Collocation Parameters Research Articles

Multistep collocation technique implementation for a pantograph-type second-kind Volterra integral equation

<p>In this research, we have elaborated high-rate multistep collocation strategies in order to concern with second-type vanishing delay VIEs. Herein, characteristics of uniqueness, existence, and regularity for both numerical and analytical solutions have been shown. To explore the solvability of the system derived from the numerical method, we have defined particular operators and demonstrated that these operators are both compact and bounded. Solvability is studied by means of the innovative compact operator concepts. The concept of convergence has been examined in greater detail, revealing that the convergence of the method is influenced by the spectral radius of the matrix generated according to the collocation parameters in the difference equation resulting from the method's error. Finally, two numerical examples are given to certify our theoretically gained results. Also, since the proposed numerical method is local in nature, it can be compared to other local methods, such as those used in reference <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. We will compare our method with <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> in the last section.</p>

COLLOCATION BASED APPROXIMATIONS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEMS

A boundary value problem for fractional integro-differential equations with weakly singular kernels is considered. The problem is reformulated as an integral equation of the second kind with respect to, the Caputo fractional derivative of y of order α, with 1 < α < 2, where y is the solution of the original problem. Using this reformulation, the regularity properties of both y and its Caputo derivative z are studied. Based on this information a piecewise polynomial collocation method is developed for finding an approximate solution zN of the reformulated problem. Using zN, an approximation yN for y is constructed and a detailed convergence analysis of the proposed method is given. In particular, the attainable order of convergence of the proposed method for appropriate values of grid and collocation parameters is established. To illustrate the performance of our approach, results of some numerical experiments are presented.

Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels

We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

Collocation Runge–Kutta–Nyström methods for solving second-order initial value problems

ABSTRACT A class of modified collocation Runge–Kutta–Nyström (RKN) methods for solving second-order initial value problem , , has been formulated and studied in the paper. The new methods are applicable to a larger class of second-order initial value problems compared to the classical collocation RKN methods and they reduce to the classical methods when is absent from the equation. Superconvergence for the new methods is attained when the set of collocation points satisfies orthogonality conditions. We proved that an s-stage modified collocation RKN method is of accuracy order of at least s for any set of collocation parameters and at most when are Gauss points. The stability function and the stability of some modified RKN methods have also been investigated. Numerical experiments are included to demonstrate the advantage of the new methods.

Numerical solution of linear fractional weakly singular integro-differential equations with integral boundary conditions

We consider a class of boundary value problems for linear fractional weakly singular integro-differential equations with Caputo fractional derivatives and integral boundary conditions. Using an integral equation reformulation of the boundary value problem, we first study the regularity of the exact solution and its Caputo derivative. Based on the obtained regularity properties and by using suitable smoothing transformations along with spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

Collocation methods for integro-differential algebraic equations with index 1

Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index $1$ into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-$1$ IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$, but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m - 1}^{(- 1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method; the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

Development of a Direct Time Integration Method Based on Quartic B-spline Collocation Method

A new version of quartic B-spline direct time integration method for dynamic analysis of structures is presented. This procedure is derived based on uniform quartic B-spline piecewise polynomial approximations and collocation method, named Alpha Quartic B-spline time integration method. In this way, at first, the method is implemented to solve the governing differential equation of motion of single-degree-of-freedom systems, and later, the proposed method is generalized for multi-degree-of-freedom systems. Stability and accuracy analysis of the proposed algorithm have been investigated completely. In the proposed algorithm by using two collocation parameters α1 and α2, unconditional stability is achieved, but a local instability is created. The best values of these two parameters have been determined not only to maintain the stability, but also to ensure the desired accuracy. For accuracy analysis, dissipation and dispersion errors have been investigated for different cases of α’s. Finally, for the proposed method, a simple step-by-step algorithm was presented. The effectiveness and robustness of the proposed algorithm in solving linear dynamic problems are demonstrated in the numerical examples.

TWO COLLOCATION TYPE METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-LOCAL BOUNDARY CONDITIONS

A class of non-local boundary value problems for linear fractional differential equations with Caputo-type differential operators is considered. By using integral equation reformulation of the boundary value problem, we study the existence and smoothness of the exact solution. Using the obtained regularity properties and spline collocation techniques, we construct two numerical methods (Method 1 and Method 2) for finding approximate solutions. By choosing suitable graded grids, we derive optimal global convergence estimates and obtain some super-convergence results for Method 2 by requiring additional assumptions on equation and collocation parameters. Some numerical illustrations for verification of theoretical results is also presented.

High-order collocation methods for nonlinear delay integral equation

The classical collocation methods based on piecewise polynomials have been studied for delay Volterra integral equations of the second-kind in Brunner (2004). These collocation methods have uniform order m for any choice of the collocation parameters and can achieve local superconvergence in the grid points by choosing the suitable collocation parameters. In this paper with the aim of increasing the order of classical collocation methods, we use a general class of multistep methods based on Hermite collocation methods and prove that this numerical method has uniform order 2m+2r for r previous time steps and m collocation points. Some numerical examples are given to show the validity of the presented method and to confirm our theoretical results.

An Improved Time Integration Algorithm: A Collocation Time Finite Element Approach

A time collocation finite element approach is employed to develop one- and two-step time integration schemes with algorithmic dissipation control capability. The newly developed time integration schemes are combined to obtain a new family of time integration algorithms using the concept employed by Baig and Bathe. The newly developed algorithm can effectively control the algorithmic dissipation by relating the collocation parameters with the spectral radius in the high frequency limit. The new algorithm provides better accuracy compared with the generalized-[Formula: see text] method for highly dissipative cases and includes the Baig and Bathe method within its family as a special case.

Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems

We construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form (D∗αy)(t)=f(t,y(t)) with Caputo type fractional derivatives D∗αy of order α>0. Using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. To illustrate the reliability of the proposed algorithms two numerical examples are given.

Spline collocation for fractional weakly singular integro-differential equations

We consider a class of boundary value problems for linear fractional weakly singular integro-differential equations which involve Caputo-type derivatives. Using an integral equation reformulation of the boundary value problem, we first study the regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the boundary value problem by suitable non-polynomial approximations is discussed. Optimal global convergence estimates are derived and a super-convergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

Smoothing transformation and spline collocation for linear fractional boundary value problems

We construct and justify a high order method for the numerical solution of multi-point boundary value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. To illustrate the reliability of the proposed method some numerical results are given.

Functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström methods

A general class of functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström (FEPTRKN) methods for solving second-order initial value problems has been studied. These methods can be considered generalized explicit pseudo two-step Runge–Kutta–Nyström (EPTRKN) methods. We proved that an s-stage FEPTRKN method has step order p=s and stage order r=s for any set of distinct collocation parameters (ci)i=1s. Super-convergence for the accuracy orders of these methods can be obtained if the collocation parameters (ci)i=1s satisfy some orthogonality conditions. We proved that an s-stage FEPTRKN method can attain accuracy order p=s+3. Numerical experiments have shown that the new FEPTRKN methods work better than do the corresponding EPTRKN methods on problems whose solutions can be well approximated by the functions in bases on which these FEPTRKN methods are developed.

Modified spline collocation for linear fractional differential equations

We propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. Theoretical results are verified by some numerical examples.

Spline collocation for nonlinear fractional boundary value problems

We consider a class of boundary value problems for nonlinear fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties and spline collocation techniques, the numerical solution of boundary value problems by suitable non-polynomial approximations is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. Theoretical results are tested by two numerical examples.

HOAPS precipitation validation with ship-borne rain gauge measurements over the Baltic Sea

ABSTRACTGlobal ocean precipitation is an important part of the water cycle in the climate system. A number of efforts have been undertaken to acquire reliable estimates of precipitation over the oceans based on remote sensing and reanalysis modelling. However, validation of these data is still a challenging task, mainly due to a lack of suitable in situ measurements of precipitation over the oceans. In this study, validation of the satellite-based Hamburg Ocean Atmosphere Parameters and fluxes from Satellite data (HOAPS) climatology was conducted with in situ measurements by ship rain gauges over the Baltic Sea from 1995 to 1997. The ship rain gauge data are point-to-area collocated against the HOAPS data. By choosing suitable collocation parameters, a detection rate of up to about 70% is achieved. Investigation of the influence of the synoptic situation on the detectability shows that HOAPS performs better for stratiform than for convective precipitation. The number of collocated data is not sufficient to validate precipitation rates. Thus, precipitation rates were analysed by applying an interpolation scheme based on the Kriging method to both data sets. It was found that HOAPS underestimates precipitation by about 10%, taking into account that precipitation rates below 0.3 mm h−1 cannot be detected from satellite information.

Collocation Solutions of a Weakly Singular Volterra Integral Equation

The discrete superconvergence properties of spline collocation solutions for a certain Volterra integral equation with weakly singular kernel are analyzed. In particular, the attainable convergence orders at the collocation points are examined for certain choices of the collocation parameters.

Superconvergence of collocation methods for a class of weakly singular Volterra integral equations

We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters, superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical examples are given which illustrate the theoretical results.

STABILITY OF PIECEWISE POLYNOMIAL COLLOCATION FOR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS

Numerical stability of the spline collocation method by piecewise polynomials for Volterra integro‐differential equations is investigated. Stability conditions depending on collocation parameters and also on parameters of certain test equation are obtained. Results of several numerical tests are presented supporting theoretical results.