Reproducing Kernel Hilbert Space Method Research Articles

A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition

Abstract In this work, we present a sophisticated operating algorithm, the reproducing kernel Hilbert space method, to investigate the approximate numerical solutions for a specific class of fractional Begley-Torvik equations (FBTE) equipped with fractional integral boundary condition. Such fractional integral boundary condition allows us to understand the non-local behavior of FBTE along with the given domain. The algorithm methodology depends on creating an orthonormal basis based on reproducing kernel function that satisfies the constraint boundary conditions so that the solution is finally formulated in the form of a uniformly convergent series in ϖ 3 [ a , b ] {\varpi }_{3}\left[a,b] . From a numerical point of view, some illustrative examples are provided to determine the appropriateness of algorithm design and the effect of using non-classical boundary conditions on the behavior of solutions approach.

Using a new implementation of reproducing kernel Hilbert space method to solve a system of second-order BVPs

Using a new implementation of reproducing kernel Hilbert space method to solve a system of second-order BVPs

Numerical Solutions of Some Classes of Partial Differential Equations of Fractional Order

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

Numerical simulation of time fractional Benjamin-Bona-Mahony-Burger equation describing propagation of long waves on the water surface

The unidirectional propagation of long waves in a certain nonlinear dispersive system is explained in the Benjamin-Bona-Mahony-Burger equation. The motivation of this manuscript is to examine the Benjamin-Bona-Mahony-Burger equation numerically and analytically by employing reproducing kernel functions. The implementation of reproducing kernel Hilbert space method produces an analytical solution in the form of a series representation, and after truncating the infinite series, an approximate solution is attained. With the support of numerical examples, the accuracy of this method has been shown through graphs and error tables. The convergence analysis of the introduced scheme has been explained by some useful theorems and lemmas. Finally, it has been observed that the approximate solution consistently converges to the exact solution with less computational overhead. The proposed method would be very effective in solving various fractional partial differential equation models.

Reproducing Kernel Method with Global Derivative

Ordinary differential equations describe several phenomena in different fields of engineering and physics. Our aim is to use the reproducing kernel Hilbert space method (RKHSM) to find a solution to some ordinary differential equations (ODEs) that are described by using the global derivative. In this research, we used the RKHSM to construct new numerical solutions for nonlinear ODEs with global derivative. The used method systematically produces analytic and approximate solutions in the series’s form. We tested three applications for showing the performance of the RKHSM.

Extension of the Reproducing Kernel Hilbert Space Method’s Application Range to Include Some Important Fractional Differential Equations

Fractional differential equations are becoming more and more indispensable for modeling real-life problems. Modeling and then analyzing these fractional differential equations assists researchers in comprehending and predicting the system they want to study. This is only conceivable when their solutions are available. However, the majority of fractional differential equations lack exact solutions, and even when they do, they cannot be assessed precisely. Therefore, in order to analyze the symmetry analysis and acquire approximate solutions, one must rely on numerical approaches. In order to solve several significant fractional differential equations numerically, this work presents an effective approach. This method’s versatility and simplicity are its key benefits. To verify the RKHSM’s applicability, the convergence analysis and error estimations related to it are discussed. We also provide the profiles of a variety of representative numerical solutions to the problem at hand. We validated the potential, reliability, and efficacy of the RKHSM by testing some examples.

An Efficient Approach for Solving Differential Equations in the Frame of a New Fractional Derivative Operator

Recently, a new fractional derivative operator has been introduced so that it presents the combination of the Riemann–Liouville integral and Caputo derivative. This paper aims to enhance the reproducing kernel Hilbert space method (RKHSM, for short) for solving certain fractional differential equations involving this new derivative. This is the first time that the application of the RKHSM is employed for solving some differential equations with the new operator. We illustrate the convergence analysis of the applicability and reliability of the suggested approaches. The results confirm that the RKHSM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed method.

Analysis of the Fractional Differential Equations Using Two Different Methods

Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods.

A modified reproducing Kernel Hilbert space method for solving fuzzy fractional integro-differential equations

The aim of this paper is to extend the application of the reproducing kernel Hilbert space method (RKHSM) to solve linear and non-linear fuzzy integro-differential equations of fractional order under Caputo's H-differentiability. The analytic and approximate solutions are given in series form in term of their parametric form in the space $W_2^2 [a,b] \bigoplus W_2^2 [a,b]$. Several examples are carried out to show the effectiveness and the absence of complexity of the proposed method

AN ITERATIVE METHOD FOR THIRD-ORDER BOUNDARY VALUE PROBLEMS

This article is intended to apply the reproducing kernel Hilbert space method (RKHSM) for solving a third order differential equation with multiple characteristics in a rectangular domain. The exact solution is expressed in form of series. The convergence of the iterative method to find the approximate solution is proven. 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 which are found to be in good agreement with each other.

Estimation and Hypothesis Test for Mean Curve with Functional Data by Reproducing Kernel Hilbert Space Methods, with Applications in Biostatistics

Functional data analysis has important applications in biomedical, health studies and other areas. In this paper, we develop a general framework for a mean curve estimation for functional data using a reproducing kernel Hilbert space (RKHS) and derive its asymptotic distribution theory. We also propose two statistics for testing the equality of mean curves from two populations and a mean curve belonging to some subspace, respectively. Simulation studies are conducted to evaluate the performance of the proposed method and are compared with the major existing methods, which shows that the proposed method has a better performance than the existing ones. The method is then illustrated with an analysis of the growth data from the National Growth and Health Study (NGHS) project sponsored by the NIH.

Application of a Machine Learning Algorithm for Evaluation of Stiff Fractional Modeling of Polytropic Gas Spheres and Electric Circuits

Fractional polytropic gas sphere problems and electrical engineering models typically simulated with interconnected circuits have numerous applications in physical, astrophysical phenomena, and thermionic currents. Generally, most of these models are singular-nonlinear, symmetric, and include time delay, which has increased attention to them among researchers. In this work, we explored deep neural networks (DNNs) with an optimization algorithm to calculate the approximate solutions for nonlinear fractional differential equations (NFDEs). The target data-driven design of the DNN-LM algorithm was further implemented on the fractional models to study the rigorous impact and symmetry of different parameters on RL, RC circuits, and polytropic gas spheres. The targeted data generated from the analytical and numerical approaches in the literature for different cases were utilized by the deep neural networks to predict the numerical solutions by minimizing the differences in mean square error using the Levenberg–Marquardt algorithm. The numerical solutions obtained by the designed technique were contrasted with the multi-step reproducing kernel Hilbert space method (MS-RKM), Laplace transformation method (LTM), and Padé approximations. The results demonstrate the accuracy of the design technique as the DNN-LM algorithm overlaps with the actual results with minimum percentage absolute errors that lie between 10−8 and 10−12. The extensive graphical and statistical analysis of the designed technique showed that the DNN-LM algorithm is dependable and facilitates the examination of higher-order nonlinear complex problems due to the flexibility of the DNN architecture and the effectiveness of the optimization procedure.

A reproducing kernel Hilbert space method for nonlinear partial differential equations: applications to physical equations

The partial differential equations (PDEs) describe several phenomena in wide fields of engineering and physics. The purpose of this paper is to employ the reproducing kernel Hilbert space method (RKHSM) in obtaining effective numerical solutions to nonlinear PDEs, which are arising in acoustic problems for a fluid flow. In this paper, the RKHSM is used to construct numerical solutions for PDEs which are found in physical problems such as sediment waves in plasma, sediment transport in rivers, shock waves, electric signals’ transmission along a cable, acoustic problems for a fluid flow, vibrating membrane, and vibrating string. The RKHSM systematically produces analytic and approximate solutions in the form of series. The convergence analysis and error estimations are discussed to prove the applicability theoretically. Three applications are tested to show the performance and efficiency of the used method. Computational results indicated a good agreement between the exact and numerical solutions.

Applications of the differential transformation to three-point singular boundary value problems for ordinary differential equations

The differential transform method is used to find numerical approximations of the solution to a class of certain nonlinear three-point singular boundary value problems. The method is based on Taylor's theorem. Coefficients of the Taylor series are determined by constructing a recurrence relation. To deal with the nonlinearity of the problems, the Faa di Bruno's formula containing the partial ordinary Bell polynomials is applied within the differential transform. The error estimation results are also presented. Four concrete problems are studied to show efficiency and reliability of the method. The obtained results are compared to other methods, e.g., reproducing kernel Hilbert space method.

Dynamical Analysis of Fractional Integro-Differential Equations

In this article, we solve fractional Integro differential equations (FIDEs) through a well-known technique known as the Chebyshev Pseudospectral method. In the Caputo manner, the fractional derivative is taken. The main advantage of the proposed technique is that it reduces such types of equations to linear or nonlinear algebraic equations. The acquired results demonstrate the accuracy and reliability of the current approach. The results are compared to those obtained by other approaches and the exact solution. Three test problems were used to demonstrate the effectiveness of the proposed technique. For different fractional orders, the results of the proposed technique are plotted. Plotting absolute error figures and comparing results to some existing solutions reveals the accuracy of the proposed technique. The comparison with the exact solution, hybrid Legendre polynomials, and block-pulse functions approach, Reproducing Kernel Hilbert Space method, Haar wavelet method, and Pseudo-operational matrix method confirm that Chebyshev Pseudospectral method is more accurate and straightforward as compared to other methods.

Enabling 5G indoor services for residential environment using VLC technology

Visible light communication (VLC) has emerged as a viable complement to traditional radio frequency (RF) based systems and as an enabler for high data rate communications for beyond-5G (B5G) indoor communication systems. In particular, the emergence of new B5G-based applications with quality of service (QoS) requirements and massive connectivity has recently led to research on the required service-levels and the development of improved physical (PHY) layer methods. As part of recent VLC standards development activities, the IEEE has formed the 802.11bb “Light Communications (LC) for Wireless Local Area Networking” standardization group. This paper investigates the network requirements of 5G indoor services such as virtual reality (VR) and high-definition (HD) video for residential environments using VLC. In this paper, we consider such typical VLC scenarios with additional impairments such as light-emitting diode (LED) nonlinearity and imperfect channel feedback, and propose hyperparameter-free mitigation techniques using Reproducing Kernel Hilbert Space (RKHS) methods. In this context, we also propose using a direct current biased optical orthogonal frequency division multiplexing (DCO-OFDM)-based adaptive VLC transmission method that uses precomputed bit error rate (BER) expressions for these RKHS-based detection methods and performs adaptive BER-based modulation-order switching. Simulations of channel impulse responses (CIRs) show that the adaptive transmission method provides significantly improved error rate performance, which makes it promising for high data rate VLC-based 5G indoor services.

A novel method for fractal-fractional differential equations

We consider the reproducing kernel Hilbert space method to construct numerical solutions for some basic fractional ordinary differential equations (FODEs) under fractal fractional derivative with the generalized Mittag–Leffler (M-L) kernel. Deriving the analytic and numerical solutions of this new class of differential equations are modern trends. To apply this method, we use reproducing kernel theory and two important Hilbert spaces. We provide three problems to illustrate our main results including the profiles of different representative approximate solutions. The computational results are compared with the exact solutions. The results obtained clearly show the effect of the fractal fractional derivative with the M-L kernel in the obtained outcomes. Meanwhile, the compatibility between the approximate and exact solutions confirms the applicability and superior performance of the method.

Reproducing kernel Hilbert space method for solving fractal fractional differential equations

Based on reproducing kernel theory, an analytical approach is considered to construct numerical solutions for some basic fractional ordinary differential equations (FODEs, for short) under fractal fractional derivative with the exponential decay kernel. For the first time, the implemented approach, namely reproducing kernel Hilbert space method (RKHSM), is proposed in terms of analytic and numerical fractal fractional solutions. Through the convergence analysis, we illustrate the high competency of the RKHSM. Our results are compared with the exact solutions, and they show us how the fractal-fractional derivative when the kernel is exponential decay affects the obtained outcomes. And, they also confirm the superior performance of the RKHSM.

A SEMI-ANALYTIC APPROACH FOR SOLVING FISHER’S REACTION-DIFFUSION EQUATION BY METHOD OF LINES USING REPRODUCING KERNEL HILBERT SPACE METHOD

Many nonlinear systems are described with the nonlinear Fisher’s reaction-diffusion equation. The purpose of this work is to propose the method of lines to find out the solution of the Fisher’s reaction-diffusion equation in one dimension with quadratic and cubic nonlinearity using reproducing kernel Hilbert space method. In this method, the partial derivatives of the space variable are discretized to get a system of ODEs in the time variable and then solve the system of ODEs using reproducing kernel Hilbert space method. Four test examples are given to demonstrate the technique’s efficacy and applicability. The results are compared with the exact and existing numerical solutions by calculating the error norms L2 and L∞ at various time levels. It has been discovered that the recommended approach is not only simple to use, but also produces superior outcomes

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and SomeError Analysis

In this paper, an efficient multi-step scheme is presented based on reproducing kernel Hilbert space (RKHS) theory for solving ordinary stiff differential systems. The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform form for a rapidly convergent series in the posed Sobolev space. Using the Gram-Schmidt orthogonality process, complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction. Consequently, by applying the standard RKHS method to each subinterval, approximate solutions that converge uniformly to the exact solutions are obtained. For this purpose, several numerical examples are tested to show proposed algorithm’s superiority, simplicity, and efficiency. The gained results indicate that the multi-step RKHS method is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.