On the Application of the Collocation Method in Spaces of p-Summable Functions to the Fredholm Integral Equation of the Second Kind

2020 ◽  
pp. 55-60
Author(s):  
Andrei P. Chegolin
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Pointwise Estimate ◽  
Spline Collocation ◽  
Third Order ◽  
The Third ◽  
Starting Point ◽  
Difference Methods ◽  
Exact Constants

This work is devoted to the study of the numerical solution by the spline collocation method of the Fredholm equation of the second kind. For numerical solutions of such problems, the classical collocation method using polynomials is not always realizable in spaces of p-summable functions for numerical solutions of such problems. It is not always possible to obtain characteristics and estimates of errors of such approximations even in the case of its implementation. In this regard, in recent years, in practice, approximations are built using finite-difference methods. The purpose of this study is to obtain estimates of the error of the obtained approximate solution in the spaces indicated above. In addition, several statements were obtained about a pointwise estimate of this error at collocation nodes in terms of the kernel norm in specially constructed spaces of functions summable over the second variable. To obtain the main results, third-order collocation splines are used, as well as integral and averaged modules of smoothness. In this case, the results obtained can become a starting point for working with collocation splines of higher orders. In the case of the third order, the exact constants involved in the estimates are obtained. These results can be extended to the case of linear, parabolic collocation splines, as well as splines of order higher than the third.

scholarly journals The exponential cubic B-spline collocation method for the Kuramoto-Sivashinsky equation

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 853-861 ◽  
Author(s):  
Ozlem Ersoy ◽  
Idiris Dag
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Spline Approximation ◽  
Algebraic Equations ◽  
Spline Collocation ◽  
Fully Integrated ◽  
B Spline ◽  
Spline Collocation Method ◽  
Sivashinsky Equation ◽  
Good Agreement

In this study the Kuramoto-Sivashinsky (KS) equation has been solved using the collocation method, based on the exponential cubic B-spline approximation together with the Crank Nicolson. KS equation is fully integrated into a linearized algebraic equations. The results of the proposed method are compared with both numerical and analytical results by studying two text problems. It is found that the simulating results are in good agreement with both exact and existing numerical solutions.

scholarly journals Numerical Simulation of the Fractal-Fractional Ebola Virus

2020 ◽  
Vol 4 (4) ◽  
pp. 49 ◽  
Author(s):  
H. M. Srivastava ◽  
Khaled M. Saad
Keyword(s):  
Numerical Simulation ◽  
Ebola Virus ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Polynomial Functions ◽  
Fractional Differentiation ◽  
Lagrange Polynomial ◽  
New Models ◽  
Difference Methods ◽  
Two Parameters

In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter ρ is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of ρ=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of ρ and k. All calculations in this work are accomplished by using the Mathematica package.

Mimetic seismic wave modeling including topography on deformed staggered grids

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. T125-T141 ◽  
Author(s):  
Josep de la Puente ◽  
Miguel Ferrer ◽  
Mauricio Hanzich ◽  
José E. Castillo ◽  
José M. Cela
Keyword(s):  
Finite Difference ◽  
Seismic Wave ◽  
Seismic Waves ◽  
Numerical Solutions ◽  
Surface Condition ◽  
Finite Difference Methods ◽  
Staggered Grid ◽  
Grid Deformation ◽  
Large Dispersion ◽  
Difference Methods

Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.

Numerical Method with the Third-Order ENO Reconstruction Satisfying Two Conversation Laws for Linear Advection Equation

2011 ◽  
Vol 130-134 ◽  
pp. 2969-2972
Author(s):  
Rong San Chen ◽  
An Ping Liu
Keyword(s):  
Numerical Solutions ◽  
Advection Equation ◽  
Finite Volume Schemes ◽  
Third Order ◽  
The Third ◽  
Linear Advection Equation ◽  
Long Time ◽  
Super Convergence ◽  
Better Than ◽  
Eno Scheme

In recent years, Mao and his co-workers developed a class of finite-volume schemes for evolution partial differential equations, see [1-5].The schemes show a super-convergence quality and have good structure-preserving property in long-time numerical simulations. In [6], Chen and Ma developed a scheme which combine the idea of paper [5] and that of the the second-order ENO scheme [7]. In this paper, we propose a scheme which extend the result of [6] and obtain the scheme using the third-order ENO reconstruction. Numerical experiments show that our scheme is robust in long-time behaviors. Numerical solutions are far better than those of [6].

NUMERICAL SOLUTIONS OF QUANTUM MECHANICAL PROBLEMS USING THE BASIS-SPLINE COLLOCATION METHOD

2001 ◽  
Vol 12 (07) ◽  
pp. 1093-1108 ◽  
Author(s):  
D. O. ODERO ◽  
J. L. PEACHER ◽  
D. H. MADISON
Keyword(s):  
Collocation Method ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Spline Collocation ◽  
Spline Collocation Method

The time-dependent and time-independent Schrödinger equations have been numerically solved for several quantum mechanical problems with known analytical solutions using the basis-spline collocation algorithm. This algorithm has been demonstrated to be efficient and versatile. The results from these calculations illustrate the necessary numerical considerations required for solving both time-independent and time-dependent Schrödinger equations and form a basis that could be used for solving these and similar problems.

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