Collocation Method Research Articles

Cubic B-spline based elastic and viscoelastic wave propagation method

The requirement for high computational efficiency in inversion imaging poses challenges to further enhancing the accuracy of imaging large elastic and viscoelastic models. Considering that forward modeling serves as a prerequisite for inversion imaging, it is essential to introduce a wave propagation simulation method that can enhance simulation accuracy without significantly compromising computational speed. In our paper, cubic B-spline is introduced into the forward modeling of the wave equation. The B-spline collocation method offers a linear complexity for computation and exhibits fourth-order convergence in terms of computational errors. We also find that it can perfectly adapt to the boundary conditions of the perfectly matched layer in the forward modeling solution of the wave simulation. Several typical numerical examples, including 2-D acoustic wave equation, 2-D elastic wave propagation, and 2-D viscoelastic propagation in homogeneous media, are provided to validate its convergence, accuracy, and computational efficiency.

Instability of penetrative convection in a vertical porous layer with non-Dirichlet temperature conditions

The linear stability of penetrative convection in a differentially heated vertical porous layer with non-Dirichlet boundary conditions on the perturbed temperature field is investigated. The penetrative convection is realized through a uniform internal heating in a porous medium. The instability-driving parameters are the Darcy-Rayleigh number, dimensionless internal heat source strength, the Prandtl-Darcy number, and the Biot number. Neutral stability curves and the critical values of the Darcy-Rayleigh number, the wave number, and the wave speed are computed numerically by employing the Chebyshev collocation method. A comprehensive analysis is conducted on the onset of thermal instability, focusing on the effects of transitioning temperature boundary conditions from Dirichlet to Neumann. A novel finding is the emergence of bi-modal or tri-modal neutral stability curves, which unveil distinct onset modes arising from the combined effects of a heat source and the transition of boundary conditions from isothermal to adiabatic. The threshold value of the Biot number, at which the transition to instability occurs, shows a significant dependence on the internal heat source strength and this threshold value diminishes as the Prandtl-Darcy number attains higher values. Moreover, increase in the Biot number advances the onset of convection, while the Prandtl-Darcy number is found to exert both stabilizing and destabilizing influences on the stability of the base flow.

Numerical solution of fractal‐fractional differential equations system via Vieta‐Fibonacci polynomials fractal‐fractional integral operators

AbstractThe main idea of this work is to present a numerical method based on Vieta‐Fibonacci polynomials (VFPs) for finding approximate solutions of fractal‐fractional (FF) pantograph differential equations and a system of differential equations. Although the presented scheme can be applied to any fractional integral, we focus on the Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio integrals with due to their privileges. To carry out the method, first, we introduce FF integral operators in the Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio senses. Then, by applying the Vieta‐Fibonacci polynomials and their FF integral operators together with the collocation method, the problem becomes reduced to a system of algebraic equations that can be solved by Mathematical software. In the presented scheme, acceptable approximate solutions are achieved by employing only a few number of the basic functions. Moreover, the error analysis of the presented method is investigated. Finally, the accuracy of the presented method is examined through the numerical examples. The proposed scheme is implemented for some famous systems of FF differential equations, such as memristor, which is a fundamental circuit element so called universal charge‐controlled mem‐element, convective fluid motion in rotating cavity, and Lorenz chaotic system.

Existence, uniqueness and Ulam–Hyers stability result for variable order fractional predator-prey system and it's numerical solution

This study presents an approximate numerical technique for solving time fractional advection-diffusion-reaction predator-prey equations with variable order (VO), where the analyzed fractional derivatives of VO are in the Caputo sense. Results for Ulam–Hyers stability are shown, as well as the existence and uniqueness of solutions. It is suggested to use a numerical approximation based on the shifted second kind of airfoil polynomials to solve the equations under consideration. A fractional derivative operational matrix with VO is derived for shifted airfoil polynomials, which will be used to compute the unknown function. The main equations are transformed into a set of algebraic equations by substituting the aforementioned operational matrix into the equations under consideration and utilizing the properties of the shifted airfoil polynomial along with the collocation points. A numerical solution is obtained by solving the acquired set of algebraic equations. To verify the accuracy and efficiency of the discussed scheme, several illustrative examples have been considered. The results obtained by the proposed method demonstrate the efficiency and superiority of the method compared to other existing methods.

An intriguing numerical strategy for Zakharov–Kuznetsov equation through graph-theoretic polynomials

This paper explores graph-theoretic polynomials to find the approximate solution of the (2+1)D Time-fractional Zakharov-Kuznetsov(TF-Z-K) equation. The Zakharov-Kuznetsov equations govern the behavior of nonlinear acoustic waves in the plasma of hot isothermal electrons and cold ions in the presence of a homogeneous magnetic field. Independence polynomials of the Ladder-Rung graph serve as the polynomial approximation for the suggested Independence Polynomial Collocation Method (IPCM). The Caputo fractional derivatives are adopted to determine the fractional derivatives in the TF-Z-K equation. The TF-Z-K equation is converted into a system of nonlinear algebraic equations using the collocation points in IPCM. The Newton-Raphson approach yields the solution of the suggested method by solving the resulting system. We’ve compared a few scenarios with the tangible outcomes to validate the procedure. Quantitative outcomes match the current findings and validate the exactness of IPCM compared t o the recent numerical and semi-analytical approaches in the literature.

Bifurcation analysis of piecewise-smooth engineering systems with delays through numeric continuation of periodic orbits

This paper presents a numeric continuation framework for periodic orbits of piecewise-smooth and hybrid dynamical systems with fixed point delays. For the numeric solution of the corresponding infinite dimensional multi-point boundary value problem, a novel discretization and interpolation scheme is developed employing Chebyshev polynomial based spectral collocation techniques. The same approach is employed for the formulation of the corresponding monodromy matrix enabling stability analysis on the found periodic orbits. Special care is attributed to the accurate detection of discontinuity induced bifurcations such as grazing and sliding, and the implemented pseudo-arclength framework is adapted to allow two parameter continuation of these critical points. The capabilities of the developed algorithms are demonstrated on a set of delayed piecewise-smooth and hybrid dynamical systems, showcasing potential engineering applications from the fields of control theory, traffic dynamics modelling, and machine tool vibrations. Finally, a detailed tutorial is attached in the appendix to accompany the open-source release of the developed codebase.

A polynomial collocation method for a class of singular fractional differential equations

In this work we consider a class of singular fractional differential equations (SFDEs). Using a suitable variable transformation we rewrite the SFDE as a cordial Volterra integral equation and propose a polynomial collocation method to find an approximate solution of the original problem. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples.

An efficient finite element computation using subparametric transformation up to cubic-order for curved triangular elements

PurposeA finite element computational methodology on a curved boundary using an efficient subparametric point transformation is presented. The proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.Design/methodology/approachOur proposed method builds upon the domain discretization into linear, quadratic and cubic-order elements using subparametric spaces and such a discretization greatly reduces the computational complexity. A unique subparametric transformation for each triangle is derived from the unique parabolic arcs via a one-of-a-kind relationship between the nodal points.FindingsThe novel transformation derived in this paper is shown to increase the accuracy of the finite element approximation of the boundary value problem (BVP). Our overall strategy is shown to perform well for the BVP considered in this work. The accuracy of the finite element approximate solution increases with higher-order parabolic arcs.Originality/valueThe proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.

An adaptive lattice Green's function method for external flows with two unbounded and one homogeneous directions

We solve the incompressible Navier-Stokes equations using a lattice Green's function (LGF) approach, including immersed boundaries (IB) and adaptive mesh refinement (AMR), for external flows with one homogeneous direction (e.g. infinite cylinders of arbitrary cross-section). We hybridize a Fourier collocation (pseudo-spectral) method for the homogeneous direction with a specially designed, staggered-grid finite-volume scheme on an AMR grid. The Fourier series is also truncated variably according to the refinement level in the other directions. We derive new algorithms to tabulate the LGF of the screened Poisson operator and viscous integrating factor. After adapting other algorithmic details from the fully inhomogeneous case [1], we validate and demonstrate the new method with transitional and turbulent flows over a circular cylinder at Re=300 and Re=12,000, respectively.

Triple solutions for unsteady stagnation flow of tri-hybrid nanofluid with heat generation/absorption in a porous medium

The present theoretical study focuses on enhancing the performance and efficiency of casting and extrusion processes. Thus, this work examines the flow characteristics and heat transfer of unsteady tri-hybrid nanofluid flow in porous media while also considering the heat source/sink effect. The governing boundary layer equations are solved via a built-in collocation method available in MATLAB software. Three distinct numerical solutions which convey the fluid flow characteristics have been identified. Notably, nanofluid composition influences boundary layer separation, with tri-hybrid nanofluid showing enhanced heat transfer when the stretching/shrinking parameter exceeds −10.53. When the sheet shrunk at −10.8, ternary and hybrid nanofluids improved thermal efficiency by 15.22 % and 20.38 %, respectively, compared to mono nanofluids.

Simulation of the SIR dengue fever nonlinear model: A numerical approach

This paper introduces a new numerical technique that utilizes the Chebyshev collocation method to solve the nonlinear SIR mathematical model for dengue disease dynamics. The main goal of this study is the development of a highly accurate and efficient numerical approach that allows for the simulation of dengue fever infection and transmission mechanisms. The suggested methodology provides a strong and adaptable tool for studying the intricate dynamics of dengue fever, which is essential for improving our comprehension and control of this significant public health problem. The numerical approach is totally programmable and can be customized to incorporate a user-friendly interface for data input, hence enhancing its accessibility and practicality for researchers and public health practitioners. The conducted numerical simulations provide evidence of the accuracy and computational efficiency of the suggested technique. This unique numerical framework is positioned as a standard for managing and regulating a diverse array of biological problems. The creation of this sophisticated numerical tool signifies significant progress in the realm of dengue fever modeling. It offers a versatile and dependable platform to investigate different epidemiological scenarios and assess the possible effects of intervention efforts. The results of this study have significant consequences for enhancing dengue monitoring, safeguarding, and control initiatives in areas where the disease is prevalent.

ENTROPY-OPTIMIZED FLOW OF COUPLE STRESSES IN AN POROUS INCLINED PIPE WITH UNIFORM MAGNETIC FIELD AND MIXED CONVENTION

Abstract The research aims to investigate the entropy-optimized flow of couple stresses in a porous inclined pipe exposed to a transverse constant magnetic field and mixed convection. It focuses on understanding the thermodynamic efficiency and fluid behaviour under convective boundary conditions, contributing to improved designs in engineering applications where such flows are relevant. The aim is to enhance heat transfer efficiency in industrial processes such as chemical reactors, heat exchangers, and geothermal systems, while also improving filtration systems for applications like water purification and oil recovery. By subjecting the flow to a uniform magnetic field and mixed convection, nonlinear governing equations arise due to mixed convection. We linearize these equations using a quasi-linearization approach and solve them using Chebyshev spectral collocation. Our analysis focuses on thermodynamic phenomena like entropy generation and the Bejan number, which have implications for the efficiency and sustainability of industrial processes. We visualize temperature and axial velocity profiles across various parameter ranges to understand the fluid's behaviour under the influence of magnetic fields and porous materials. As the magnetic parameter increases, there is a decrease in fluid velocity and temperature. However, the opposite tendency is seen for the couple stress viscosity ratio parameter. We also observe irreversibility dominating heat transfer at the pipe wall, while fluid friction irreversibility dominates around the pipe's centre. This research contributes to advancing our understanding of thermodynamic processes in complex fluid systems and has practical implications for optimizing industrial processes and developing more efficient filtration systems.

NUMERICAL SOLUTION OF NONLINEAR SCHRODINGER EQUATION USING COLLOCATION METHOD WITH CHEBYSHEV POLYNOMIALS

The nonlinear Schrodinger equation (NSE) is a partial differential equation (PDE) with numerous applications in quantum mechanics. Analytical methods for the solution of NSE are almost impossible due to their complexity. Thus, the need for a numerical scheme to seek the approximate solution of the NSE. Hence, this paper considered the numerical solution of the NSE via the spectral collocation method (SCM) with Chebyshev orthogonal polynomials of the first kind. The scheme was efficiently constructed to solve the NSE equation with the aid of MAPLE 18, and numerical evidence compared with Adomian Decomposition Method (ADM), Homotopy Analysis Transform Method (HATM) and Residual Power Series Method (RPSM) as available in the literature. The findings show that the SCM is an effective solver for NSE with rapid convergence to the exact solution as the parameter �varies.

Linear stability analysis of a Couette-Poiseuille flow: A fluid layer overlying an anisotropic and inhomogeneous porous layer

We investigate the temporal stability analysis of a two-layer flow inside a channel that is driven by pressure. The channel consists of a fluid layer overlying an inhomogeneous and anisotropic porous layer. The flow contains a Couette component due to the movement of the horizontal impermeable upper and lower walls binding the two layers. These walls of the channel move at an identical speed but in opposite directions. The flow dynamics for the porous medium are modelled by the Darcy-Brinkman equations, and the Navier-Stokes equations are employed to describe the motion within the fluid layer. The hydrodynamic instability of infinitesimal disturbance is investigated using Orr-Sommerfeld analysis. The corresponding eigenvalue problem is derived and solved numerically using the Chebyshev polynomial-based spectral collocation method. Results reveal that stability features are strongly affected by the axial and spatial permeability variations of the porous medium. Further, the ratio of the depth of the fluid layer to the porous layer and the strength of the Couette component play a crucial role. The destabilization of the perturbed system is noticed by strengthening the Couette flow component. The combined impact of increasing the anisotropy parameter and depth ratio, decreasing Darcy number, and reducing the inhomogeneity factor stabilizes the system. This facilitates us to have greater control over the instability characteristics of such fluid-porous configuration by suitably adjusting various flow parameters. The outcome will be beneficial in relevant applications for enhancing or suppressing the instability of perturbation waves, as preferable.

Trends analysis of Evapotranspiration and responses to soil moisture and wind speed over the Taklimakan Desert, China

The Tarim Basin is a typical hyper-arid area, 57% of which is occupied by the Taklimakan desert. Because evapotranspiration (ET) is a key factor in determining water demand, accurate estimation of ET is critical for efficient water management in arid regions. In this study, the spatiotemporal dynamics of ET across the Tarim Basin from 2001 to 2020 were simulated based on a revised Surface Energy Balance System (SEBS) algorithm, and the impacts of driving factors on ET trends in the Taklimakan desert were quantified. The findings indicated that the mean annual ET in the entire basin was slightly greater than that in the desert (302 mm vs. 295 mm), although the decreasing rate of annual ET across the entire basin was slower than that in the desert (−0.27 mm/year vs. −0.50 mm/year). Strong increasing trends were found in croplands, whereas decreasing trends, which accounted for 56.5% of the basin, were primarily concentrated in the central and southern parts of the desert. Correlation analysis revealed that wind speed (WS) and soil moisture (SM) were the dominant factors, with a negative contribution of more than 35% and a positive contribution of more than 45%, respectively, to ET trends. Compared with the ERA5-Land and complementary relationship ET using the extended triple collocation method, SEBS ET provided detailed information and a lower random error standard deviation. Our findings contribute to the understanding of how dominant factors impact ET in hyper-arid regions and provide a reference for water management.

A weighted stabilized lagrange interpolation collocation method for boundary condition identification in 3D electromagnetic inverse scattering

The identification of boundary conditions in electromagnetic inverse scattering is of importance in various engineering applications, ranging from geophysical exploration to wireless communication. Conventional numerical methods solving this problem often suffer from the iterative process, leading to inefficiencies and non-convergence. This paper introduces a weighted scheme of the stabilized Lagrange interpolation collocation method (weighted SLICM) to resolve this problem. Weighted SLICM efficiently integrates governing equations, boundary conditions, and measurement conditions using a weighted least squares approach, offering a straightforward single-step solution and obviating the need for iterative processes in traditional methods like the finite element method. By incorporating regularization techniques, weighted SLICM decreases measurement errors which are unavoidable in engineering problems, thereby ensuring high efficiency and accuracy. In addition, characterized as a strong-form collocation method that relies solely on point information but not on grid connectivity, the weighted SLICM is readily extendible to complex three-dimensional applications in electromagnetic inverse scattering. Extensive simulations of benchmark problems show its ability to achieve accurate and stable results in boundary condition identification in electromagnetic inverse scattering problems including 1D, 2D, and 3D environments, highlighting the effectiveness of the weighted SLICM in navigating complex engineering challenges and substantially enriching research methodologies in this area.

A spectral collocation scheme for solving nonlinear delay distributed-order fractional equations

The basic purpose of this paper is to implement a variant of the Jacobi–Gauss collocation scheme for solving nonlinear delay (in difference format) fractional differential equations of distributed-order type. At the first stage of the solving procedure, the Legendre–Gauss quadrature rule is implemented to approximate the main problem by a difference delay fractional differential equation of multi-term type. So, one can apply the Jacobi–Gauss collocation approach for localizing the resulting multi-term fractional differential equation with the difference delay factor. Moreover, some special cases of Jacobi–Gauss quadrature rules can be used for approximating the involved integrals in the aforementioned problem. By considering the Lagrange interpolating polynomials, as the numerical solutions, the solution of the main problem can be approximated by solving the associated system of nonlinear algebraic equations in terms of unknown Lagrange multipliers. Convergence analysis of the problem is investigated rigorously under some mild conditions at a spectral rate. Extensive test problems are considered to justify the theoretical analysis.

Fibonacci Wavelet Collocation Method for Solving Dengue Fever SIR Model

The main focus in this manuscript is to find a numerical solution of a dengue fever disease model by using the Fibonacci wavelet method. The operational matrix of integration has been obtained using Fibonacci wavelets. The proposed method is called Fibonacci wavelet collocation method (FWCM). This biological model has been transformed into a system of nonlinear algebraic equations by using the Fibonacci wavelet collocation scheme. Afterward, this system has been solved by using the Newton–Raphson method. Finally, we provide evidence that our results are better than those obtained by various current approaches, both numerically and graphically, demonstrating the method’s accuracy and efficiency.

Crack propagation arrest by the Joule heating in micro/nano-sized structures

The Joule heating technique is utilized to arrest the crack propagation so to protect some important structures being further damaged. In this approach there are created compression stresses at the crack tip vicinity due to a specific temperature distribution induced by the electric current. The Joule heating is amulti-physical problem which is applied here to micro/nano-sized structures. The size-effects are considered for description of both the elastic and the thermal fields. The heat conduction can’t be well described by classical Fourier’s law in nano-sized structures. A novel gradient theory is developed in such structures adopting the size effect of heat conduction. Besides, the strain gradients and double stresses are introduced into the constitutive laws. Then, the governing equations are given by partial differential equations with order of derivatives being higher than in classical theory. The principle of virtual work is the base of weak formulation and derivation of the discretized finite element equations for ageneral 2d boundary value problem with cracks. The collocation mixed FEM with large strain gradients is developed for numerical solution, where the C0 continuous approximation is applied independently to displacements and strains and also to temperature and temperature gradients. Kinematic constraints between strains and displacements are satisfied by the collocation method at selected interior points. The collocation mixed FEM is applied to solve 2D crack problems with Joule heating.

Deep neural network homogenization of multiphysics behavior for periodic piezoelectric composites

We present a physically informed deep neural network homogenization theory for identifying homogenized moduli and local electromechanical fields of periodic piezoelectric composites. The theory employs a multi-network model to represent solutions to stress equilibrium and electrostatic partial differential equations in the inclusion and the matrix phases, leading to a more accurate depiction of field variables. Satisfaction of periodicity conditions for hexagonal and cubic symmetries are efficiently tackled using a series of sinusoidal functions with known periods and adjustable parameters. Additionally, the theory applies fully trainable weights to the collocation points. These trainable weights, trained concurrently with the neural network weights, compel the neural networks to enhance their performance when faced with large local stress/deformation gradients. The predictive capability of the proposed theory is illustrated by comparison with finite-element solutions for composites reinforced with unidirectional fiber, spherical particle, or weakened by an ellipsoidal cavity over a wide range of volume fractions.