Spline Collocation Method Research Articles

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.

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.

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

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

An orthogonal spline collocation method for singularly perturbed parabolic reaction–diffusion problems with time delay

An orthogonal spline collocation method for singularly perturbed parabolic reaction–diffusion problems with time delay

BDF2 ADI orthogonal spline collocation method for the fractional integro-differential equations of parabolic type in three dimensions

In this paper, we are concerned with constructing a fast and an efficient alternating direction implicit (ADI) scheme for the fractional parabolic integro-differential equations (FPIDE) with a weakly singular kernel in three dimensions (3D). Our constructed scheme is based on a second-order backward differentiation formula (BDF2) for temporal discretization, orthogonal spline collocation (OSC) method for spatial discretization and a second-order fractional quadrature rule proposed by Lubich for the Riemann-Liouville fractional integral. The stability and convergence of the constructed numerical scheme are derived. Finally, some numerical examples are given to illustrate the accuracy and validity of the BDF2 ADI OSC method. Based on the obtained results, the numerical results are in line with the theoretical ones.

Spline collocation methods for solving some types of nonlinear parabolic partial differential equations

Spline collocation methods for solving some types of nonlinear parabolic partial differential equations

A Method for Solving the Fredholm Integral Equation of the First Kind

The paper considers a numerical method for solving the Fredholm integral equation of the first kind. The essence of the method is to replace the original equation with the corresponding regularized equation of the second kind, which is then solved by the modified spline collocation method. The solution in this case is represented as a linear combination of minimal splines. The coefficients at the splines are computed using local approximation (in some cases, quasi-interpolation) methods. Results of numerical experiments are presented and show that on model problems, the method proposed yields sufficiently accurate approximations, and the approximation accuracy can be improved by using minimal nonpolynomial splines and related functionals.

Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation

Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation

The numerical solution of nonlinear delay Volterra integral equations using the thin plate spline collocation method with error analysis

The numerical solution of nonlinear delay Volterra integral equations using the thin plate spline collocation method with error analysis

A priori mesh grading in collocation solution of noncompact Volterra integral equations with diagonal singularity

This work is devoted to some theoretical results concerning the collocation method on a graded mesh for a class of noncompact cordial Volterra integral equations whose kernels possess diagonal singularity depending on a real parameter . The existence and uniqueness with regularity properties of the solution are analysed. Our main focus is to choose the mesh points in such a way that the highest possible order of convergence is attained. To do this, we consider the spline collocation method with a graded mesh by an optimal scaling parameter which gives higher order of convergence. Due to the instability of the error behaviours for large values of grading exponent, choosing the optimal value of is a crucial issue from a numerical point of view. We will provide a sufficient condition on the grading exponent which controls the grading and characterizes the density of the mesh points. The global convergence as well as the optimal superconvergence results of the solutions on such a graded mesh are also investigated. Finally, some results of numerical experiments are reported which verify the validity of the theoretical results.

An ADI orthogonal spline collocation method for a new two-dimensional distributed-order fractional integro-differential equation

An ADI orthogonal spline collocation method for a new two-dimensional distributed-order fractional integro-differential equation

Analyze the Flow Behavior for MHD Power-Law Fluids

Objective: This study investigates the Numerical solution of laminar boundary layer flow of Magnetohydrodynamics (MHD) model for power-law fluid over a continuous moving surface in the presence of a transverse magnetic. Methods: The governing partial differential equation for the flow was transformed into non-linear ordinary differential equation using the Group theoretic method. Firstly, we convert this non-linear ordinary differential equation (ODE) into linear by using quasilinearization process. This linear ODE was solved numerically by applying the Spline collocation method suggested by Bickley. Findings: The solution for displacement profile and velocity profile were obtained as functions of the magnetic parameters. The effect of the magnetic parameters was discussed graphically. We used MATLAB software for finding the outcomes. Novelty: The main goal of this article is to analyze boundary layer flow of Magneto hydrodynamics (MHD) model for power-law fluid over a continuous moving surface in the presence of a transverse magnetic. The conservation equations of mass, momentum and energy are converted into ordinary differential equations along with boundary conditions by appropriate similarity transformations and solved by applying Spline Collocation Method. The convergence of solutions is important for providing the developing linear functions of solutions, which is a benefit of the Spline Collocation Method. These research findings are applicable, for example, in predicting skin friction and heat transfer rate over a stretching sheet, which has implications in technological and manufacturing industries such as polymer extrusion. Comparisons with previously published works are made, and the results show a high level of agreement. This type of research is applicable to work in fire dynamics in insulation, solar collection systems, recovery of petroleum products, etc. Keywords: Power-Law Fluids; Magnetic Field; Nonlinear Differential Equation; Quasilinearization; Bickley’s Method; Linear Equations

Hypersingular Integral Equations of Prandtl’s Type: Theory, Numerical Methods, and Applications

In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl’s type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous method for solving nonlinear operator equations, which is stable for perturbations of the coefficients and the right-hand sides of equations.

A Cubic Spline Collocation Method to Solve a Nonlinear Space-Fractional Fisher’s Equation and Its Stability Examination

This article seeks to show a general framework of the cubic polynomial spline functions for developing a computational technique to solve the space-fractional Fisher’s equation. The presented approach is demonstrated to be conditionally stable using the von Neumann technique. A numerical illustration is given to demonstrate the proposed algorithm’s effectiveness. The novelty of the present work lies in the fact that the results suggest that the presented technique is accurate and convenient in solving such problems.

Fourth order orthogonal spline collocation methods for two-point boundary value problems with interfaces

Fourth order orthogonal spline collocation methods for two-point boundary value problems with interfaces

Stability of two-step spline collocation methods for initial value problems for fractional differential equations

This paper analyzes the numerical stability of a class of two-step spline collocation methods for initial value problems for fractional differential equations. The stability region is characterized in terms of the eigenvalues of a power series, which depends on the method parameters. We provide the stability regions for several choices of the method parameters. Numerical experiments prove the effectiveness of the stability results both in a constant stepsize framework and in the case of graded meshes.

A high-order unconditionally stable numerical method for a class of multi-term time-fractional diffusion equation arising in the solute transport models

In this paper, we study a high-order unconditionally stable numerical method to approximate the class of multi-term time-fractional diffusion equations. This type of problem appears in the modelling of transport of certain quantities such as heat, mass, energy, solutes in ground water and soils. The multi-term time-fractional derivative is approximated by using the Crank–Nicolson method for the Caputo's time derivative. The space derivative is approximated by using the collocation method based on quintic B-spline basis functions. We have established the stability and convergence analysis of the proposed numerical scheme thoroughly, and it is shown that the order of convergence in space variable is almost four and in the time variable is . To prove the accuracy and efficiency of the developed method, we consider four numerical examples and perform the numerical simulation. The developed algorithm works well andvalidate the theoretical results. The developed method is fourth-order convergent in the space variable, which is almost two orders of magnitude higher than the other spline collocation methods.

Rannacher time-marching with orthogonal spline collocation method for retrieving the discontinuous behavior of hedging parameters

In this paper, we extensively study the orthogonal spline collocation method known as the spline collocation at Gauss points with Rannacher’s time-marching scheme for free boundary value option pricing problems. Such financial problems commonly feature non-smooth payoff functions that cause inaccuracies in approximating the solution and its derivatives. As a result, unlike the problems with the smooth initial data, the quadratic convergence is not realized by the Crank-Nicolson time-stepping scheme for these problems. Furthermore, the non-smoothness in the initial condition leads to severe degradation in the convergence rates and spurious oscillations near the discontinuity. The rationale is that classical schemes strongly rely on the smoothness of the initial data. A rigorous time-marching scheme referred to as Rannacher time-stepping scheme is introduced for the American option’s price diagnosed by a linear complementarity problem to smoothen the data. Moreover, with careful analysis, second and fourth orders of convergence are established for the present scheme in temporal and spatial directions, respectively. The numerical results for three test problems are presented in tables and graphs to validate the theory. These results show that the present scheme achieves higher accuracy and sufficiently restores the expected behavior.

An optimal two-step quadratic spline collocation method for the Dirichlet biharmonic problem

An optimal two-step quadratic spline collocation method for the Dirichlet biharmonic problem

Approximation of the Riesz–Caputo Derivative by Cubic Splines

Differential problems with the Riesz derivative in space are widely used to model anomalous diffusion. Although the Riesz–Caputo derivative is more suitable for modeling real phenomena, there are few examples in literature where numerical methods are used to solve such differential problems. In this paper, we propose to approximate the Riesz–Caputo derivative of a given function with a cubic spline. As far as we are aware, this is the first time that cubic splines have been used in the context of the Riesz–Caputo derivative. To show the effectiveness of the proposed numerical method, we present numerical tests in which we compare the analytical solution of several boundary differential problems which have the Riesz–Caputo derivative in space with the numerical solution we obtain by a spline collocation method. The numerical results show that the proposed method is efficient and accurate.