are satisfied. Based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be constructed in the reproducing kernel spaces spanned by the generalized Jacobi basis polynomials. Some new error estimates for application of the method are established. The convergence analysis is established theoretically. The proposed method is successfully used for solving an option pricing problem arising in financial modelling. The ideas and techniques presented in this paper will be useful for solving many other problems.

Performance of multi-trait genomic selection for Eucalyptus robusta breeding program

In forest tree genetic improvement, multi-trait genomic selection (GS) may have advantages in improving the accuracy of the genotype estimation and shortening selection cycles. For the breeding of Eucalyptus robusta, one of the most exotic planted species in Madagascar, volume at 49 months (V49), total lignin (TL), and holo-cellulose (Holo) were considered. For GS, 2919 single nucleotide polymorphisms (SNP) were used with the genomic best linear unbiased predictor (GBLUP) method, which was as efficient as the reproducing kernel Hilbert space (RKHS) and elastic net methods (EN), but more adapted to multi-trait modeling. The efficiency of individual I model, including the genomic data, was much higher than the provenance effect P model. For example, with V49, mean goodness-of-fit was: rI_Full = 0.79, rP_Full = 0.37 for I and P, respectively. The prediction accuracies using the cross-validation procedure were lower for V49: rI = 0.29 rP = 0.28. The genetic gains resulting from the indexes associating (V49, TL) and (V49, Holo) were higher using I than for the P model; for V49, the relative genetic gain was 37 and 20%, respectively, with 5% of selection intensity. The single-trait approach was as efficient as the multi-trait approach given the weak correlations between V49 and TL or Holo. The I model also brings greater diversity: for V49 the number of provenances represented in a selected population was two and three with the P model, and 6 and 16 with the I model.

Linear convergence rates for extrapolated fixed point algorithms

ABSTRACTWe establish linear convergence rates for a certain class of extrapolated fixed point algorithms which are based on dynamic string-averaging methods in a real Hilbert space. This applies, in particular, to the extrapolated simultaneous and cyclic cutter methods. Our analysis covers the cases of both metric and subgradient projections.

Derivative Free Regularization Method for Nonlinear Ill-Posed Equations in Hilbert Scales

Abstract In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

Approximate Solutions of Second-Order Obstacle Problem of Fractional-order Using Reproducing Kernel Hilbert Space Method

In this paper, reproducing kernel Hilbert space method is employed to approximate the solution and its derivative to second-order boundary-value problems associated with obstacle, unilateral, and contact problems. The analytical solution is represented in the form of a series in the reproducing kernel Hilbert space. The 𝑛-term approximation is obtained and proved to converge to the analytical solution. Moreover, the reproducing kernel Hilbert space method is modified to obtain an approximate solution for the obstacle problem, replacing second derivative by Caputo fractional derivative 𝐷𝑎𝛼,𝛼∈(1,2). A numerical example is given to demonstrate the computation efficiency of the proposed method.