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1

Numerical study for a second order Fredholm integro-differential equation by applying Galerkin-Chebyshev-wavelets method

Henka Youcef, Lemita Samir, Aissaoui Mohamed-Zine

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 4

28--39

EN

In the present paper, we apply the Galerkin method using Chebyshev wavelets to approximate the exact solution for a second order Fredholm integro-differential equation with initial conditions. This numerical method gives us a nonlinear algebraic system that would be solved using the Picard successive approximations technique. Furthermore, we show the validity and the ability of the proposed method through some illustrative examples.

2

A novel hybrid iterative method for applied mathematical models with time-efficiency

Jamali Khalid, Solangi Muhammad Anwar, Qureshi Sania

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 3

19--29

EN

Non-linear phenomena appear in many fields of engineering and science. Research on numerical methods is continually extending with the improvement of the latest computing tools. In today’s computational field, one requires maximum achievement in a minimum amount of time. Therefore, there is a need to modify the Newton-type method to achieve higher-order convergence to solve non-linear equations. While the modified methods are expected to be higher-order convergent, the minor computational information and the maximum time efficiency are also important factors. We propose a three-step hybrid iterative method having a non-linear nature. Per iteration, the method requires three function evaluations and three first-order derivatives. The method is theoretically proven to be tenth-order convergent. The mathematical results of the proposed strategy to solve models from fluid dynamics, electric field, and real gases demonstrated better performance. In light of error analysis, computational productivity, and CPU times, the proposed method’s performance is compared to the famous Newton and a recently proposed tenth-order method.

3

Nonlinear mechanics of a compliant beam system undergoing large curvature deformation

Akano Theddeus Tochukwu, Olayiwola Patrick Shola

Archive of Mechanical Engineering

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2020

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LXVII, nr 4

471--489

EN

A compliant beam subjected to large deformation is governed by a multifaceted nonlinear differential equation. In the context of theoretical mechanics, solution for such equations plays an important role. Since it is hard to find closed-form solutions for this nonlinear problem and attempt at direct solution results in linearising the model. This paper investigates the aforementioned problem via the multi-step differential transformation method (MsDTM), which is well-known approximate analytical solutions. The nonlinear governing equation is established based on a large radius of curvature that gives rise to curvature-moment nonlinearity. Based on established boundary conditions, solutions are sort to address the free vibration and static response of the deforming flexible beam. The geometrically linear and nonlinear theory approaches are related. The efficacy of the MsDTM is verified by a couple of physically related parameters for this investigation. The findings demonstrate that this approach is highly efficient and easy to determine the solution of such problems. In new engineering subjects, it is forecast that MsDTM will find wide use.

4

Pionowe drgania własne osiowosymetrycznej sztywnej bryły zagłębionej w inercyjnej półprzestrzeni sprężystej

Sienkiewicz Z.

Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture

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2017

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z. 64, nr 2/I

119--128

PL

Przedstawiono analizę pionowych drgań własnych masywnej osiowosymetrycznej sztywnej bryły zagłębionej w jednorodnej inercyjnej półprzestrzeni sprężystej. Zespoloną sztywność półprzestrzeni z więzami nałożonymi przez sztywną bryłę otrzymano z rozwiązania mieszanego osiowosymetrycznego zagadnienia brzegowego dynamicznej teorii sprężystości metodą elementów brzegowych w dziedzinie częstości. Część rzeczywista zespolonej sztywności pionowej reprezentuje sztywność i inercję podłoża, część urojona przedstawia tłumienie związane z rozchodzeniem się fal w półnieskończonym ośrodku sprężystym (tłumienie radiacyjne). Współczynniki sztywności i tłumienia półprzestrzeni są funkcjami częstości drgań. Częstość drgań własnych sztywnej bryły z więzami nałożonymi przez inercyjną półprzestrzeń sprężystą jest pierwiastkiem nieliniowego równania charakterystycznego. Analizę drgań własnych przeprowadzono stosując parametry bezwymiarowe: współczynnik zagłębienia bryły w podłożu, współczynnik masy, współczynnik częstości oraz współczynnik tłumienia radiacyjnego. Przedstawiono zależność współczynnika częstości drgań własnych i współczynnika tłumienia od współczynnika masy i współczynnika zagłębienia. Wyznaczono również współczynniki częstości drgań własnych bryły przy pominięciu tłumienia radiacyjnego oraz w przypadku bryły zagłębionej w półprzestrzeni nieinercyjnej, której pionowa sztywność statyczna jest granicą dynamicznego współczynnika sztywności półprzestrzeni przy częstości dążącej do zera. Różnice między współczynnikami częstości reprezentują wpływ tłumienia radiacyjnego oraz inercji półprzestrzeni.

EN

An analysis of vertical eigenvibration of a massive axisymmetric rigid body embedded in a uniform elastic half-space is presented. The complex-value stiffness of the half-space with the constrains imposed by the rigid body has been obtained from the solution of a mixed axisymmetric boundary value problem of the dynamic elasticity by the boundary element metod in the frequency domain. The real part of the complex-valued stiffness represents the stiffnes and interia of the medium while the imaginary part describes the damping due to energy dissipated by waves propagating away from the foundation (radiation damping). Stiffness and damping coefficients of the half-space are frequency dependent. Eigenfrequency of the rigid body with the constrains imposed by the inertial elastic halfspace is the root of nonlinear characteristic equation. The analysis of the eigenvibration has been realized using the dimensionless parameters: embedment ratio, mass ratio, frequency ratio and radiation damping ratio. Variation of dimensionless eigenfrequency and damping ratio with the mass and embedment ratios are presented. Dimensionless eigenfrequencies at neglected radiation damping and in the case of a massless elastic medium are also computed. The differences between the damped and undamped eigenfrequencies represent the effects of radiation damping and interia of the half-space.

5

Shape identification in nonlinear boundary problems solved by pies method

Zieniuk E., Bołtuć A., Szerszeń K.

Acta Mechanica et Automatica

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2014

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Vol. 8, no. 1

16--21

EN

The paper presents the strategy for identifying the shape of defects in the domain defined in the boundary value problem modelled by the nonlinear differential equation. To solve the nonlinear problem in the iterative process the PIES method and its ad-vantages were used: the efficient way of the boundary and the domain modelling and global integration. The identification was performed using the genetic algorithm, where in connection with the efficiency of PIES we identify the small number of data required to the defect’s definition. The strategy has been tested for different shapes of defects.

6

An application of Newton-type iterative method for the approximate implementation of Lavrentiev regularization

George S, Pareth S

Journal of Applied Analysis

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2013

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Vol. 19, nr 2

181--196

EN

Motivated by the two-step directional Newton method considered by Argyros and Hilout (2010) for approximating a zero X* of a differentiable function F defined on a convex set D of a Hilbert space H, we consider a two-step Newton–Lavrentiev method (TSNLM) for obtaining an approximate solution to the nonlinear ill-posed operator equation F(x)=f, where F : D(F) ⊆ X → X is a nonlinear monotone operator defined on a real Hilbert space X. It is assumed that F(∧x)=f and that the only available data are fδ with ||f- fδ||≤δ. We prove that the TSNLM converges cubically to a solution of the equation F(x)+α(x-x0)= fδ (such solution is an approximation of ∧x) where x0 is the initial guess. Under a general source condition on x0-∧x, we derive order optimal error bounds by choosing the regularization parameter α according to the balancing principle considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

7

Oscillation criteria for third order nonlinear difference equations

Grace S. R., Agarwal R. P., Aktas M. F.

Fasciculi Mathematici

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2009

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Nr 42

39-51

EN

We shall establish some new criteria for the oscillation of third order nonlinear difference equations of the form ...[wzór]

8

Matematiceskoe modelirovanie nelinejnoj dinamiki i massoperenosa

Holpanov L.

Czasopismo Techniczne. Mechanika

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2008

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R. 105, z. 2-M

95-112

EN

In the message the methods of the decision of nonlinear dynamics and mass transfer are presented. The methods are illustrated on examples of a wave liquid films flow, mass transfer in a wave films, and mass transfer involving the inlet portion in different regimes. The laws of occurrence of self-organizing and chaos are submitted.

9

Rozwiązywanie równań nieliniowych metodami numerycznymi

Ufnalski W.

LAB Laboratoria, Aparatura, Badania

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2005

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R. 10, nr 1

42-48

10

Solving the nonlinear problem of gas-lubricated hybrid circular bearings. Part II: unsteady case

Nedelcu S., Postelnicu A.

International Journal of Applied Mechanics and Engineering

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2002

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Vol. 7, no 2

465-478

EN

In the paper, a dynamic analysis of gas-lubricated hybrid circular bearings is made. The mathematical model is the Reynolds equation in unsteady regimes along with the boundary conditions for a multiple connected domain. Within the hypothesis of a periodic relative motion of bearing surfaces, the method of small perturbations is used. The equations of the model are solved numerically using a difference finite method and finally, the curves of variation of the critical mass versus the eccentricity are obtained.

11

Uwagi na temat dynamiki mechanizmów

Skalmierski B.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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2001

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z. 17

195-200

PL

Analiza dynamiki mechanizmów wymaga rozwiązywania nieliniowych równań różniczkowych. Są nimi równania Lagrange ’a , które są równaniami drugiego rzędu. Są nimi równania Lagrange ’a , które są równaniami drugiego rzędu. W postaci tensorowej zapiszemy je następująco [wzór] W szczególnym przypadku, układów o jednym stopniu swobody redukują się one do postaci [2, 3]: [wzór]

EN

Mechanism dynamics analysis demands the solution of the complicated non-linear differential equations. These equations are derived on the basis of Lagrange equations of the second order and they are the tensor equations in the form: [formula] For the systems of single degree of freedom the system of equations (A) is reduced to a single equation in the form: [formula]

12

Optimization problems with convex epigraphs. Application to optimal control

Kryazhimskii A. V.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 4

773-801

EN

For a class of infinite-dimensional minimization problems with nonlinear equality constraints, an iterative algorithm for finding global solutions is suggested. A key assumption is the convexity of the "epigraph", a set in the product of the image spaces of the constraint and objective functions. A convexification method involving randomization is used. The algorithm is based on the extremal shift control principle due to N.N. Krasovskii. An application to a problem of optimal control for a bilinear control system is described.

13

Periodicity of the Population Size of an Age-Dependent Model With a Dominant Age Class by Means of Impulsive Perturbations

Covachev V.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

147-155

EN

For an age-dependent model with a dominant age class an w-periodic regime of the population size is sought by means of impulsive perturbations. For both noncritical and critical cases of first order the problem is reduced to operator systems solvable by a convergent simple iteration method.

14

On the nonexistence of positive solution of Laplace equation in half-space Rn with a nonlinear Neumann boundary condition

Binh D. T. T., Long N. T.

Demonstratio Mathematica

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2000

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Vol. 33, nr 2

365-372

15

Recent advances in solvers for nonlinear algebraic equations

Dent D., Paprzycki M., Kucaba-Piętal A.

Computer Assisted Mechanics and Engineering Sciences

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2000

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Vol. 7, No. 4

493-505

EN

In this paper the performance of four solvers for systems of nonlinear algebraic equations applied to a number of test problems with up to 250 equations is discussed. These problems have been collected from research papers and from the Internet and are often recognized as ``standard'' tests. Solver quality is assessed by studying their convergence and sensitivity to simple starting vectors. Experimental data is also used to categorize the test problems themselves. Future research directions are summarized.

16

Finite element method for a nonlinear problem

Bognar G.

Computer Assisted Mechanics and Engineering Sciences

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2000

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Vol. 7, No. 4

471-478

EN

We consider the nonlinear eigenvalue problem of a nonlinear partial differential equation under Dirichlet boundary condition in a two-dimensional space. The classical solutions are given for rectangular domains. We give numerical solutions obtained by finite element method for the first eigenvalue and eigenfunctions and we analyze the error in the approximate finite element solutions.

17

A certain approximate solutions of nonlinear flutter equation

Grzędziński J.

Engineering Transactions

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1999

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Vol. 47, iss. 1

39-55

EN

A nonlinear integro-differential flutter equation of a thin airfoil placed in an incompressible flow is solved by two different methods. The first method involves the center-manifold reduction and gives the asymptotic limit cycle amplitude and frequency in terms of power series expansions. The second method replaces the integro-differential equation by an approximate set of first-order ordinary differential equations which are solved by using bifurcation and continuation software package. A comparison of these two methods shows that the domain of a good agreement between them varies significantly depending on the parameters of the problem.