Collocation Method Research Articles

Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

This article introduces two numerical methods to address boundary value problems associated with secondorder and fractional differential equations. These methods employ two parameters related to shifted Gegenbauer polynomials as their basis functions. The process involves establishing a differentiation operational matrix for the shifted Gegenbauer polynomials. Subsequently, the initial/boundary value problems for ordinary and fractional differential equations are transformed into a system of equations through the Galerkin, collocation, and tau methods. The convergence analysis is ensured by leveraging theorems pertaining to the shifted Gegenbauer polynomials. To validate the accuracy of the approach, numerous numerical examples are presented.

Numerical approximation of the typhoid disease model via Genocchi wavelet collocation method

AbstractIn this paper, we have considered the fractional typhoid disease model and obtained the numerical approximation of the model via the innovative wavelet scheme called the Genocchi wavelet collocation method (GWCM) with the help of Caputo fractional derivative for the fractional order. The approach under consideration is a powerful tool for obtaining numerical solutions to fractional-order nonlinear differential equations. The GWCM approach yields accurate solutions that are very close to exact solutions for highly nonlinear problems by avoiding data rounding and just computing a few terms. The Genocchi wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main benefit of this method lies in its capability to reduce the computational complexity associated with solving systems of ODEs, resulting in accurate and efficient solutions. The results of the developed technique, the RK4 method, and the ND solver have been compared. The numerical outcomes demonstrate that the implemented technique is incredibly effective and precise for solving the Typhoid model of fractional order. This paper contributes to numerical analysis by introducing the Genocchi wavelet method as a robust tool for solving biological models.

Retrievals of aerosol optical depth over the western North Atlantic Ocean during ACTIVATE

Abstract. Aerosol optical depth was retrieved from two airborne remote sensing instruments, the Research Scanning Polarimeter (RSP) and Second Generation High Spectral Resolution Lidar (HSRL-2), during the National Aeronautics and Space Administration (NASA) Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE). The field campaign offers a unique opportunity to evaluate an extensive 3-year dataset under a wide range of meteorological conditions from two instruments on the same platform. However, a long-standing issue in atmospheric field studies is that there is a lack of reference datasets for properly validating field measurements and estimating their uncertainties. Here we address this issue by using the triple collocation method, in which a third collocated satellite dataset from the Moderate Resolution Imaging Spectroradiometer (MODIS) is introduced for comparison. HSRL-2 is found to provide a more accurate retrieval than RSP over the study region. The error standard deviation of HSRL-2 with respect to the ground truth is 0.027. Moreover, this approach enables us to develop a simple, yet efficient, quality control criterion for RSP data. The physical reasons for the differences in two retrievals are determined to be cloud contamination, aerosols near the surface, multiple aerosol layers, absorbing aerosols, non-spherical aerosols, and simplified retrieval assumptions. These results demonstrate the pathway for optimal aerosol retrievals by combining information from both lidars and polarimeters for future airborne and satellite missions.

Impact of a floating flexible plate on the stability of double-layered falling flow

The hydrodynamic stability behavior of a two-layer falling film is explored with a floating flexible plate on the top layer surface. The stress balance at the surface is modeled using a modified membrane equation. There is an insoluble surfactant at the liquid-liquid interface of the flow system. The linear instability of perturbation waves is captured by numerically solving the generalized Orr-Sommerfeld eigenvalue problem using the Chebyshev spectral collocation technique. Four different types of linear stability modes are identified: surface mode (SM), interface mode (IM), interface surfactant mode (ISM), and shear mode (SHM). The floating flexible plate has an inferior impact on all the existing modes in the longwave zone. However, a significant influence is noticed for the finite wave numbers. The surface and interface wave instabilities can be suppressed in the smaller wavenumber zone by imposing higher structural rigidity and uniform thickness of the flexible plate. The stabilizing nature of the surface mode becomes more powerful when the top layer viscosity is dominant. A new interfacial instability emerges when the top layer is less viscous than the lower one. At moderate Reynolds numbers, the behavior of interface mode is different in two different zones mr<1 and mr>1, where m and r denote the viscosity and density ratio, respectively. Further, the unstable shear modes induced by the top and bottom layers are detected under the low inclination angles with a strong inertia force. The occurrence of the shear modes requires the viscosity and density of the lower layer to be much higher than those of the upper layer. The influence of characteristic parameters of the flexible plate on the lower layer shear mode is not very sensitive. Finally, the competition between the different modes for dominance of stability boundaries is also discussed.

Numerical solutions of Schrödinger–Boussinesq system by orthogonal spline collocation method

This paper is concerned with the numerical solutions of Schrödinger–Boussinesq (SBq) system by an orthogonal spline collocation (OSC) discretization in space and Crank–Nicolson (CN) type approximation in time. By using the mathematical induction argument and standard energy method, the proposed CN+OSC scheme is proved to be unconditionally convergent at the order O(τ2+h4) with mesh-size h and time step τ in the discrete L2-norm. We devise a new computation method based on the orthogonal diagonalization techniques (ODT) to realize the proposed CN+OSC scheme. In order to compare the performance of ODT, we devise an alternating direction implicit (ADI) method to compute the CN+OSC scheme for high spatial dimension SBq system. As an alternative implementation, the new method ODT not only exhibits more accurate numerical results, but also demonstrates stronger invariance preserving ability. Numerical results are reported to verify the error estimates and the discrete conservation laws.

A novel perspective for simulations of the Modified Equal-Width Wave equation by cubic Hermite B-spline collocation method

In the current study, the Modified Equal-Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B-splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms L2 and L∞ as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.

Adaptive deep homogenization theory for periodic heterogeneous materials

We present an adaptive physics-informed deep homogenization neural network (DHN) approach to formulate a full-field micromechanics model for elastic and thermoelastic periodic arrays with different microstructures. The unit cell solution is approximated by fully connected multilayers via minimizing a loss function formulated in terms of the sum of residuals from the stress equilibrium and heat conduction partial differential equations (PDEs), together with interfacial traction-free or adiabatic boundary conditions. In comparison, periodicity boundary conditions are directly satisfied by introducing a network layer with sinusoidal functions. Fully trainable weights are applied on all collocation points, which are simultaneously trained alongside the network weights. Hence, the network automatically assigns higher weights to the collocation points in the vicinity of the interface (particularly challenging regions of the unit cell solution) in the loss function. This compels the neural networks to enhance their performance at these specific points. The accuracy of adaptive DHN is verified against the finite element and the elasticity solution respectively for elliptical and circular cylindrical pores/fibers. The advantage of the adaptive DHN over the original DHN technique is justified by considering locally irregular porous architecture where pore–pore interaction makes training the network particularly slow and hard to optimize.

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.

Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation

Abstract In the current study, we delve into examining the movement of a nanofluid within a Williamson boundary layer, focusing on the analysis of heat and mass transfer (HMT) processes. This particular flow occurs over a sheet that undergoes nonlinear stretching. A significant facet of this investigation involves the incorporation of both the magnetic field and the influence of viscous dissipation within the model. The sheet is situated within a porous medium, and this medium conforms to the Darcy model. Since more precise outcomes are still required, the model assumes that both fluid conductivity and viscosity change with temperature. In this research, we encounter a system of extremely nonlinear ordinary differential equations that are treated through a numerical technique, specifically by employing the spectral collocation method. Graphical representations are used to illustrate how the relevant parameters impact the nanoparticle volume fraction, velocity, and temperature profiles. The study involves the computation and analysis of the effect of physical parameters on the local Sherwood number, skin friction coefficient, and local Nusselt number. Specific significant findings emerging from the present study highlight that the rate of mass transfer is particularly influenced by the thermophoresis factor, porous parameter, and Williamson parameter, showing heightened effects, while conversely, the Brownian motion parameter demonstrates an opposing pattern. The results were computed and subjected to a comparison with earlier research, indicating a notable degree of conformity and accord.

Mixed Convection Flow of a Casson Fluid Past a Thin Needle

Mixed Convection Flow of a Casson Fluid Past a Thin Needle

Numerical methods for highly oscillatory Volterra integral equations with general oscillators

In this paper, collocation methods for linear Volterra integral equations with highly oscillatory trigonometric kernels are considered. Different from the cases with linear oscillator, we need a further action to obtain a practical scheme for general oscillators. To deal with it, the moment free Filon type method is employed in the exact collocation equation resulting to a fully discretized one. In order to conduct the convergence analysis, we firstly study the highly oscillatory integrals occurring in the error equations. Both the theoretical and numerical results show that our method not only converges with respect to step length but also has an asymptotic order which could be two. The asymptotic order indicates that the numerical results will become more accurate as the increase of frequency.

Quasinormal modes by improved matrix method and weighted residual method

Abstract This work discusses the Improved Matrix Method and Weighted Residual Method for studying the quasinormal modes (QNMs) of black holes. In the first method, by utilizing Jordan decomposition, the improved matrix method avoids the calculation of inverse matrices in the original matrix method. This significantly saves computational resources and time needed to compute the coefficient square matrices of the derivatives. In the second method, we illustrate the effectiveness of the weighted residual method in QNMs calculations by using the collocation method and Galerkin method as examples. We compared the results of both methods with the Generalized Horowitz-Hubeny Method and the WKB approximation. The methods proposed in this paper demonstrates superior precision. Additionally, we provide five criteria to exclude extraneous roots from the methods.&amp;#xD;

On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application

AbstractThe Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation.

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&amp;#39;s ``best&amp;#39;&amp;#39; 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&amp;#39;&amp;#39; 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.

Approximate solution of Fredholm integral equations of the second kind through multinode Shepard operators

An efficient numerical method based on the Shepard method has been developed to deal with Fredholm integral equations of the second kind. The procedure involves expressing the unknown function as a linear combination of Shepard basis functions. Subsequently, a collocation method is used to get the matrix equation yielding the coefficients of such an approximate solution. The effectiveness of the proposed approach is illustrated by some examples.

Troesch’s problem: A numerical study with cubic trigonometric B-spline method

This research paper presents a numerical investigation of Troesch’s problem using the collocation method based on the cubic trigonometric B-Spline technique. Troesch’s problem is a well-known BVP that arises in the field of mathematical physics and engineering. The objective of this study is to provide an accurate and efficient numerical solution to Troesch’s problem. Our method utilizes the properties of cubic trigonometric B-splines with collocation technique to approximate the solution of Troesch’s problem. The method is employed to satisfy the boundary conditions, resulting in a system of algebraic equations. A comprehensive error analysis is conducted to assess the accuracy and convergence of the proposed scheme. The obtained numerical results are compared with the exact solution, where available, and with other existing numerical methods to validate the accuracy and efficiency of the proposed technique.The significance and relevance of this work lie in its ability to provide highly accurate solutions to Troesch’s problem. Furthermore, the proposed method exhibits good convergence properties and demonstrates superiority over other numerical techniques in terms of accuracy and efficiency. These findings have practical implications in the field of mathematical physics and engineering, where precise solutions to Troesch’s problem can greatly enhance the understanding and optimization of various systems.

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.