Orthogonal Spline Collocation Method Research Articles

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.

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.

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

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

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.

A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation

In this paper, we consider an orthogonal spline collocation (OSC) method to solve the fourth-order multi-term subdiffusion equation. The L1 method on graded meshes is employed in temporal direction. In the spatial direction, we develop the orthogonal spline collocation method. The stability and convergence of the above mentioned method are proved theoretically, and it's still valid if α1→1−. A new robust error bound is obtained, which is a more natural and desirable result and does not blow up as α1→1−. Some numerical results are also provided to verify our theoretical analysis and demonstrate the effectiveness of the OSC method.

A fourth order orthogonal spline collocation method Interface boundary value problem

Objective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two different sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem. To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our findings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy finding of the study. Keywords: Helmholtz problem; Orthogonal spline collocation techniques (OSC); Discontinuous data; Super Convergence; Piecewise cubic Hermite basis functions; Almost block diagonal (ABD) structure

One-parameter orthogonal spline collocation methods for nonlinear two-dimensional Sobolev equations with time-variable delay

This paper deals with the numerical solutions of initial–boundary value problems (IBVPs) of nonlinear two-dimensional Sobolev equations with time-variable delay. For solving this type of IBVPs, two classes of one-parameter orthogonal spline collocation (OSC) methods are constructed. For the obtained one-parameter OSC methods, their errors in L2- and H1-norm are analyzed and thus the corresponding error estimates are derived. With some numerical experiments, the computational effectiveness and theoretical accuracy of the methods are further confirmed. Moreover, a numerical comparison shows that the both kinds of one-parameter OSC methods have almost the same computational efficiency and accuracy under the same parameter and stepsizes.

A highly accurate algorithm for retrieving the predicted behavior of problems with piecewise-smooth initial data

A numerical scheme is constructed for the second-order parabolic partial differential equation with piecewise smooth initial data. The scheme comprises an orthogonal spline collocation strategy with the Rannacher time-marching. The irregular behavior of the underlying initial conditions of such differential equations results in inaccurate approximations due to the quantization error. For such problems, even the A-stable Crank-Nicolson scheme yields only first-order convergence in the temporal direction, with oscillations near the discontinuity. Applying mathematical perspective to dampen these oscillations, we present a highly accurate orthogonal spline collocation method with a smooth but straightforward time-marching scheme that significantly improves the convergence order. Through rigorous analysis, the present conjunctive scheme's convergence in the spatial direction is shown fourth-order (in L2 and L∞-norms) and third-order (in H1-norm), and it is shown second-order in the temporal direction. The performance and robustness of the contributed scheme are conclusively demonstrated with two test examples.

Orthogonal spline collocation method for the two-dimensional time fractional mobile-immobile equation

Orthogonal spline collocation method for the two-dimensional time fractional mobile-immobile equation

A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation

A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation

Unit Heads Amidst COVID 19 Pandemic: Work-Life Balance, Talent Management Skills and Institutional Performance

Purpose: The study aims to determine the level of work-life balance, talent management skills, and institutional performance of Northern Iloilo Polytechnic State College (NIPSC) key officials during the COVID 19 pandemic. Methods: This study is descriptive-correlational research using purposive samplings. The 97 key officials from the 7 campuses were selected as respondents of this study. The researcher adopted a questionnaire focused on work balance, talent management skills, and institutional performance. To analyze the data, statistical tools like frequency count, percentage, mean, standard deviation for descriptive, and for inferential were t-test, one-way analysis of variance, and Pearson’s r was utilized. Results: The results revealed that the work-life balance of different units’ heads was ”Highly Balanced”. While the talent management skills were ”Highly Managed”. And, the level of key officials’ institutional performance was ”Outstanding.” Furthermore, the findings also indicate that there are no significant differences in the level of work-life balance, talent management skills, and institutional performance of school officials. The school key officials’ highly balanced lifestyle was able them to cater to the needs of their work and personal life in this catastrophe. Their experiences as key officials improved their management skills and impacted and influenced them to cope with the crisis. Novelty: Higher Education Institutions (HEIs) unit heads should know work-life balance for teachers and management skills to inspire colleagues for institutional performance in this virus crisis. This pandemic teaches key officials to become dynamic and visionary to manage the new educational policies by implementing different modalities and training for teachers. The four mandates such as instruction, research, extension services, and production of the college are prospering despite health protocols restrictions. Keywords: Northern Iloilo Polytechnic State College; key Officials; Pandemic; WorkLife Balance; Management Skills; Institutional Performance 1 Introductionbjective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two different sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem. To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our findings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy finding of the study. Keywords: Helmholtz problem; Orthogonal spline collocation techniques (OSC); Discontinuous data; Super Convergence; Piecewise cubic Hermite basis functions; Almost block diagonal (ABD) structure

The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions

A high-order method based on orthogonal spline collocation (OSC) method is formulated for the solution of the fourth-order subdiffusion problem on the rectangle domain in 2D with sides parallel to the coordinate axes, whose solutions display a typical weak singularity at the initial time. By introducing an auxiliary variable v=Δu, the fourth-order problem is reduced into a couple of second-order system. The L1 scheme on graded mesh is considered for the Caputo fractional derivatives of order α∈(0,1) by inserting more grid points near the initial time. By virtue of some properties, such as complementary discrete convolution kernel and discrete fractional Grönwall inequality, we establish unconditional stability and convergence for the original unknown u and auxiliary variable v. Some numerical experiments are provided to further verify our theoretical analysis.

Unconditional convergence of linearized orthogonal spline collocation algorithm for semilinear subdiffusion equation with nonsmooth solution

AbstractA linearized orthogonal spline collocation (OSC) method withC1splines of degree≥3on a suitable graded mesh is formulated and analyzed for approximate solution of an initial‐boundary‐value problem of semilinear subdiffusion equations with nonsmooth solutions in time. The sharp error estimate in theL2norm is established without any restriction on the relative temporal and spatial mesh sizes. Such unconditional convergence results are proved by including the typical singularity of the solution near the timet = 0. Results of numerical experiments support the analytical results.

An Efficient Spline Collocation Method for a Nonlinear Fourth-Order Reaction Subdiffusion Equation

The nonlinear fourth-order reaction–subdiffusion equation whose solutions display a typical initial weak singularity is considered. A new analytical technique is introduced to analyze orthogonal spline collocation (OSC) method based on L1 scheme on graded mesh. By introducing a discrete convolution kernel and discrete fractional Gronwall inequality, convergence of the scheme is proved rigorously. This novel analytical technique can provide new insights in analyzing other time fractional fourth-order differential equations with weakly singular solutions.

The Crank–Nicolson orthogonal spline collocation method for one-dimensional parabolic problems with interfaces

One-dimensional parabolic problems with interfaces are solved using a method in which orthogonal spline collocation (OSC) is employed for the spatial discretization and the Crank–Nicolson method for the time-stepping. The derivation of the method is described in detail for the case in which cubic monomial basis functions are used in the development of the OSC discretization. With this background, the method is easily extended to monomials of higher degree. No matter the degree of the monomials, the method requires the solution of an almost block diagonal linear system at each time step using an existing algorithm at a cost of O(N2) on a spatial partition of N subintervals. The results of extensive numerical experiments involving examples from the literature are presented. Both cubic and quartic basis functions are employed. In each case, the results exhibit optimal global accuracy in the L∞,L2 and H1 spatial norms and second order accuracy in time. Moreover, superconvergence is observed at the nodes.

An extrapolated CN-WSGD OSC method for a nonlinear time fractional reaction-diffusion equation

In this paper, a new method is proposed to solve the time-fractional reaction-diffusion equation with nonlinear variable coefficients. Based on the second-order weighted and shifted Grünwald difference (WSGD) method for time Riemann-Liouville fractional derivative, we formulate and analyze the extrapolated Crank-Nicolson (ECN) method in time and orthogonal spline collocation (OSC) method in space. We prove the optimal global convergence order by a novel choice of the first step solution uh1. Finally, some numerical results are given to verify the convergence rate in the theoretical analysis.

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

AbstractOrthogonal spline collocation is implemented for the numerical solution of two‐dimensional Helmholtz problems with discontinuous coefficients in the unit square. A matrix decomposition algorithm is used to solve the collocation matrix system at a cost of O(N2 log N) on an N × N partition of the unit square. The results of numerical experiments demonstrate the efficacy of this approach, exhibiting optimal global estimates in various norms and superconvergence phenomena for a broad spectrum of wave numbers.

Observability Inequalities for Hermite Bi-cubic Orthogonal Spline Collocation Methods of 2-D Integro-differential Equations in the Square Domains

We will consider the problem of observability inequality of the Hermite bi-cubic orthogonal spline collocation space semi-discretizations of the 2-D integro-differential equations in the square domains. We prove the uniform (with respect to mesh-size) observability inequality in a subspace of solutions generated by the low frequencies of the negative part, and the middle frequencies of the positive part. Our method uses previously known uniform observability inequalities in the 1-d case and a dyadic spectral time decomposition. Some numerical results are presented to illustrate our theoretical analysis.