The method of normal spline collocation

Gorbunov, V. K.

doi:10.1016/0041-5553(89)90059-1

Cited by 8 publications

(7 citation statements)

References 4 publications

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“…The minimal norm element of this intersection is named the normal spline x m . It is known [4], the sequence of the normal splines x m converges to a normal solution x 0 in the norm (3) when the maximum step of the grid (4) tends to zero.…”

Section: The Normal Spline Methodsmentioning

confidence: 99%

“…In [4], [5] there was developed a variational normal spline (NS) method for the problem (1), (2). The cases of infinite intervals [0, ∞) and (−∞, ∞) were also covered.…”

Section: Introductionmentioning

confidence: 99%

“…One can find the results of the realization of the NS method for ODEs under l ∈ {1, 2} in [4], [5]. The method was realized for arbitrary integer l in [9] where general view of reproducing kernels were created.…”

Section: Introductionmentioning

confidence: 99%

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Development of the Normal Spline Method for Linear Integro-Differential Equations

Gorbunov

Petrischev

Sviridov

2003

Lecture Notes in Computer Science

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The normal spline method is developed for the initial and boundary-value problems for linear integro-differential equations, probably being unresolved with respect to the derivatives, in Sobolev spaces of the arbitrary smoothness. It allows to solve a high-order systems without the reduction to first-order ones. The solving system can be arbitrary degenerate (with high differentiation index or irreducible to normal form). The method of nonuniform collocation grid creation for stiff problems is offered. Results of numerical solution to test problems are demonstrated. Supported by Russian Foundation for Basic Research, project N o 01-01-00731

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Section: The Normal Spline Methodsmentioning

confidence: 99%

“…In [4], [5] there was developed a variational normal spline (NS) method for the problem (1), (2). The cases of infinite intervals [0, ∞) and (−∞, ∞) were also covered.…”

Section: Introductionmentioning

confidence: 99%

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Development of the Normal Spline Method for Linear Integro-Differential Equations

Gorbunov

Petrischev

Sviridov

2003

Lecture Notes in Computer Science

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“…1. the choice of the coordinate functions ϕ jk (t) (7) and their number; 2. the smoothness condition for the connection of the approximate solution in the nodes t j , j=1, 2, …, N−1 (the number q in (8)); 3. the number of collocation points l in (10) and their location on the interval [t k−1 , t k ]; 4. the choice of the objective function.…”

Section: The Collocation-variation Approachmentioning

confidence: 99%

Collocation‐variation difference schemes for differential‐algebraic equations

Bulatov

Solovarova

2018

Math Methods in App Sciences

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We consider the initial value problem in linear differential-algebraic equations and highlight the difficulties of the construction of numerical methods for these problems. We propose one-step difference schemes based on the collocation-variational approach and show the principal difference between these algorithms and the known difference schemes. The analysis of the particular cases of such schemes are given. KEYWORDS differential-algebraic equations, difference schemes | INTRODUCTIONA number of applied problems can be described by interrelated ODEs and algebraic equations. These systems are called differential-algebraic equations. Examples of the problems can be found in previous works. [1][2][3][4][5] It is assumed that the initial condition is given in such a way that the problem under consideration has a solution.Both the qualitative theory and the theory of numerical methods of DAEs have been developed since the 1970s to 1980s. This is due to the fact that the construction and the justification of numerical methods for solving DAEs is a more complex task than the development of methods for initial value problem in ODEs solved with respect to derivatives. At present, there are several approaches to the numerical solution of the problem under consideration.1. Application of various projectors, the generalized inverse of a matrix, the operation of differentiation for reduction of the original problem to a simpler problem, for example, to an ODE solved with respect to derivatives. A bibliography on this topic is presented in other studies. 3,6 2. The method of normal splines, 7 which is similar to the least squares method. 8 3. The method of solution continuation with respect to the parameter. 9 4. Using the so-called extended systems that are based on the following idea: The original DAE and several its total derivatives are combined in a single system. By applying nonsingular transformations to this system, we obtain a first-order ODE. Further research and an extensive bibliography on this topic are presented in previous works. 2,4 This approach can be used for the construction of special difference schemes for DAEs.

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“…An b.v. / i.v. problem for (5.7) in a regular as well as in a degenerated case can be effective resolved by means of the normal spline-collocation (NS) method, developed in our works [9,10,16]. This method concluding in the introducing of arbitrary collocation net and minimization of a norm of the Hilbert-Sobolev In [4] it is shown, that the system under condition of γ + δ t = 1, T ∈ [0, 1] has the unique solution at any right part f ∈ C 2 2 [0, 1].…”

Section: Example Of Singular Problem In Odementioning

confidence: 99%

Regularization of degenerated equations and inequalities under explicit data parameterization

Gorbunov

2001

Journal of Inverse and Ill-posed Problems

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The regularization problem of degenerated nonlinear system of equations/inequalities in Banach spaces is considered. The proposed approach is based on the explicit parametrization of input data and on the utilization of the multi-valued mapping techniques. Two regularization methods are suggested: the extended minimization and the regularized partial penalty function one. According to the first method an ill-posed problem should be replaced by a search of the minimal norm element in the join of solution sets of the family of problems that are equivalent to the initial one with respect to input data accuracy. The second one is based on separation from the initial system some regular part. Suggested approach is illustrated on some examples (analytical and numerical), including a singular system of ordinary differential and finite equations.

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The method of normal spline collocation

Cited by 8 publications

References 4 publications

Development of the Normal Spline Method for Linear Integro-Differential Equations

Development of the Normal Spline Method for Linear Integro-Differential Equations

Collocation‐variation difference schemes for differential‐algebraic equations

Regularization of degenerated equations and inequalities under explicit data parameterization

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