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1
Content available Symmetry analysis, exact solutions and conservation laws of the nonlinear time-fractional sharma-tasso-olever equation
Yu Jicheng, Feng Yuqiang
Journal of Applied Mathematics and Computational Mechanics
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2024
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Vol. 23, nr 4
120--132
EN
The Lie symmetry analysis method (LSAM) is applied to obtain all Lie symmetries of the nonlinear time-fractional Sharma-Tasso-Olever equation. The studied fractional partial differential equation (FPDEs) is reduced to some fractional ordinary differentia equations (FODEs), of which some exact solutions including the convergent power series solution are obtained. The dynamic behaviors of these exact solutions are presented graphically. In addition, the conservation laws for the obtained symmetries are constructed by Ibragimov’s theory.
2
Multiple solitons, periodic solutions and other exact solutions of a generalized extended (2 + 1)-dimensional Kadomstev-Petviashvili equation
Humbu Isaac, Muatjetjeja Ben, Motsumi Teko Ganakgomo, Adem Abdullahi Rashid
Journal of Applied Analysis
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2024
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Vol. 30, nr 1
197--208
EN
This paper aims to study a generalized extended (2 + 1)-dimensional Kadomstev-Petviashvili (KP) equation. The KP equation models several physical phenomena such as shallow water waves with weakly nonlinear restoring forces. We will use a variety of wave ansatz methods so as to extract bright, singular, shock waves also referred to as dark or topological or kink soliton solutions. In addition to soliton solutions, we will also derive periodic wave solutions and other analytical solutions based on the invariance surface condition. Moreover, we will establish the multiplier method to derive low-order conservation laws. In order to have a better understanding of the results, graphical structures of the derived solutions will be discussed in detail based on some selected appropriate parametric values in 2-dimensions, 3-dimensions and contour plots. The findings can well mimic complex waves and their underlying properties in fluids.
3
Lie symmetry, exact solutions and conservation laws of time fractional Black-Scholes equation derived by the fractional Brownian motion
Yu Jicheng
Journal of Applied Analysis
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2024
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Vol. 30, nr 1
137--145
EN
The Black-Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black-Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black-Scholes equation.
4
Content available On the numerical discretization of optimal control problems for conservation laws
Schäfer Aguilar Paloma, Schmitt Johann Michael, Ulbrich Stefan, Moos Michael
Control and Cybernetics
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2019
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Vol. 48, No. 2
345--376
EN
We analyze the convergence of discretization schemes for the adjoint equation arising in the adjoint-based derivative computation for optimal control problems governed by entropy solutions of conservation laws. The difficulties arise from the fact that the correct adjoint state is the reversible solution of a transport equation with discontinuous coefficient and discontinuous end data. We derive the discrete adjoint scheme for monotone difference schemes in conservation form. It is known that convergence of the discrete adjoint can only be expected if the numerical scheme has viscosity of order O(h) with appropriate 0 < α < 1, which leads to quite viscous shock profiles. We show that by a slight modification of the end data of the discrete adjoint scheme, convergence to the correct reversible solution can be obtained also for numerical schemes with viscosity of order O(h) and with sharp shock resolution. The theoretical findings are confirmed by numerical results.
5
On the solutions and conservation laws of a two-dimensional Korteweg de Vries model: multiple exp-function method
Adem A. R.
Journal of Applied Analysis
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2018
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Vol. 24, nr 1
27--33
EN
Under investigation in this paper is a two-dimensional Korteweg de Vries model, which is a spacial extension of the Korteweg de Vries model. An infinite number of nonlocal conservation laws are given which indicate the integrability of this model. Exact soliton solutions are then respectively derived by means of the multiple exp-function method.
6
Content available Global existence for strong solutions of viscous Burgers equation. (1) The bounded case
Unterberger J.
Control and Cybernetics
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2017
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Vol. 46, no. 2
109--136
EN
We prove that the viscous Burgers equation (∂t−∆)u(t, x)+( u •∇)u(t, x) = g(t, x), (t, x) ∈ R+ × Rd (d ≥ 1) has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term g are smooth and bounded together with their derivatives. Such solutions may have infinite energy. The proofdoes not rely on energy estimates, but on a combinationof the maximumprinciple and quantitative Schauder estimates. We obtain precise bounds on the sup norm of the solution and its derivatives, making it plain that there is no exponential increase in time. In particular, these bounds are time-independent if g is zero. To get a classical solution, it suffices to assume that the initial condition and the forcing term have bounded derivatives up to order two.
7
Content available remote Thermal Waves, Second Sound. Works of Witold Kosiński
Saxton K., Saxton R.
Mathematica Applicanda
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2015
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Vol. 43, No. 2
291--303
EN
This paper is dedicated to Witold Kosiński. Our contribution to this special issue will concentrate on the properties of thermal waves, one of many scientific interests of our friend and collaborator, and this article is dedicated to his memory. Working together with Witold was always an insightful and pleasant experience, and it benefited all of his coworkers including the authors of this note. His scope of research was broad, spanning many disciplines and applications. Here we focus on a few of those aspects to which he applied a deep knowledge of continuum thermodynamics and its mathematical foundations.
PL
Niniejsza praca jest poświęcona pamięci naszego przyjaciela Witolda Kosińskego. Chcielibyśmy przedstawić jego najważniejsze osiągnięcia w dziedzinie propagacji fal termicznych, termodynamiki i teorii hiperbolicznych układów różniczkowych. Zakres badań Kosińskiego był bardzo bogaty. Obejmował wiele dyscyplin na pograniczu mechaniki, matematyki i teorii komputerowych. Współpraca z Witoldem była owocna, zawsze wypełniona entuzjazmem i wzajemnym szacunkiem. Autorzy tego artykułu jak i inni współpracownicy Witolda korzystali z jego wiedzy i zawodowego doświadczenia. Bedzie nam go bardzo brakowało.
8
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.
9
Content available Conservation Laws in Physics. Emmy Noether’s Theorem
Manolea D.
International Letters of Chemistry, Physics and Astronomy
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2014
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Vol. 10
55-61
EN
The study is explanatory-interpretative and argues the practical character of Physics. It starts from premise that formation of a correct conception of the world begins with the understanding of physics. It is one of the earliest chapters of human knowledge, studying the material world from the microscopic level of the particles to the macroscopic level of the celestial body. As an example for the practical importance of applying the laws of physics take the set of physical laws of conservation, in particular, it explains the practical impact of Emmy Noether's Theorem.
10
Content available remote An approximate analytic solution for isentropic flow by an inviscid gas model
Az-Zo'bi E. A.
Archives of Mechanics
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2014
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Vol. 66, nr 3
203--212
EN
The aim of the present analysis is to apply the modified decomposition method (MDM) for the solution of isentropic flow of an inviscid gas model (IFIG). The modification form based on a new formula of Adomian’s polynomials (APs). The new approach provides the solution in the form of a rapidly convergent series with easily computable components and not at grid points. The proof of convergence of MDM applied to such systems is introduced with a bound of the error. Using suitable initial values, the solution of the system has been calculated and represented graphically. An analytic continuous solution with high accuracy was obtained.
11
Content available Isospectral integrability analysis of dynamical systems on discrete manifolds
Blackmore D., Prykarpatsky A. K., Prykarpatsky Y. A.
Opuscula Mathematica
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2012
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Vol. 32, no. 1
41-66
EN
It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco-Tu and Riemann-Burgers dynamical systems.
12
Content available remote Connecting Euler and Lagrange systems as nonlocally related systems of dynamical nonlinear elasticity
Bluman G., Ganghoffer J. F.
Archives of Mechanics
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2011
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Vol. 63, nr 4
363-382
EN
Nonlocally related systems for the Euler and Lagrange systems of twodimensional dynamical nonlinear elasticity are constructed. Using the continuity equation, i.e., conservation of mass of the Euler system to represent the density and Eulerian velocity components as the curl of a potential vector, one obtains the Euler potential system that is nonlocally related to the Euler system. It is shown that the Euler potential system also serves as a potential system of the Lagrange system. As a consequence, a direct connection is established between the Euler and Lagrange systems within a tree of nonlocally related systems. This extends the known situation for one-dimensional dynamical nonlinear elasticity to two spatial dimensions.
13
Content available remote Komplementarność fizycznych i chemicznych praw zachowania w aspekcie układów elektrolitycznych
Michałowski T.
Wiadomości Chemiczne
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2007
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[Z] 61, 7-8
625-640
EN
The charge, concentration and electron balances are closely related to other, more elementary rules of conservation of a matter in a closed system, separated from the environment by diathermal walls. The conservation rules can be formulated for the elements, electrons and protons. Among others, the generalised electron balance (GEB) concept presented and applied in some author's papers [1-7, 14-16] is derived from the elementary rules of conservation and exemplified by some batch and dynamic (titration) systems of a different degree of complexity. Some elementary rules of conservation are interdependent. This interdependency of the resulting balances and formulation of the set of independent relationships will be considered with the help of some examples, where the complex nature of the system, exemplified by the formation of aqua-complexes by both ionic and neutral species, will also be taken into account. Among others, the dynamic system is exemplified by titration of KIO_3 + HCl + H_2SeO_3 + HgCl_2 with ascorbic acid (C6H8O6). The degree of complexity of this system is evidenced by more than 40 equilibrium constants involved in the related balances.
14
Content available remote Balance errors in numerical solutions of shallow water equations
Gąsiorowski D.
TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk
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2007
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Vol. 11, No 4
329-340
EN
An analysis of the conservative properties of shallow water equations is presented, focused on the consistency of their numerical solution with the conservation laws of mass and momentum. Two different conservative forms are considered, solved by an implicit box scheme. Theoretical analysis supported with numerical experiments is carried out for a rectangular channel and arbitrarily assumed flow conditions. The improper conservative form of the dynamic equation is shown not to guarantee a correct solution with respect to the conservation of momentum. Consequently, momentum balance errors occur in the numerical solution. These errors occur when artificial diffusion is simultaneously generated by a numerical algorithm.
15
On some problems in mathematical modeling of turbulent flow
Burnat M., Moszyński K.
Journal of Technical Physics
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2007
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Vol. 48, no 3-4
171-192
EN
The paper presents a new, rather elementary mathematical model of turbulent fluid flow with intensive self-mixing. The flow is described in this model by the (in general non-invertible) mappings Φt,t0 : ω -->S(t, t0,ω), ω, S ⊂ R³, t > t0, such that ◛ ∩ ω² = ∅ in general not implies S(t, to, ◛) ∩ S(t, t0, ω²) = ∅. This modeling allows a new approach to certain problems of turbulent flow. For example, there are possibilities of describing the unpredictable explosions of turbulence in a calm, laminar flow.
16
Content available remote Thermodynamic aspects of variational principles for fluids with heat flow
Sieniutycz S.
Archives of Thermodynamics
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2005
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Vol. 26, no. 2
45-71
EN
Processing some known results of nonequilibrium statisctical mechanics we focus on nonequilibrium corrections Δs to entropy s of the fluid in terms of the nonequilibrium density distribution function, f. To evaluate corrections Δe to the energy e or kinetic potential L we apply a relationship that links energy and entropy representations of thermodynamics. We also evaluate coefficients of wave model of heat conduction, such as: relaxation time, propagation speed and thermal inertia. With corrections to L we formulate a quadratic Lagrangian and a variational principle of Hamilton's (least action) type for a fluid with heat flux, or other random-type effect, in the field or Eulerian representation of fluid motion. We discuss canonical and generalized conservation laws and show that satisfaction of the second law of thermodynamics under the constraint of canonical conservation laws.
17
Two-waves nonlinear interactions for hyperbolic systems of balance laws
Barbera E., Giambo S.
Journal of Technical Physics
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2002
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Vol. 43, no 2
165-185
EN
In this paper we study the weakly nonlinear interaction of two waves whose propagation is governed by hyperbolic systems of balance laws. The method used here makes use of nonlinear phase variables and consists in a perturbation analysis. It is applied to an Eulerian gas and to a gas described by extended thermodynamics with thirteen moments.
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