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1
Content available A numerical scheme for time-fractional fourth-order reaction-diffusion model
Koç Dilara Altan
Journal of Applied Mathematics and Computational Mechanics
|
2023
|
Vol. 22, nr 2
15--25
EN
In fractional calculus, the fractional differential equation is physically and theoretically important. In this article an efficient numerical process has been developed. Numerical solutions of the time fractional fourth order reaction diffusion equation in the sense of Caputo derivative is obtained by using the implicit method, which is a finite difference method and is developed by increasing the number of iterations. The advantage of the implicit difference scheme is unconditionally stable. The stability analysis and convergency have been proven. A numerical example has been presented, and the validity of the method is supported by tables and graphics.
2
Content available Nonlinear Interaction of Modes in a Planar Flow of a Gas with Viscous and Thermal Attenuation
Perelomova Anna
Archives of Acoustics
|
2019
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Vol. 44, No. 3
551--559
EN
The nonlinear interaction of wave and non-wave modes in a gas planar flow are considered. Attention is mainly paid to the case when one sound mode is dominant and excites the counter-propagating sound mode and the entropy mode. The modes are determined by links between perturbations of pressure, density, and fluid velocity. This definition follows from the linear conservation equations in the differentia form and thermodynamic equations of state. The leading order system of coupling equations for interacting modes is derived. It consists of diffusion inhomogeneous equations. The main aim of this study is to identify the principle features of the interaction and to establish individual contributions of attenuation (mechanical and thermal attenuation) in the solution to the system.
3
Content available Development of the Jackson and Hunt Theory for Rapid Eutectic Growth
Wołczyński W.
Archives of Metallurgy and Materials
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2018
|
Vol. 63, iss. 1
65--72
EN
A development of the Jackson-Hunt’s theory is delivered. Contrary to Jackson-Hunt’s theory for ideally coupled growth the current description is dealing with the coupled eutectic growth which is more realistic than an ideal course of eutectic structure formation. Thus, the undercooling of every eutectic phase is not equal to each other. A new boundary condition is introduced to solve the diffusion equation. According to this condition, the eutectic concentration is always maintained at the triple point of the solid / liquid (s/l) interface. Therefore, the solution to diffusion equation is given separately for both lamellae. The mass balance is satisfied by the current solution. Both thermodynamic equilibrium and mechanical equilibrium are assumed to be situated at the triple point of the s/l interface, only. A protrusion of the leading phase over the wetting phase is defined mathematically due to the mass balance fulfilment. The current description is associated with the asymmetrical phase diagrams. Finally, the current description is applied to interpretation of the rapid eutectic growth. Therefore, Aziz’s concept for the changes of partition ratio versus growth rate is introduced into the description. As a result, the rapid formation of the eutectic structure is described by the oscillatory mode. Interpretation of the oscillatory mode of the eutectic structure formation is illustrated in the arbitrary eutectic phase diagram. The eutectic structure, obtained through the detonation gas spraying onto the steel substrate (rapid solidification) is delivered to illustrate the present description.
4
Content available The practical stability of the discrete, fractional order, state space model of the heat transfer process
Oprzędkiewicz K., Gawin E.
Archives of Control Sciences
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2018
|
Vol. 28, No. 3
463--482
EN
In the paper the practical stability problem for the discrete, non-integer order model of one dimmensional heat transfer process is discussed. The conditions associating the practical stability to sample time and maximal size of finite-dimensional approximation of heat transfer model are proposed. These conditions are formulated with the use of spectrum decoposition property and practical stability conditions for scalar, positive, fractional order systems. Results are illustrated by a numerical example.
5
Content available Inverse Model for the Solute Micro-Field Formation during Self-Propagating High Temperature Reaction
Wołczyński W.
Archives of Metallurgy and Materials
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2017
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Vol. 62, iss. 1
141--147
EN
A new thermodynamic description for the self-propagating high temperature synthesis (SHS - reaction) is presented in the “inverse” version. This description is worked out for the diffusion barrier, thickness of which is at the limit, i.e. its value is infinitesimally small. The solution to the diffusion equation delivered in the description can be easily extended for the diffusion barrier of a greater thickness. The Ni/Al multi-layers system is treated as a virtual eutectic alloy solidifying with the rate equal to that involved by the self-propagating reaction. It is suggested to inverse the curves obtained for solidification in order to characterize the melting completed by the formation of the AlNi - intermetallic phase required in the self-propagating synthesis.
6
Content available General remarks on dynamic projection method
Leble S.
TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk
|
2016
|
Vol. 20, No 2
113--130
EN
A brief history and a mathematical description of the dynamic projection operators technique is presented. An example of the general Cauchy problem for evolution equations in 1+ 1 dimensions is studied in detail. A boundary regime propagation is formulated in terms of operators and illustrated by the simplest one-dimensional diffusion equation. The problem of temperature waves is discussed.
7
Content available A non integer order, state space model for one dimensional heat transfer proces
Oprzedkiewicz K., Gawin E.
Archives of Control Sciences
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2016
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Vol. 26, no. 2
261--275
EN
In the paper a new, state space, non integer order model for one dimensional heat transfer process is presented. The model is based on known semigroup model. The derivative with respect to time is described by the non integer order Caputo operator, the spatial derivative is described by integer order operator. The elementary properties of the state operator are proven. The solution of state equation is calculated with the use of Laplace transform. Results of experiments show, that the proposed model is more accurate than analogical integer order model in the sense of square cost function.
8
Content available Liquefaction of saturated soil and the diffusion equation
Sawicki A., Sławińska J.
Studia Geotechnica et Mechanica
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2015
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Vol. 37, nr 2
39--44
EN
The paper deals with the diffusion equation for pore water pressures with the source term, which is widely promoted in the marine engineering literature. It is shown that such an equation cannot be derived in a consistent way from the mass balance and the Darcy law. The shortcomings of the artificial source term are pointed out, including inconsistencies with experimental data. It is concluded that liquefaction and the preceding process of pore pressure generation and the weakening of the soil skeleton should be described by constitutive equations within the well-known framework of applied mechanics. Relevant references are provided.
9
Content available Producibility of the ion-exchange method in manufacturing gradient refractive index in glass
Rogoziński R.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2014
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Vol. 62, nr 4
655-665
EN
This paper presents the technological aspect of application of the ion exchange method in producing gradient refractive index in glass. The possibility of predictable and repeatable producing of the changes in glass refraction with the use of this method has been presented, as well as the method of in situ control of the process of diffusion doping of glass based on the measurement of the temperature. This method is based on simultaneous (to the carried process) solving the nonlinear diffusion equation modeling the spatio-temporal changes in normalized concentration of the admixture ions in glass. For this purpose the knowledge of temperature characteristics of diffusion coefficients of exchanged ions is used. The result of such control of diffusion processes is information on the current (temporary) refractive index profile of the resulting waveguide. The presented method of control has been confirmed by experimental results, which concern modeling and measurements of planar waveguide structures of slab type. The proposed methodology can also be used to control the diffusion processes of producing another type of two- and three-dimensional gradient structures. According to the author’s knowledge the method mentioned above has not been described in literature before.
10
Content available remote Determination of a fundamental solution for diffusion equation in cylindrical co-ordinate system with temperature-dependent source term
Borowska J., Klekot J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
|
2012
|
Vol. 11, nr 2
15-24
EN
In the paper the diffusion equation with temperature - dependent source function describing the heat conduction problem in axisymmetrical domain is considered. The fundamental solution is determined and the results obtained by means of the boundary element method are compared with the analytical solution.
11
Content available remote Metoda kolokacji rozwiązywania równań różniczkowych typu parabolicznego
Povstenko Y.
Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie
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2011
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Nr 1(1)
43--50
PL
W artykule omówiono metodę spektralną rozwiązywania równań typu parabolicznego. Opisano procedurę numeryczną i przedstawiono wyniki eksperymentu numerycznego.
EN
In this paper the spectral method for solving equations of parabolic type is discussed. Numerical procedure is described, and results of numerical experiment are presented.
12
Content available remote Possibility to use the diffusion equation to heat flow modeling in a composting process
Sobieski W.
Journal of Applied Computer Science
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2011
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Vol. 19, nr 2
111-123
EN
The paper proposes a new concept to use the diffusion equation for heat flow estimation in a composting process. Only general ideas are described, without the full mathematical model and without experiments. The lack of the detailed description stems from the fact that such an approach was not (as it seems) used earlier, and the details are to be developed. The article describes the preparatory work in this field. In the first part of the article, derivation of the diffusion equation is presented together with using an approach consistent with modern Computational Fluid Dynamics, in which all fluid phenomena in a Control Volume are considered. In the second part of this article, the basic information about implementation and needed programming techniques are described. It is emphasized that the diffusion equation is adequate not only for the temperature flows, but for other phenomena as well. The main purpose of this study is to create a general template for 3D modeling of phenomena which are diffusive in nature.
13
Content available Solutions to Time-Fractional Diffusion-Wave Equation in Spherical Coordinates
Povstenko Y.
Acta Mechanica et Automatica
|
2011
|
Vol. 5, no. 2
108-111
EN
Solutions to time-fractional diffusion-wave equation with a source term in spherical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time t, the finite Fourier transform with respect to the angular coordinate , the Legendre transform with respect to the spatial coordinate , and the Hankel transform of the order n+1/2 with respect to the radial coordinate . In the central symmetric case with one spatial coordinate the obtained results coincide with those studied earlier.
14
Content available remote Optical tomography benchmark: analytical solution of 2D regions with clear layer gap
Sikora J., Filipowicz S.F., Nita K.
Przegląd Elektrotechniczny
|
2005
|
R. 81, nr 2
103-106
EN
The paper presents an analytical solution of the clear region immersed in the highly scattering one. In the clear region diffusion equation is not valied, so the problem is solved with the aid of non-local boundary conditions, connecting highly scattering regions. This problem is very important in Optical Tomography as it is a numerical model of the thin CSF layer in the human brain. The analytical solution is an excellent benchmark problem so we can compare the numerical solution like FEM or BEM with the analytical one presented in this paper.
PL
Praca prezentuje analizę przypadku, gdy występuje podobszar przezroczysty, w którym nie obowiązuje równanie dyfuzji, zgodnie z którym, światło jest rozprzestrzeniane w środowiskach silnie rozpraszających (aproksymacja równania Boltzmana). W tym przypadku obszary rozdzielone taką warstwą (na przykład cienka warstwa płynu rdzeniowo-mózgowego CSF), są analizowane za pomocą tzw. nielokalnych warunków brzegowych, W pracy przedstawiono dwuwymiarowy przykład, rozpatrzono dwa przypadki takich warunków brzegowych i podano rozwiązanie dokładne, które może być pomocne przy weryfikacji symulacji numerycznej przypadków bardziej złożonych.
15
Content available remote Correction of the diffusion equation
Kholeif I. A., El-Naggar B. B.
Mechanics and Mechanical Engineering
|
2004
|
Vol. 7, nr 2
99-107
EN
A test problem is investigated and indicate that the conventional foundation of Fick's law and the resulting diffusion equation admit mass transfer at relatively high velocity. This contradicts nature and two independent corrections are made: 1. The front beyond which matier cannot reach; advances with a characteristic speed dependent on the diffusing substance and the medium; 2. Relativistic type correction in which time dilation and length contraction is taken in consideration. In both cases solutions are obtained and discussed.
16
Content available remote Some nonlinear diffusion equation with three nonlinearities
Tal-Figiel B.
Fasciculi Mathematici
|
2004
|
Nr 34
105-120
EN
The subject of the paper is a construction of the classical solution to the nonlinear diffusion equation , in the domain , satisfying the limit conditions <2 formulas>.
17
Content available remote The Use of the Collocation Method for Solving the Diffusion Equation under Two Mass Specifications
Derkach S.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2003
|
Vol. 9
31--33
EN
In this paper the diffusion equation [formula] with the initial condition [formula] is considered.
18
Content available remote New integral equation approach to solution of diffusion equation
Sladek V., Sladek J., Keer van R.
Computer Assisted Mechanics and Engineering Sciences
|
2002
|
Vol. 9, No. 4
556-572
EN
The paper concerns the theoretical derivation of a new formulation for solution of the initial-boundary value problems for the diffusion equation. The global and local integral equations are derived by using the fundamental solution for the Laplace differential operator. Assuming certain approximations with respect to spatial variable, we obtain a set of the ordinary differential equations (ODE) with continuous time variable. Standard methods for the time integration can be applied to these ODEs. Besides a review of the one step theta-method we propose a new integral equation method for solution of a set of linear ODEs. The paper deals also with the numerical implementation of the global and local integral equations yielding the ODEs.
19
Multiple elliptical-Gaussian-Density annealing as a tool for finding the most stable structures. Application to Lennard-Jones atomic clusters
Pillardy J., Piela L.
Polish Journal of Chemistry
|
1998
|
Vol. 72, nr 7S
1849-1857
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