On approximate solution of boundary-value problems by the least squares method

Чуйко, С. М.

doi:10.1007/s11072-009-0053-9

Cited by 30 publications

(15 citation statements)

References 2 publications

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“…To determine an approximate solution of the periodic problem for the Duffing-type equation (8), we use the least-squares method [8][9][10][11]. Thus, we set ' 1 .t / D OEsin t sin 3t sin 5t sin 7t sin 9t sin 11t :…”

Section: Sufficient Conditionmentioning

confidence: 99%

“…According to the iterative scheme [8], the first approximation to the solution of the 2 -periodic problem for the Duffing-type equation (8) We set ' 2 .t / D OEsin t sin 3t sin 5t sin 7t sin 9t sin 11t sin 13t sin 15t :…”

Section: Sufficient Conditionmentioning

confidence: 99%

“…According to the iterative scheme [8], the third approximation to the solution of the 2 -periodic problem for the Duffing-type equation (8) In [7], we have obtained the following approximation to the solution of the 2 -periodic problem for the Duffing-type equation (8):…”

Section: Sufficient Conditionmentioning

confidence: 99%

“…The accuracy of the obtained approximation to the solution of the periodic problem for the Duffing-type equation (8) The accuracy of three approximations to the solution of the 2 -periodic problem for the Duffing-type equation (8) obtained by the iterative scheme [8] is characterized by the discrepancies k . "/ WD y 00 k .t; "/ C y k .t; "/ D " The procedure of reduction of the Fredholm boundary-value problem to the critical case of the first order proposed in the present paper can also be used in the case of boundary-value problems for functional differential equations [1,2,9,11,12].…”

Section: Sufficient Conditionmentioning

confidence: 99%

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Reduction of the Fredholm Boundary-Value Problem to the Critical Case of the First Order

2015

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UDC 517.9We establish necessary and sufficient conditions for the existence of solutions and propose an iterative scheme for the determination of solutions of the Fredholm weakly nonlinear boundary-value problem for a system of ordinary differential equations in the case of multiple roots of the equation for generating constants. Statement of the ProblemWe study the problem of construction of the solution z.t; "/ W z.;We seek the solution of problem (1) in a small neighborhood of the solution of the generating problemHere, A.t/ is an .n n/ matrix and f .t / is an n -dimensional column vector whose elements are real functions continuous on the segment OEa; b and`z./ is a linear bounded vector functional`z./ W C OEa; b ! R m : Assume that the nonlinearities Z.z; t; "/ and J.z.; "/; "/ of the Fredholm .m ¤ n/ problem (1) are twice continuously differentiable with respect to the unknown z in a small neighborhood of the generating solution and with respect to the small parameter " in a small positive neighborhood of zero. In addition, we also assume that the vector function Z.z; t; "/ is continuous in the independent variable t on the segment OEa; b: We investigate the critical case .P Q 6 D 0/ under the assumption thatIn this case, the generating problem has an .r D n n 1 / -parameter family of solutionsHere, X.t/ is a normal .X.a/ D I n / fundamental matrix of the homogeneous part of the generating system, Q D`X./ is an .m n/ matrix, rank Q D n 1 ; X r .t / D X.t /P Q r ; P Q r is an .n r/ matrix formed by r

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Section: Sufficient Conditionmentioning

confidence: 99%

Section: Sufficient Conditionmentioning

confidence: 99%

Section: Sufficient Conditionmentioning

confidence: 99%

Section: Sufficient Conditionmentioning

confidence: 99%

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Reduction of the Fredholm Boundary-Value Problem to the Critical Case of the First Order

2015

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“…The aim of the present paper is to construct a modified iterative procedure for finding solutions of the boundaryvalue problem (7), (8) by analogy with [6] with the use of the least-squares method [7], which guarantee higher accuracy for a smaller number of iterations. Let ' 1 .…”

Section: Iterative Schemementioning

confidence: 99%

On the approximate solution of autonomous boundary-value problems by the least-squares method

Чуйко

Starkova

2009

Nonlinear Oscill

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Using the least-squares method, we construct a new iterative procedure for finding solutions of an autonomous weakly nonlinear boundary-value problem in the critical case in the form of a generalized Fourier polynomial expansion. Statement of the ProblemWe investigate the problem of the construction of solutions z.t; "/W z. ; "/ 2 C 1 OEa; b."/; z.t; / 2 C OE0; " 0 ; b. / 2 C OE0; " 0 ; of the autonomous boundary-value problem dz dt D Az C f C "Z.z; "/;`z. ; "/ D˛C "J.z. ; "/; "/;˛2 R m :We seek a solution of the Noetherian .m ¤ n/ problem (1) in a small neighborhood of theHere, A is a constant n n matrix, Z.z; "/ is a nonlinear vector function continuously differentiable with respect to the unknown z in a small neighborhood of a solution of the generating problem and continuously differentiable with respect to the small parameter " on the segment OE0; " 0 ;`z. ; "/ is a linear vector functional, J.z. ; "/; "/ is a nonlinear vector functional,`z. ; "/; J.z. ; "/; "/W C OEa; b."/ ! R m ; and the latter functional is continuously differentiable with respect to the unknown z and the small parameter " in a small neighborhood of a solution of the generating problem and on the segment OE0; " 0 : In the critical case .P Q ¤ 0/; under the condition P Q d f˛ `KOEf . /g D 0; problem (2) has the family of solutions [1, 2] z 0 .t; c r / D X r .t /c r CGOEf I˛.t /; c r 2 R r : Here, Q D`X. / is an m n matrix, rank Q D n 1 ; n n 1 D r; P Q is an m m orthoprojector matrix, P Q W R m ! N.Q /; X.t/ is the normal .X.a/ D I n / fundamental matrix of the homogeneous part of the differential system (2), X r .t/ D X.t/P Q r ; P Q r is an n r matrix composed of r linearly independent columns of the n n orthoprojector matrix, P Q W R n ! N.Q/; P Q d is a d m matrix composed of d D m n 1 linearly independent rows of the orthoprojector matrix P Q ; GOEf I˛.t / D X.t /Q C f˛ `KOEf . /g C KOEf .t/ is the generalized Green operator of problem (2), Q C is the Moore-Penrose pseudoinverse matrix [1],

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On the approximate solution of an autonomous boundary-value problem by the Newton–Kantorovich method

2013

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We establish necessary and sufficient conditions for the existence of solutions of an autonomous Noetherian boundary-value problem for a system of second-order ordinary differential equations in the critical case. For the construction of solutions of a nonlinear Noetherian boundary-value problem in the critical case, we propose an iterative scheme that combines the Newton-Kantorovich method and the leastsquares technique. The efficiency of the proposed method is demonstrated in the analysis of a periodic problem for a Liénard-type equation.

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On approximate solution of boundary-value problems by the least squares method

Cited by 30 publications

References 2 publications

Reduction of the Fredholm Boundary-Value Problem to the Critical Case of the First Order

Reduction of the Fredholm Boundary-Value Problem to the Critical Case of the First Order

On the approximate solution of autonomous boundary-value problems by the least-squares method

On the approximate solution of an autonomous boundary-value problem by the Newton–Kantorovich method

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