Collocation Method Research Articles

An efficient computational method for solving the fractional form of the European option price PDE with transaction cost under the fractional Heston model

This paper introduces a new method for precise option pricing analysis by utilizing the fractional form of the option price partial differential equation (PDE) and implementing a collocation technique based on the first Fermat polynomial sequence. The key innovation of this study is the exploration of the fractional formulation of European option pricing within the context of the fractional Heston model, which employs a collocation scheme utilizing Fermat polynomials to generate operational matrices characterized by numerous zeros, thereby enhancing computational efficiency. To achieve this, we first derive the option price PDE and convert it to its fractional form in the Caputo sense. We then solve the fractional PDE using fractional Fermat functions, expressing the solution as a series of multivariate Fermat functions with unknown coefficients. Following this, we compute the operational matrices for the Caputo fractional derivative and related partial derivatives, demonstrating how this computational framework transforms the primary problem into a nonlinear system of equations. Additionally, we conduct a convergence analysis of the collocation method. We conclude by presenting several numerical examples that illustrate the applicability and effectiveness of the proposed method, with its robust theoretical foundations and successful numerical tests indicating significant potential for practical applications in finance.

Fourth-order phase field modelling of brittle fracture with strong form meshless method

This study aims to find a solution for crack propagation in 2D brittle elastic material using the local radial basis function collocation method. The staggered solution of the fourth-order phase field and mechanical model is structured with polyharmonic spline shape functions augmented with polynomials. Two benchmark tests are carried out to assess the performance of the method. First, a non-cracked square plate problem is solved under tensile loading to validate the implementation by comparing the numerical and analytical solutions. The analysis shows that the iterative process converges even with a large loading step, whereas the non-iterative process requires smaller steps for convergence to the analytical solution. In the second case, a single-edge cracked square plate subjected to tensile loading is solved, and the results show a good agreement with the reference solution. The effects of the incremental loading, length scale parameter, and mesh convergence for regular and scattered nodes are demonstrated. This study presents a pioneering attempt to solve the phase field crack propagation using a strong-form meshless method. The results underline the essential role of the represented method for an accurate and efficient solution to crack propagation. It also provides valuable insights for future research towards more sophisticated material models.

Numerical solution of time-dependent two-parameter singularly perturbed problems via trigonometric quintic B-spline collocation technique

Abstract This paper introduces a novel algorithm for solving time-dependent two-parameter singularly perturbed parabolic convection-diffusion-reaction equations with Dirichlet boundary conditions. The algorithm is formulated using the Crank-Nicolson (CN) scheme for the temporal derivative discretization. Then, the Trigonometric Quintic B-spline (TQBS) is applied to approximate the state variable and its spatial derivatives on nonuniform collocation points. We conducted a comprehensive convergence analysis and stability of the proposed method and proved that the scheme achieved a parameter uniform convergence of approximately fourth order in space and second order in time. To make additional evidence to support the theoretical findings and further assess the proposed method, we implemented the numerical algorithm to solve three test examples. Furthermore, using these test examples, we demonstrated the parameter-uniform convergence of the proposed numerical scheme.

Error Analysis of a Collocation Method on Graded Meshes for Nonlocal Diffusion Problems with Weakly Singular Kernels

Error Analysis of a Collocation Method on Graded Meshes for Nonlocal Diffusion Problems with Weakly Singular Kernels

A Rapid Numerical Method for Nonlinear Generalized Time-Fractional Kawahara Equations via Domination Polynomials of Complete Graph.

Abstract The key objective of this study is to examine the nonlinear Time-fractional generalized Kawahara equation (N-TF-GKE), a significant nonlinear equation that appears in a variety of physical domains, including ion-acoustic waves in plasmas, shallow water waves, and nonlinear acoustics. This Equation exhibits complex dynamic properties, most notably the appearance of shock waves, solitary waves, and chaotic solutions. We use a graph theoretic polynomial approach to obtain numerical solutions of the N-TF-GKE. We present the dominance polynomials of complete graphs (DPC), a new class of graph polynomials, to develop an efficient operational integration matrix to approximate the N-TF-GKE. The N-TF-GKE transformed into a collection of nonlinear algebraic equations with the standard collocation points. We acquire the numerical solutions for the Domination polynomial using Newton's approach. We validate the efficiency and robustness of our domination polynomial collocation method (DCM) with a set of numerical experiments and comparative analyses.

Application of linear Legendre multi-wavelets collocation method for solution of fourth order boundary value problems

Application of linear Legendre multi-wavelets collocation method for solution of fourth order boundary value problems

Wavelet-based collocation methods for solving multi-dimensional integro-differential systems

This paper presents a comprehensive investigation of wavelet-based collocation methods for solving multi-dimensional integro-differential systems. The research addresses the challenging problem of finding numerical solutions to integro-differential equations (IDEs) that are essential in various scientific and engineering applications. We propose novel numerical approximations for infinitely dimensional problems, including fractional Cattaneo-Rayleigh waves and nonlinear diffusion problems defined on unbounded domains. The study introduces a unified computational strategy that directly applies to IDEs without discretization, particularly focusing on the integer and fractional-order Cattaneo-Rayleigh model of thermoelasticity in one-dimensional zonal regions. Through numerical experiments, we demonstrate that our wavelet collocation approach achieves high efficiency and superior accuracy compared to traditional methods. The results show particularly strong performance in handling systems with differential physical singularities, functional physical singularities, and vector-valued physical singularities. Our approach provides a valuable toolbox for addressing complex modeling challenges in biology, mechanical signal processing, and digital communications where partial integro-differential equations naturally arise.

Numerical solution based on the Haar wavelet collocation method for partial integro-differential equations of Volterra type

In this paper, a numerical investigation of a class of parabolic Volterra integro-differential equations has been carried out. Basically, the finite difference method associated with the Haar wavelet collocation technique is pursued. The main idea relies on semi-discretizing the parabolic-VIDEs by considering the finite differences in time and approximating the integral term within the trapezoidal rule. Afterwards, using a uniform mesh, the spatial derivative is approximated by the Haar wavelet method. Stability and convergence of the proposed approach are established in the frame of Sobolev space. Numerical examples are used to illustrate the effectiveness of this approach under different test scenarios. The error analysis, using both L 2 and L ∞ norms, shows that the method has high efficiency computationally with a great deal of accuracy. It is observed that these numerical results are in excellent agreement with the analytical solution. In addition, the methods and results obtained in this study are compared with those reported in recent literature.

Nonlinear Meshfree Collocation Method for Solving Double-Diffusive Natural Convection in a Porous Enclosure

A meshfree collocation method formulated by using reproducing kernel shape functions with Newton–Raphson method is presented with application to analyzing a three-phase coupling system in a porous enclosure. To tackle such a highly nonlinear phenomenon caused by double-diffusive natural convection, direct collocation using local approximation in conjunction with a two-step iteration solver is employed to stably and efficiently solve this problem. It has been shown that a determined and well-conditioned system is established under the collocation framework with reproducing kernel approximation. By conducting the parametric study, the numerical results have demonstrated the robustness of the proposed method in analyzing the porous enclosure by considering various geometry, boundary conditions, and governing parameters.

Conservation laws, traveling wave solutions and wavelet solution for the two-component Novikov equation

Abstract The objective of this present article is to study the two-component Novikov equation. This
 equation is also known as Geng-Xue equation and is characterized by cubic nonlinearities. By
 applying the multiplier method, new conserved quantities are found, and some traveling wave
 solutions are derived based on the (G′/G )-expansion method. Furthermore, a numerical solution
 is determined by employing the Haar wavelet collocation method. In order to confirm the
 accuracy, this numerical solution is compared with the solution obtained by (G′/G )-expansion
 method. Additionally, the convergence analysis of the Haar wavelet collocation method is also
 presented.

A Fast Chebyshev Collocation Method for Stability Analysis of a Robotic Machining System with Time Delay

A Fast Chebyshev Collocation Method for Stability Analysis of a Robotic Machining System with Time Delay

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

A spline-based framework for solving the space–time fractional convection–diffusion problem

In this study we consider a spline-based collocation method to approximate the solution of fractional convection–diffusion equations which include fractional derivatives in both space and time. This kind of fractional differential equations are valuable for modeling various real-world phenomena across different scientific disciplines such as finance, physics, biology and engineering.The model includes the fractional derivatives of order between 0 and 1 in space and time, considered in the Caputo sense and the spatial fractional diffusion, represented by the Riesz-Caputo derivative (fractional order between 1 and 2). We propose and analyze a collocation method that employs a B-spline representation of the solution. This method exploits the symmetry properties of both the spline basis functions and the Riesz-Caputo operator, leading to an efficient approach for solving the fractional differential problem. We discuss the advantages of using Greville Abscissae as collocation points, and compare this choice with other possible distributions of points. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

A Fast Collocation Method with Inscribed Polygonal Approximation of the Peridynamic Neighborhood for Bond-Based Linear Viscoelastic Peridynamic Models

A Fast Collocation Method with Inscribed Polygonal Approximation of the Peridynamic Neighborhood for Bond-Based Linear Viscoelastic Peridynamic Models

Solar sail transfer strategy for long-duration multi-objective mission in cislunar space utilizing solar radiation and reflectivity control device

Abstract This study delves into the complexities of optimizing solar sail trajectories for long-duration multi-objective missions in cislunar space, focusing on orbital transfers between halo orbits and distant retrograde orbits. It presents an innovative transfer strategy that integrates solar radiation propulsion and reflectivity control devices to effectively adjust thrust force magnitude and direction, thereby improving maneuver capabilities. The optimization method is founded on the coupled dynamics of a solar sail, aiming to minimize the transfer time of flight using the Hermite-Simpson collocation method while ensuring optimal performance. Through thorough analysis and comparison, the research strives to determine the most efficient trajectory for solar sail orbital transfers in cislunar space.

A Chebyshev collocation method for directly solving two-dimensional ocean acoustic propagation in linearly varying seabed.

It is one of the most concerning problems in hydroacoustics to find a method that can calculate the acoustic propagation accurately and adapt to the variation of the seabed. Currently, the one-dimensional spectral method has been employed to address the simplified ocean acoustic propagation model successfully. However, due to the model's application limitations and approximation error, it poses challenges when attempting to solve real-world ocean acoustic fields. Hence, there is a crucial need to develop a direct solution method for the two-dimensional Helmholtz equation of ocean acoustic propagation, without relying on a simplified model. In previous work, we achieved successful solutions for the two-dimensional Helmholtz equation within a rectangular domain, utilizing a collocation-type spectral method. Taking into account the fluctuations in the actual seabed, we introduce a Chebyshev collocation spectral method to directly tackle the two-dimensional ocean acoustic propagation problem, which could solve the case of a seabed with linear variation, sound velocity variation and inhomogeneous medium situation. After comparative verification, the calculation result of the two-dimensional spectral method is more accurate than traditional mature models such as Kraken and COUPLE. By eliminating model constraints and enlarging the solution range, this spectral method holds immense potential in real marine environments.

Super-convergence and Stability Analysis of the Finite Element Orthogonal Collocation Method for Time-Fractional Telegraph Equation

Super-convergence and Stability Analysis of the Finite Element Orthogonal Collocation Method for Time-Fractional Telegraph Equation

Vieta–Lucas polynomials-based collocation simulation to analyze the solvent fraction factor in active and passive control flow induced by torsional motion

The present research explores the Boger fluid flow past a stretching cylinder with torsional motion in the presence of the magnetic field. It is assumed that the cylinder rotates continuously around its axis and that the starting point's position along the axis correlates with the cylinder wall's expansion rate. Additionally, the consequence of active and passive control of nanoparticles, activation energy, thermophoresis, and Brownian motion effects are considered. Similarity variables transform the governing partial differential equations into non-dimensional ordinary differential equations (ODEs). Furthermore, the Vieta–Lucas polynomials-based collocation method (V-LPBCM) is employed to solve the resulting ODEs. The V-LPBCM outcomes of Nusselt and Sherwood numbers are compared with Runge–Kutta Fehlberg's fourth-fifth-order scheme for validation purposes. The impact of various dimensionless parameters on the different profiles is depicted in the graphical representation. The increase in values of the magnetic parameter, the ratio of relaxation time, and the Reynolds number decline the velocity profile. The velocity profile increases as the values of the solvent fraction parameter rise. The thermal profile increases as the heat source/sink, and thermophoretic parameters rise. The increase in values of activation energy parameter increases the thermal profile.

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods like PINNs rely on auto differentiation and sampling collocation points, leading to a lack of interpretability and lower accuracy than traditional numerical methods. As a result, we propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.