Collocation Method Research Articles

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

This paper presents a new method to numerically solve the nonlinear time-fractional generalized Kawahara equations (NTFGKE) with uniform initial boundary conditions (IBCs). A class of modified shifted eighth-kind Chebyshev polynomials (MSEKCPs) is introduced to satisfy the given IBCs. The proposed method is based on using the operational matrices (OMs) for the ordinary derivatives (ODs) and the fractional derivatives (FDs) of MSEKCPs. These OMs are employed together with the spectral collocation method (SCM). Our presented algorithm enables the extraction of efficient and accurate numerical solutions. The convergence of the suggested method and the error analysis have been developed. Three numerical examples are presented to demonstrate the applicability and accuracy of our algorithm. Some comparisons of the presented numerical results with other existing ones are offered to validate the efficiency and superiority of our approach. The presented tables and graphs demonstrate that the proposed approach produces approximate solutions with high accuracy.

Reinforcement learning for optimal control of linear impulsive systems with periodic impulses

This paper focuses on the finite- and infinite-horizon optimal control problems of linear impulsive systems with periodic impulses under the quadratic performance index. Necessary and sufficient conditions for the optimal impulsive system are derived in terms of hybrid Riccati equations by utilizing the variational method and the collocation method combined with a time-varying Lyapunov function. Different from the existing model-based impulsive control schemes, three reinforcement learning (RL)-based algorithms are proposed to solve the optimal impulsive controller and hybrid controller for the impulsive system without the exact knowledge of the system dynamics. The asymptotical stability of the impulsive control system and the convergence of the RL-based algorithms are proved rigorously. Finally, a numerical simulation illustrates the effectiveness of the proposed control methods.

A high-order B-spline collocation method for solving a class of nonlinear singular boundary value problems

A high-order B-spline collocation method for solving a class of nonlinear singular boundary value problems

An innovative Fibonacci wavelet collocation method for the numerical approximation of Emden-Fowler equations

This article presents a novel approach using the Fibonacci wavelet collocation method (FWCM) for the numerical solution of Emden-Fowler-type equations. The Emden-Fowler equations are a class of nonlinear differential equations that arise in various fields of science and engineering, particularly in astrophysics and fluid dynamics. Due to their nonlinear and singular nature, the conventional approaches to solving these equations encounter difficulties. This method is particularly effective in handling problems with singularities, as it adapts naturally to the local behaviour of the solution. Here, we introduced the Fibonacci wavelet collocation method as a powerful numerical technique for tackling Emden-Fowler-type equations. The Fibonacci wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main advantage of this approach lies in its ability to reduce the computational complexity associated with solving Emden-Fowler equations, resulting in accurate and efficient solutions. Comparative analyses with other established numerical methods reveal its superior accuracy and convergence rate performance. Further, several examples demonstrate the method's flexibility when dealing with different singularity levels. This paper contributes to numerical analysis by introducing the Fibonacci wavelet method as a robust tool for solving Emden-Fowler-type equations.

An Efficient Numerical Scheme for Fractional Order Mathematical Model of Cytosolic Calcium Ion in Astrocytes

The major aim of this article is to obtain the numerical solution of a fractional mathematical model with a nonsingular kernel for thrombin receptor activation in calcium signals using two numerical schemes based on the collocation techniques. We present the computational solution of the considered fractional model using the Laguerre collocation method (LCM) and Jacobi collocation method (JCM). An operational matrix of the fractional order derivative in the Caputo sense is needed for the recommended approach. The computational scheme converts fractional differential equations (FDEs) into an algebraic set of equations using the collocation method. The technique is used more quickly and successfully than in other existing schemes. A comparison between LCM and JCM is also presented in the form of figures. We obtained very good results with a great agreement between both the schemes. Additionally, an error analysis of the suggested procedures is provided.

Employing CNPS and CPS approaches to calculate numerical roots of ninth-order linear and nonlinear boundary value problems

The extremely accurate findings of ninth-order linear and nonlinear BVPs are described in this paper. CNPS and CPS are used to find out the numerical roots of linear and nonlinear, ninth-order BVPs. The proposed methods transform the ninth-order BVPs into a system of linear equations. The algorithms developed in this study not only provide approximate solutions to the BVPs using CNPS and CPS, but they also estimate the derivatives from the first to the ninth-order of the analytical solution simultaneously. These approaches are implemented across five diverse problems, showcasing the effectiveness of these techniques by utilizing step sizes of [Formula: see text] and [Formula: see text]. In order to gauge the accuracy of the methods, a comparison is drawn between the outcomes and the exact solutions, which are then presented in tabular and graphical format. The precision of these techniques is demonstrated through a detailed investigation and is found to be superior to the CBS, the Petrov–Galerkin Method (PGM) using Splines functions as basis functions and Septic B Splines functions as weight functions, the collocation method for ninth-order BVPs by Quintic B Splines and Sextic B Splines as evidenced by the comparison of AEs of CPS and CNPS with these alternative methods.

Mathematical analysis of a combustible viscoelastic material in a cylindrical channel taking into account induced electric field: A spectral approach

Abstract In spite of the enormous applications of heating combustible materials due to exothermic chemical reactions, scientists and engineers still face a problem with these materials’ tendency to self-ignition, which can result in significant property damage if serious precautions are not taken. Therefore, the thermal decomposition of combustible viscoelastic material in a cylindrical channel is investigated in this study. With a third-order constitutive model considered for viscoelastic flow behavior, momentum and energy-balanced governing equations are provided. The chemical reaction of the material is assumed exothermic and thus follows Arrhenius’s kinetics. A numerical solution is provided for the boundary value problem via the bivariate spectral collocation method, and the impact of kinetics parameters on the combustible viscoelastic material is investigated. Our findings show that self-ignition is delayed with higher values of activation energy parameter ( ε \varepsilon ) and third-order parameter ( γ \gamma ), as well as lower values of magnetic field parameter ( M M ), current density parameter ( δ \delta ), and electrical conductivity exponent ( m m ).

Convective heat mechanism in Williamson nanoliquid over an escalating surface through an interface with viscous heating

This analysis explains the magneto-hydrodynamic flow on Williamson nanofluids previous stretching surface surrounded by the permeable media. The apt magnetic field was suggested for the angle of the axial direction of the flow. Anyhow, this flow phenomenon was characterized into the added heat source/sink and conjunction of radiating heat. The impacts of convective heating and viscous heating by expanding surface were again the significant feature of the analysis. This originality arises by the combination of the cross-diffusion effects of reverse behavior on the thermophoresis and Brownian motion. This form sketched into the aforesaid phenomenon was modified into the nonlinear ordinary form by the appropriate assumptions on comparison transformations. Therefore, the sets of equations were controlled for the numerical access using Lobatto-IIIa collocation method applicable to this Matlab bvp4c shooting process. This parametric performance of many components about their statistical values was given numerical imitations graphically by the rate coefficients in tabular forms. The validation and the compliance of the current result were acquired by the past study on the specific case. Further, the significant results of this analysis were: This non-Newtonian Williamson parameter combination of that magnetizing property diminishes the fluid velocities. In addition, the important influence of both viscosity parameter and radiation parameter of heating process was noted.

Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation

Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation

Impact of wall-slip on the soluto-marangoni instability in a two-fluid system in a channel–Creeping flow scenario

The role of wall-slip on the soluto-Marangoni instability that emerges in a two-fluid creeping two-dimensional Poiseuille flow is assessed within the framework of Orr-Sommerfeld analysis and species-transport system. Different wall-slip scenarios, namely S1 (slip at the upper wall), S2 (slip at the lower wall), S1S2 (symmetric slip at the upper and lower walls) are considered. The system instability is characterized by identifying unstable and stable zones in the m-n (viscosity ratio - depth ratio) plane for each of the slip conditions at the wall using asymptotic analysis for long-wave disturbances. Numerical simulations based on Chebyshev spectral collocation method reveals the emergence of two modes (M1 and M2) and corresponding stability features are displayed for typical values of Marangoni numbers, n, m and slip lengths. The occurrence of short-wave modes and the features of exchanging dominance between the short-wave modes in the presence of wall-slip for different slip condition are elucidated and the physical mechanism responsible for triggering the instability in the system are revealed through energy budget analysis. The channel system hosting less viscous fluid adjacent to the upper wall demonstrates interesting characteristics such as (among other things): stabilizing/destabilizing the emerging M1/M2 modes in system with no-slip by imposing S1/S2/S1S2 wall conditions and increasing the slip length; exchange of dominance of unstable modes beyond a threshold slip length for S1 slip type. The study thus presents possible means of stabilizing/destabilizing a creeping channel flow system by designing the walls of the channel with appropriate slip type relevant to the application.

Mittag-Leffler kernel-based oversampling collocation method for fractional initial value problems with contaminated data

Mittag-Leffler RKFs were introduced by Rosenfeld et al. based on the RKFs, a numerical approach called kernelized ABM was developed for solving fractional initial value problems (FIVPs). However, the accuracy of the obtained approximate solution degraded when the fractional order tends to small values. By employing the Mittag-Leffler RKFs, we develop an oversampling collocation technique for Caputo FIVPs with contaminated data. The method can yield higher accurate approximated solution even though the order of the considered FIVPs is small. Also, the approach can improve the numerical instabilities and reduce the influences of noise data.

Numerical assessment of hyperbolic type double interface problems via Haar wavelets

In this manuscript, we have addressed wave propagation challenges within heterogeneous media and hyperbolic interface model. A hybrid numerical approach is introduced for solving these problems, combining the finite difference method and Haar wavelet collocation method. The method entails approximating the second-order space partial derivative through a truncated Haar wavelet series, while the temporal derivative is approximated using finite difference method. For linear hyperbolic interface model, the resulting algebraic systems are solved using the Gauss elimination technique. In the case of non-linear problems, the nonlinearity is addressed through the quasi-Newton linearization formula. To assess the accuracy of the proposed technique, we compute the computational rate of convergence, root mean square errors and maximum absolute errors employing various collocation points. The proposed method perform very well and produces a stable solution if sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, whereas, the other existing method loses its accuracy in such cases. The numerical experiments, stability and rate of convergence both theoretical and computational, confirm the accuracy and diverse applicability of the method.

On the existence, uniqueness, stability, and numerical aspects for a novel mathematical model of HIV/AIDS transmission by a fractal fractional order derivative

In this study, we explore a mathematical model of the transmission of HIV/AIDS. The model incorporates a fractal fractional order derivative with a power-law type kernel. We prove the existence and uniqueness of a solution for the model and establish the stability conditions by employing Banach’s contraction principle and a generalized α-ψ-Geraghty type contraction. We perform stability analysis based on the Ulam–Hyers concept. To calculate the approximate solution, we utilize Gegenbauer polynomials via the spectral collocation method. The presented model includes two fractal and fractional order derivatives. The influence of the fractional and fractal derivatives on the outbreak of HIV is investigated by utilizing real data from the Cape Verde Islands in 1987–2014.

Three-step Collocation Method for Solution of Third Derivative Initial Value Problem

In this research paper, we introduce a novel three-step block method designed to directly tackle third-order initial value problems. This method is crafted through an interpolation and collocation approach, leveraging power series analysis. We conduct a comprehensive examination of the proposed method, ensuring it meets all requisite conditions for rigorous analysis. To assess its efficacy and validity, we employ the method on both highly stiff linear and nonlinear initial value problems, juxtaposing our findings with established approaches in the literature. Our results highlight that the new method exhibits faster convergence compared to existing methods, underscoring its superior performance. Initial value problems, Interpolation and collocation procedure, Three step, Third order

A Numerical Method for the Solution of Linear Volterra Integral Equations

The goal of this paper is to present a numerical method for solving the Volterra integral equations using the collocation approximation method with a power series polynomial. The modeled problem is converted into an algebraic equation system and solved using the standard collocation points. After establishing the approach's uniqueness and convergence, numerical examples were used to evaluate its effectiveness. The results show that the method outperforms others. Approximate solution, Integral equations, Polynomial power series, Volterra

Irreversibility Analysis for Magnetized and Nonlinear Convection Flow of Carreau Fluid Past a Nonlinear Surface with Higher-Order Adherence Factor

Studying entropy generation allows scientists and engineers to optimize processes, design more efficient systems, and develop sustainable technologies. These studies aim towards minimizing energy losses, reducing environmental impacts, and advancing the frontiers of thermodynamics and energy science. As such, the current investigation expunges the dynamics of heat, flow, entropy and irreversibility analysis for a magnetized and dissipative Carreau fluid flow over a nonlinearly stretching sheet. The model is subject to thermal and electrical conductivity variations, higher-order adherence factors and nonlinear convective motion over an inclined nonlinear surface. Herein, the heat source in the flow system is considered space and temperature-dependent. The mathematical model presented in PDE systems is transformed to ODEs via similarity transformation. Numerically, the solutions to the distribution profiles are approximated via Chebyshev Collocation Method (CCM) with the pseudospectra (differentiation matrix) technique. In the limited case, the results here benchmark existing literature. Our findings identified that thermal irreversibility dominated the entire flow system, further entropy generation and irreversibility enhancement requires more thermal energy. We as well elucidate that alteration of both heat source modulation affects the heat transfer coefficient while Joule heating effect generates inner energy which induces the Carreau fluid energy. Carreau fluid, Entropy generation Momentum slip, Nonlinear Convection, Pseudo spectral method

Determining Fast Battery Charging Profiles Using an Equivalent Circuit Model and a Direct Optimal Control Approach

This work used an electrical equivalent circuit model combined with a temperature model and computational optimal control methods to determine minimum time charging profiles for a lithium–ion battery. To effectively address the problem, an optimal control problem formulation and direct solution approach were adopted. The results showed that, in most cases studied, the solution to the battery’s fast-charging problem resembled the constant current–constant voltage (CC-CV) charging protocol, with the advantage being that our proposed approach optimally determined the switching time between the CC and CV phases, as well as the final time of the charging process. Considering path constraints related to the terminal voltage and temperature gradient between the cell core and case, the results also showed that additional rules could be incorporated into the protocol to protect the battery against under/over voltage-related damage and high-temperature differences between the core and its case. This work addressed several challenges and knowledge gaps, including emulating the CC-CV protocol using a multi-phase optimal control approach and direct collocation methods, and improving it by including efficiency and degradation terms in the objective function and safety constraints. To the authors’ knowledge, this is the first time the CC-CV protocol has been represented as the solution to a multi-phase optimal control problem.

Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples.

New Computational Methods Using Seventh Derivative Type for the Solution of First Order Initial Value Problems

In this research, a class of implicit block methods of a seventh derivative type are examined through interpolation and collocation techniques using finite power series as the basis function. The discrete schemes, which are implicit two-point block methods, are obtained by carefully and unevenly choose collocation points that ensure better methods’ stability via test. However, these schemes require seventh derivative functions unlike other existing numerical formulae. The new methods are found, investigated and proven to be convergent and A-stable. The implementation of methods is achieved by using Newton Raphson’s method. Experiments show the efficiency and accuracy of the developed formulae on different class of first-order initial value problems, including SIR, growth models and Prothero-Robinson oscillatory problem and with comparison to such existing methods. In addition, it is observed that uneven and positioning of collocation points greatly influence the efficiency and accuracy of numerical methods.

Assessment of Local Radial Basis Function Collocation Method for Diffusion Problems Structured with Multiquadrics and Polyharmonic Splines

This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and the initial value test is associated with the Dirichlet jump problem on a square. The spectra of the free parameters of the method, i.e., node density, timestep, shape parameter, etc., are analyzed in terms of the average error. It is found that the use of MQs is less stable compared to PHSs for irregular node arrangements. For MQs, the most suitable shape parameter is determined for multiple cases. The relationship of the shape parameter with the total number of nodes, average error, node scattering factor, and the number of nodes in the local subdomain is also provided. For regular node arrangements, MQs produce slightly more accurate results, while for irregular node arrangements, PHSs provide higher accuracy than MQs. PHSs are recommended for use in diffusion problems that require irregular node spacing.