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Numerical Analysis and Applications

2014 year, number 1

On the numerical solution to loaded systems of ordinary differential equations with non-separated multipoint and integral conditions

Kamil Rajab Aidazade1, Vagif Maarif Abdullaev2
1Azerbaijan State Oil Academy, 20, Azadlyg ave., AZ1010, Baku, Azerbaijan
2Cybernetics Institute of Azerbaijan National Academy of Sciences, 9, AZ1141, Azerbaijan
Keywords: ordinary loaded differential equations system, non-separated conditions, integral conditions, non-local multipoint conditions, sequential shift operation

Abstract >>
We propose a numerical method of solving systems of linear non-autonomous ordinary loaded differential equations with non-separated multipoint and integral conditions. This method is based on the operation of convolution of integral conditions to local conditions. This approach allows reducing the solution to the original problem to solving the Cauchy problem with respect to a system of ordinary differential equations and to linear algebraic equations. Numerous computational experiments on several test problems with application of the formulas and schemes of the numerical solution have been carried out. The results of the experiments have shown a sufficiently high efficiency of the approach described.

A method of optimal real-time computation of a linear system with retarded control

Vladimir Aleksandrov
Sobolev Institute Mathematics of SB RAS, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia
Keywords: optimal control, speed, switching moment, retardation, adjoint system, phase trajectory

Abstract >>
A new method of solving time-optimal control problems in real time has been developed. The method is based on the following: 1) approximating the attainability sets with a family of hyperplanes; 2) subdividing the whole computational process into the computations performed beforehand and those that are carried out while the control takes place; 3) integrating differential equations only over the displacement intervals of the control completion moment and the switching moments. The labor-intensive characteristic of the method is evaluated. Characteristics of calculating the optimal control of a linear system with retarded control in real time are considered. The results of modeling and numerical estimations are presented.

New modified optimal families of King's and Traub-Ostrowski's method

Ramandeep Behl1, V. Kanwar2, Kapil K. Sharma3
1School of Mathematics & Computer Applications, Thapar University, Patiala-147 004, India
2University Institute of Engineering and Technology, Panjab University, Chandigarh-160 014, India
3Department of Mathematics, South Asian University Akbar Bhavan, Chayankya Puri, New Delhi, India
Keywords: nonlinear equations, Newton's method, King's family, Traub-Ostrowski's method, Jarratt's method, optimal order of convergence, efficiency index

Abstract >>
Based on quadratically convergent Schröder's method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King's family of fourth-order methods and Traub-Ostrowski's method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King's method and Traub-Ostrowski's method are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King's family, Traub-Ostrowski's method, and Jarratt's method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.

A sensitivity functionals in variational inequalities of mechanics and their application to duality schemes

Ellina Mihailovna Vikhtenko1, Nadezhda Nikolaevna Maksimova2, Robert Viktorovich Namm3
1Pacific National University, Khabarovsk, Tihookeanskaya str., 136
2Amur State University, Blagoveshchensk, Ignatievskoe Highway 21
3Computing Center FEB RAS, Khabarovsk, Kim Yu Chen str., 65
Keywords: scalar Signorini problem, duality scheme, modified Lagrangian functional, sensitivity functional

Abstract >>
Characteristic properties of a sensitivity functional in the variational inequalities mechanics on an example of a scalar Signorini problem are investigated. Applications of sensitivity functionals in duality schemes are considered.

On a posteriori approximation of a set of solutions to a system of quadratic equations with the use of the Newton method

Mikhail Yurjevich Kokurin, Alexander Ivanovich Kozlov
Mary State University, Space them. 1 Lenin, Yoshkar-Ola
Keywords: quadratic operator, the Newton method, a posteriori estimate, numerical range, convex hull

Abstract >>
For quadratic systems of algebraic equations we propose an algorithm generating a posteriori estimates of a convex hull of a set of solutions using the results of a step of the Newton method. Results of numerical tests are given.

A family of highly stable second derivative block methods for stiff IVPs in ODEs

R.I. Okuonghae, M.N.O. Ikhile
Department of Mathematics, University of Benin, P.M.B 1154, Benin City, Edo state, Nigeria
Keywords: block methods, continuous methods, collocation and interpolation, boundary locus, A(О±)-stability, stiff IVPs

Abstract >>
This paper considers a class of highly stable block methods for the numerical solution of initial value problems (IVPs) in ordinary differential equations (ODEs). The boundary locus of the proposed parallel one-block, r–output point algorithms shows that the new schemes are A-stable for output points r = 2(2)8 and A(α)-stable for output points r = 10(2)20, where r is the number of processors in a particular block method in the family. Numerical results of the block methods are compared with a second derivative linear multistep method in [8].

Semilocal convergence for the Super-Halley's method

M. Prashanth, D.K. Gupta, S. Singh
Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, India
Keywords: nonlinear operator equations, П‰-continuity condition, recurrence relations, R-order of convergence, a priori error bounds

Abstract >>
The semilocal convergence of Super-Halley's method for solving nonlinear equations in Banach spaces is established under the assumption that the second Frёchet derivative satisfies the ω-continuity condition. This condition is milder than the well known Lipschitz and Hölder continuity conditions. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where the Lipschitz and Hölder continuity conditions fail. Difficult computation of the second Frёchet derivative is also avoided by replacing it with a divided difference containing only the first Frёchet derivatives. A number of recurrence relations based on two parameters are derived. A convergence theorem is established to estimate a priori error bounds along with the domains of existence and uniqueness of the solutions. The R–order of convergence of the method is shown to be at least three. Two numerical examples are worked out to demonstrate the efficiency of our method. It is observed that in both examples the existence and uniqueness regions of solution are improved when compared with those obtained in [7].