Collocation Method Research Articles

An innovative Vieta–Fibonacci wavelet collocation method for the numerical solution of three-component Brusselator reaction diffusion system of fractional order

An innovative Vieta–Fibonacci wavelet collocation method for the numerical solution of three-component Brusselator reaction diffusion system of fractional order

A wavelet collocation method for fractional Black–Scholes equations by subdiffusive model

AbstractIn this investigation, we propose a numerical method based on the fractional‐order generalized Taylor wavelets (FGTW) for option pricing and the fractional Black–Scholes equations. This model studies option pricing when the underlying asset has subdiffusive dynamics. By applying the regularized beta function, we give an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FGTW. An error analysis of the numerical scheme for estimating solutions is performed. Finally, we conduct a variety of numerical experiments for several standard examples from the literature to assess the efficiency of the proposed method.

High-efficiency scattering field modeling in metallic components: a machine-learning-inspired approach

We present a highly efficient method for characterizing the scattering field distribution of surface plasmon polaritons in metallic components by combining the eXtended Pseudospectral Frequency-Domain (X-PSFD) method with an iterative, machine-learning-inspired procedure. Shifting away from traditional matrix operations, we utilize the “Adam” optimizer—an effective and swift machine learning algorithm—to solve the scattering field distribution. Our method encompasses the derivation of the associated cost function and gradient differentiation of the field, leveraging spectral accuracy at Legendre collocation points in the Helmholtz equation. We refine the total-field/scattered-field (TF/SF) formulation within the X-PSFD framework for optimized incident field management and employ Chebyshev–Lagrange interpolation polynomials for rapid, accurate computation of broad-band results. To ensure global accuracy, we introduce unique physical boundary conditions at subdomain interfaces. Demonstrating our method’s robustness and computational efficiency, we model perfect electric conductors (PECs) and silver nanocylinders, and we apply our approach to analyze the excited electric field on subtly distorted metallic surfaces, particularly plasmonic structures, thereby validating its wide-ranging effectiveness.

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature.

Collocation methods for nonlinear Volterra integral equations with oscillatory kernel

This work is devoted to the numerical solution of second kind nonlinear Volterra integral equations with highly oscillatory kernel. We use a collocation appraoch by discretizing the oscillatory integrals in the collocation equation using a Filon-type quadrature rule. We investigate the convergence of the numerical method in terms of step length h and frequency ω. As h decreases, the suggested technique converges with order d, while its asymptotic order as the frequency increase, is at least 1 and may reach 2 in some cases. Numerical experiments validate theoretical results.

Magnetic field influence on Casson fluid flow in rotating convection

The stability of buoyant flow in an infinite extended vertical fluid layer bounded by impermeable conducting isothermal rigid walls, known as magnetic field influence on Casson fluid flow in rotating convection, is investigated. A system of governing equations (Navier–Stokes, heat, and induction ones) is solved with isothermal rigid boundary conditions. When the majority of electrically conducting fluids are extremely small, the stability of governing equations can be simplified by taking the smallness of magnetic Prandtl number into account. In linear stability, the Chebyshev collocation method is used to solve numerically the system of eigenvalue problems. The Casson fluid parameter, Chandrasekhar number, magnetic Prandtl number, and Taylor number all have destabilizing effects on the system's basic velocity and basic magnetic field, resulting in instability. The critical Rayleigh number (Rc), critical wave number (ac), and critical wave speed (cc) are calculated using the influence of governing parameters. The Casson fluid parameter and magnetic Prandtl number were found to stabilize stationary disturbances in neutral stability curves.

Quantile-based sequential optimization and reliability assessment method under random and interval hybrid uncertainty

Under random and interval hybrid uncertainties, solving hybrid reliability based design optimization (HRBDO) can acquire an optimal balance between structural performance and reliability. Since solving HRBDO includes a triple nested framework involving minimum analysis of performance function (PF), failure probability constraint analysis and design parameter optimization, the computational complexity of HRBDO is high, especially for dealing with complex structures. Therefore, a quantile-based sequential optimization and reliability assessment method (QSORA) is proposed for reducing the computational complexity of HRBDO. In the proposed QSORA for HRBDO, failure probability constraint is firstly transformed into minimum PF (MPF) quantile one corresponding to target failure probability. Then, approximating the difference between PF and its target quantile at current iteration by that at previous one, the failure probability constraint analysis is decoupled from the design parameter optimization. Moreover, by approximating the minimum point of the PF with respect to the interval input in the current iteration by that in the previous one, the minimum analysis of PF is separated from the design parameter optimization. By the separation of minimum analysis and failure probability constraint analysis from the design parameter optimization in the proposed QSORA, the triple nested framework of HRBDO is decoupled sequentially as the deterministic design optimization, the minimum analysis of the PF and the target MPF quantile estimation, and this way of reconstructing the HRBDO from the triple nested framework to three single-loop frameworks can significantly enhance the efficiency of solving HRBDO. Furthermore, the MPF quantile at the current design parameter is estimated by stochastic collocation based statistical moment method, in which the stochastic collocation method is employed to efficiently estimate the MPF moment to approximate the probability density function of MPF. The efficiency and accuracy of the QSORA are validated by four numerical and engineering examples finally.

[formula omitted]- and [formula omitted]-stability of Runge–Kutta collocation methods

This paper deals with stability of classical Runge–Kutta collocation methods. When such methods are embedded in linearly implicit methods as developed in [13] and used in [14] for the time integration of nonlinear evolution PDEs, the stability of these methods has to be adapted to this context. For this reason, we develop in this paper several notions of stability, that we analyze. We provide sufficient conditions that can be checked algorithmically using just the collocation points of the method to ensure that these stability notions are fulfilled by a given Runge–Kutta collocation method. We also introduce examples and counterexamples used in [14] to highlight the necessity of these stability conditions in this context.

Crack propagation simulation and overload fatigue life prediction via enhanced physics-informed neural networks

The fatigue crack growth simulation and life prediction of structures are implemented in this paper based on the physics-informed neural networks (PINNs). Firstly, the enhanced PINNs are proposed by introducing the crack tip asymptotic displacement fields, so that the crack tip stress intensity factors can be calculated accurately even when the number of collocation points is small and the distribution grid is regular. The enhanced PINNs essentially transform the solution of elastic body containing crack into the optimization of minimizing the constructed loss functions, and can invert the fracture parameters. Then, an automatic crack propagation simulation method is developed based on the enhanced PINNs. The network architecture and the overall node distribution can be unchanged during the crack propagation process, and only new crack surfaces need to be processed and corresponding loss functions need to be modified. Because the nodal refinement around crack-tip is not required, this simulation method is convenient and can accurately predict the mixed-mode crack propagation path. Finally, the fatigue crack growth life algorithm considering overload is developed, where the effect of each overload can be captured by the cycle-by-cycle method. Based on this algorithm, the retardation behavior can be characterized and the fatigue life of structure under the load spectrum with periodic overloads can be accurately predicted. The sufficient examples are given to verify the feasibility and accuracy of the method proposed in this paper.

Semi-analytic PINN methods for boundary layer problems in a rectangular domain

Singularly perturbed boundary value problems pose a significant challenge for their numerical approximations because of the presence of sharp boundary layers. These sharp boundary layers are responsible for the stiffness of solutions, which leads to large computational errors, if not properly handled. It is well-known that the classical numerical methods as well as the Physics-Informed Neural Networks (PINNs) require some special treatments near the boundary, e.g., using extensive mesh refinements or finer collocation points, in order to obtain an accurate approximate solution especially inside of the stiff boundary layer. In this article, we modify the PINNs and construct our new semi-analytic SL-PINNs suitable for singularly perturbed boundary value problems. Performing the boundary layer analysis, we first find the corrector functions describing the singular behavior of the stiff solutions inside boundary layers. Then we obtain the SL-PINN approximations of the singularly perturbed problems by embedding the explicit correctors in the structure of PINNs or by training the correctors together with the PINN approximations. Our numerical experiments confirm that our new SL-PINN methods produce stable and accurate approximations for stiff solutions.

Numerical solution of Fisher’s equation through the application of Haar wavelet collocation method

This research article presents algorithm that employs the Haar wavelet collocation method (HWCM) to solve the convection diffusion reaction (CDR) model, specifically Fisher’s equation (F.E.). The proposed method is suitable for addressing both linear and nonlinear problems. The Haar wavelet method is a numerical technique that utilizes wavelet basis functions to provide an approximation of the solution for the Fisher’s equations. Additionally, a forward difference scheme is utilized to approximate the time derivative while the highest order space derivative is replaced by Haar function. The performance of the technique is demonstrated through some examples investigated in this study. The maximum absolute errors are computed for varying numbers of collocation points. The HWCM provides superior accuracy, flexibility, convenience, and computational efficiency compared to alternative methods.

Exact Solution of Couple Burgers’ Equation using Cubic B-Spline Collocation Method for Fluid Suspension/Colloid under the Influence of Gravity

In this research, Cubic B-Spline plane method was used to solve numerically the one-dimensional Burger’s Equation with initial condition , boundary conditions; . The cubic trigonometric B-spline was used for interpolating the solutions at each time and using the Von-Neumann method to check the stability. The obtained numerical result showed that the method was efficient, robust and reliable for solving Burgers’ Equation accurately even involving high Reynolds numbers for which the exact solutions have failed.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

Entropy production on mixed convection stagnation point flow of nonlinear radiative fourth-grade hybrid nanofluid through a stretchable Riga surface

The purpose of the current study is to inspect entropy generation, mixed convective stagnation point flow, and thermal transfer features of nonlinear radiative fourth-grade hybrid nanofluid (NF) confined by a convectively heated Riga surface. The heat transport is examined with the existence of two disparate heat source modulations, variable thermal conductivity, and viscous dissipation. The original flow equations are first transmuted using appropriate transformations into non-dimensional ordinary differential equations, which are then solved via an overlapping grid-based spectral collocation scheme. The upshots of various pertinent parameters on velocity, temperature, entropy generation, and valuable engineering quantities are deliberated. Pivotal results obtained reveal that speedily flow of fluid and skin friction can be accelerated by strong magnetic force, rising mixed convection, and material parameters. Also, convective boundary conditions, along with nonlinear radiation and fluctuating thermal conductivity, are recommended for boosting fluid temperature, rates of entropy generation and heat transport. Thermal mechanism in hybrid NF is dominant over simple NF, which implies that performance of hybrid NF is better than that of simple NF. The outcomes of this study can be useful in enriching thermal performance of the working fluid, assisting in diagnosing causes of incompetency in thermal systems, and discovering suitable means of minimizing entropy generation with the intention of mitigating the loss of useful and scarce energy resources.

Generalized log orthogonal functions spectral collocation method for two dimensional weakly singular Volterra integral equations of the second kind

AbstractIn this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples.

Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

Stability analysis and numerical evaluations of a COVID-19 model with vaccination

A novel (nonlinear) mathematical model for the transmission of Coronavirus 19 (COVID-19) with eight compartments and considering the impact of vaccination is examined in this manuscript. The qualitative behavior of the system such as the boundedness of solutions, the basic reproduction number, and the stability of the equilibrium points is investigated in detail. Some domestic real data collected from the Kerman University of Medical Science (KUMC) is used to estimate the parameters of the proposed model. We predict the dynamical behavior of the system through numerical simulations based on a combined spectral matrix collocation methodology. In this respect, we first linearize the nonlinear system of equations by the method of quasilinearization (QLM). Hence, the shifted version of Chebyshev polynomials of the second kind (SCPSK) is utilized along with the domain-splitting strategy to acquire the solutions of the system over a long time interval. The uniform convergence and upper bound estimation of the SCPSK bases are proved in a rigorous manner. Moreover, the technique of residual error functions is used to testify the accuracy of the QLM-SCPSK method. The presented numerical results justify the robustness and good accuracy of the QLM-SCPSK technique. The achieved numerical orders of convergence indicate that the QLM-SCSK algorithm has exponential rate of convergence. Using the linearization technique in one hand and the domain-splitting strategy on the other hand, enable us to predict the behaviour of similar disease problems with high accuracy and maximum efficiency on an arbitrary domain of interest.

Non-Polynomial Collocation Spectral Scheme for Systems of Nonlinear Caputo–Hadamard Differential Equations

In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct a non-polynomial spectral collocation scheme, describe its effective implementation, and derive its convergence analysis in both L2 and L∞. In addition, we provide numerical results to support our theoretical analysis.

Reactant-Product Decoupling Technique Using the Intermediate Coordinate Method.

Although the reactant-product decoupling (RPD) technique was proposed over two decades ago, it remains an efficient approach for calculating product state-resolved information on some simple direct reactions using the quantum wave packet method. In the past, usually the RPD technique employed the collocation method to transform the wave function between reactant and product arrangements, which requires quite large computational efforts. In this work, the intermediate coordinate (IC) method is employed to realize the RPD technique. Numerical examples demonstrate that this new IC RPD (IRPD) technique has superior computational efficiency compared with the original method employing the collocation method. Especially, the new IRPD technique significantly saves disk space and computer memory. To illustrate the features of our new method, the total reaction probabilities of the H + H2, H + Br2, and F + H2 reactions with J = 0 and the differential cross sections of the H + H2 and F + H2 reactions at a series of collision energy are calculated and presented. With this efficient and effective new RPD technique, the Li + HF reaction, which involves sharp resonances with long-range wave functions in the van der Waals wells in both the reactant and product arrangements, is also calculated with several J at the product state-resolved level to reveal the ability of the RPD technique for describing resonance wave functions. With these numerical examples, it is found that, for the reaction with resonances, the RPD approach should be applied carefully. Otherwise, it is very possible that the resonances could disappear with the application of the RPD technique.