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1
Exact solutions for the total variation denoising problem of piecewise constant images in dimension one
Cristoferi Riccardo
Journal of Applied Analysis
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2021
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Vol. 27, nr 1
13--33
EN
A method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented. The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies. Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.
2
Content available remote Differential Equation Chazy of the Third Order with Exact Solutions Expressed via the Elliptic Functions
Chichurin A.
Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie
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2017
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Nr 2(14)
29--46
EN
The procedure of integrating the third-order Chazy differential equation with six singularities is presented, when the first six coefficients are equal to ±1/2, ± 1, ± 2. For five third-order differential equations it is shown that their general solutions are expressed via the elliptic functions. Two-parameter families of solutions in elliptic functions and one-parameter families of solutions in elementary functions are presented. The proposed integration method can be realized for the differential Chazy equation in those cases when the first six coefficients have values for which it is possible to solve a special algebraic equation of the fifth degree in radicals.
PL
W artykule rozpatrzona została metoda rozwiązania równania różniczkowego Chazy’ego trzeciego rzędu, zawierającego sześć osobliwości w przypadku, gdy sześć pierwszych współczynników równa się ± 1/2, ± 1, ± 2. Dla pięciu równań różniczkowych trzeciego rzędu znaleziono ich ogólne rozwiązania za pomocą funkcji eliptycznych. Przedstawione zostały rodziny rozwiązań dwuparametrycznych za pomocą funkcji eliptycznych oraz rodziny rozwiązań jednoparametrycznych za pomocą funkcji elementarnych. Proponowana metoda rozwiązania może być zrealizowana dla równania różniczkowego Chazy’ego w tych przypadkach, gdy pierwsze sześć współczynników ma wartości, dla których istnieje możliwość rozwiązania szczególnego równania algebraicznego piątego stopnia za pomocą pierwiastków.
3
Content available Combined algorithm for finding conservation laws and implectic operators for the Boussinesq-Burgers nonlinear dynamical system and its finite dimensional reductions
Kindybaliuk A., Prytula M.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 3
85--99
EN
In the article the combined algorithm for finding conservation laws and implectic operators has been proposed. Using the Novikov-Bogoyavlensky method the finite dimensional reductions have been found. The structure of invariant submanifolds has been examined. Having analyzed phase portraits of Hamiltonian systems, partial periodical solutions have been found.
4
Content available remote Influences of the moving velocity and material property on frictionless contact problem of orthotropic materials indented by a moving punch
Zhou Y. T., Lee K. Y., Jang Y. H.
Archives of Mechanics
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2013
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Vol. 65, nr 3
195--217
EN
In analyzing the contact behavior of a material indented by a moving punch, of much importance are the contributions of the moving velocity and material property. The present paper develops a smoothly moving contact model for orthotropic materials indented by a rigid punch. Based on fundamental solutions of each eigenvalue case, the mixed boundary-value problem is reduced to a Cauchy type singular integral equation by applying the Galilean transformation and Fourier transform. Particularly, the exact solution of the obtained singular integral equation is presented, and closed-form expressions of the physical quantities are given for a flat punch and a cylindrical punch. Figures are plotted to show the influences of the moving velocity, material properties and other loadings on the contact behaviors and to reveal the surface damage mechanism, which may provide useful guidelines for material’s designing and optimization.
5
Content available Exact solutions for free vibrations of a steel-concrete bridge
Longa L. D., Morassi A., Rotaris A.
Journal of Theoretical and Applied Mechanics
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2011
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Vol. 49 nr 3
705-705
EN
This paper deals with a class of free vibrations for a two-span, two-lane steel-concrete bridge. The deck structure is modeled as a thin, homogeneous, orthotropic plate stiffened by beams running along the longitudinal direction of the bridge. The method of separation of variables is used to find exact solutions for a class of free vibrations of the structure. A comparison between analytical and experimental natural frequencies and vibration modes of the bridge is presented and discussed.
PL
Praca dotyczy analizy pewnej klasy drgań swobodnych dwuprzęsłowego, dwujezdniowego mostu stalowo-betonowego. Przęsło zamodelowano jako cienką, jednorodną, ortotropową płytę usztywnioną belkami zamocowanymi w kierunku wzdłużnym mostu. Do znalezienia dokładnych rozwiązań problemu drgań swobodnych modelu zastosowano metodę rozdzielenia zmiennych. Na koniec zaprezentowano i przedyskutowano porównanie częstości własnych i postaci drgań własnych modelu z wynikami zarejestrowanymi na rzeczywistej konstrukcji.
6
Content available remote Exact solutions for the longitudinal flow of a generalized Maxwell fluid in a circular cylinder
Siddique I.
Archives of Mechanics
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2010
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Vol. 62, nr 4
305-317
EN
This paper deals with the longitudinal flow of a generalized Maxwell fluid in an infinite circular cylinder, due to the longitudinal variable time-dependent shear stress that is prescribed on the boundary of the cylinder. The fractional calculus approach in the constitutive relationship model of a Maxwell fluid is introduced. The velocity field and the resulting shear stress are obtained by means of the Laplace and finite Hankel transforms and satisfy all the imposed initial and boundary conditions. The solutions corresponding to ordinary Maxwell fluids as well as those for Newtonian fluids are obtained as limiting cases of our general solutions. Finally, the influence of the fractional coefficient on the velocity and shear stress of the fluid is analyzed by graphical illustrations.
7
Content available Application of Waterloo MAPLE 9.5 and Wolfram MATHEMATICA 5.1 software for analytic solving of certain nonlinear partial differential equations of physics
Stępień Ł.T.
Computer Science
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2008
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Vol. 9
147-155
EN
In the current paper some applications of the packet MAPLE (v. 9.5) for analytic solying of certain nonline partial differential eguations have been presented. Additionally, for graphic presentation of the found solutions packet MATHEMATICA (v. 5.1) has been applied.
PL
W niniejszej pracy zostały przedstawione zastosowania pakietu MAPLE (wersja 9.5 do analitycznego znajdywania rozwiązań wybranych nieliniowych równań różniczkowych cząstkowych. Do prezentacji graficznej tych rozwiązań użyto pakietu MATHEMATICA (wersje 5.1).
8
Content available remote Exact solutions for unsteady incompressible viscous fluid flows
Jamil M., Ahmed F., Khan N. A.
International Journal of Applied Mechanics and Engineering
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2007
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Vol. 12, no 4
975-988
EN
Two-dimensional, unsteady, laminar equations of motions of an incompressible fluid with variable viscosity are considered. The problem investigated is the flow for which the vorticity distribution is proportional to the stream function perturbed by a generalized uniform stream making an angle with the positive x-axis. Employing transformation variables, the goveming Navier-Stocks Equations (NSE) are transformed into steady state equations and then simple ordinary differential equations and a class of exact solutions are obtained. Several graphs of physical interest of streamline are also displayed and discussed.
9
On exact solutions on some nonlinear diffusion equation
Icha A.
Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska
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2005
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z. 29
133-138
EN
The generalized conditional symmetry method is useful tool in finding symmetry reductions and constructing exact solutions of some nonlinear evolution equations. This paper considers a general form of the equation u,= K(t,u,ux), where the subscripts denote differentiation with respect to indicated variables. The construction of some exact solutions of the nonlinear diffusion equation with power law diffusivity is investigated by using the generalized conditional symmetry approach.
PL
Ważna metodą w badaniu nieliniowych równań różniczkowych cząstkowych jest podejście oparte na koncepcji uogólnionej symetrii warunkowej. Celem pracy jest zaprezentowanie jednolitego ujęcia zagadnienia w celu znalezienia nowych, ściłych rozwiązań nieliniowego równania ewolucyjnego postaci u = K(t,u,ux), gdzie indeksy oznaczają różniczkowanie względem wskazanych zmiennych. Wykorzystując metodę uogólnionej symetrii warunkowej znaleziono przykładowe, scisłe rozwiązanie nieliniowego równania dyfuzji z potęgowym współczynnikiem dyfuzji.
10
Content available remote Exact N-envelope-soliton solutions of the Hirota equation
Shu J.-J.
Optica Applicata
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2003
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Vol. 33, nr 2-3
539-546
EN
We discuss some properties of the soliton equations of the type ś u/ś t = S[u, u], where S is a nonlinear operator differential in x, and present the additivity theorems of the class of the soliton equations. On using the theorems, we can construct a new soliton equation through two soliton equations with similar properties. Meanwhile, exact N-envelope-soliton solutions of the Hirota equation are derived through the trace method.
11
Content available remote Exact solutions to an incompressible second-grade fluid flow equations
Naeem R. K., Zia S. S.
International Journal of Applied Mechanics and Engineering
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2003
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Vol. 8, no 1
71-85
EN
Employing complex variables and complex functions the exact solutions to equations governing the motion of an incompressible second-grade fluid are determined.
12
Exact solutions for free longitudinal vibration of bars with non-uniform Cross-sections
Li Q. S.
Applied Mechanics and Engineering
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2000
|
Vol. 5, no 3
521-541
EN
Using appropriate transformations, the differential equations of free longitudinal vibration of bars with variably distributed mass and axial stiffness are reduced to Bessel?s equations or ordinary differential equations with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the distributions of mass and axial stiffness. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step non-uniform bar are derived and used to obtain the general solution and the frequency equation of a multi-step non-uniform bar with various boundary conditions, including non-classical cases. This approach which combines the transfer matrix method and closed form solutions of one step bars leads to a single frequency equation for any number of steps.
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