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Artykuły w czasopismach na temat „Non-solvable system of equations”

Autor: Grafiati

Data publikacji: 3 czerwca 2025

Data aktualizacji: 13 lipca 2025

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1

KARA, MERVE, and YASIN YAZLIK. "ON A SOLVABLE SYSTEM OF NON-LINEAR DIFFERENCE EQUATIONS WITH VARIABLE COEFFICIENTS." Journal of Science and Arts 21, no. 1 (2021): 145–62. http://dx.doi.org/10.46939/j.sci.arts-21.1-a13.

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In this paper, we show that the system of difference equations can be solved in the closed form. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, we obtain periodic solutions of aforementioned system.

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2

Stević, Stevo, Bratislav Iricanin, and Witold Kosmala. "Symmetric nonlinear solvable system of difference equations." Electronic Journal of Qualitative Theory of Differential Equations, no. 49 (2024): 1–16. http://dx.doi.org/10.14232/ejqtde.2024.1.49.

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We show the theoretical solvability of the system of difference equations x n + k = y n + l y n − c d y n + l + y n − c − d , y n + k = x n + l x n − c d x n + l + x n − c − d , n ∈ N 0 , where k ∈ N , l ∈ N 0 , l < k , c , d ∈ C and x j , y j ∈ C , j = 0 , k − 1 ¯ . For several special cases of the system, we give some detailed explanations on how some formulas for their general solutions can be found in closed form, that is, we show their practical solvability. To do this, among other things, we use the theory of homogeneous linear difference equations with constant coefficients and the product-type difference equations with integer exponents, which are theoretically solvable.

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3

Bombelli, L., W. E. Couch, and R. J. Torrence. "Solvable systems of wave equations and non-Abelian Toda lattices." Journal of Physics A: Mathematical and General 25, no. 5 (1992): 1309–27. http://dx.doi.org/10.1088/0305-4470/25/5/032.

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4

Stanzhytskyi, O. M., R. E. Uteshova, M. Mukash, and V. V. Mogylova. "Application of the method of averaging to boundary value problems for differential equations with non-fixed moments of impulse." Carpathian Mathematical Publications 14, no. 2 (2022): 304–26. http://dx.doi.org/10.15330/cmp.14.2.304-326.

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The method of averaging is applied to study the existence of solutions of boundary value problems for systems of differential equations with non-fixed moments of impulse action. It is shown that if an averaged boundary value problem has a solution, then the original problem is solvable as well. Here the averaged problem for the impulsive system is a simpler problem of ordinary differential equations.

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5

Stević, Stevo, Josef Diblík, Bratislav Iričanin, and Zdeněk Šmarda. "On a solvable system of rational difference equations." Journal of Difference Equations and Applications 20, no. 5-6 (2013): 811–25. http://dx.doi.org/10.1080/10236198.2013.817573.

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6

Stević, Stevo. "On a solvable rational system of difference equations." Applied Mathematics and Computation 219, no. 6 (2012): 2896–908. http://dx.doi.org/10.1016/j.amc.2012.09.012.

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7

Zhu, Changrong. "From homoclinics to quasi-periodic solutions for ordinary differential equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 5 (2015): 1091–114. http://dx.doi.org/10.1017/s0308210515000189.

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We consider the quasi-periodic solutions bifurcated from a degenerate homoclinic solution. Assume that the unperturbed system has a homoclinic solution and a hyperbolic fixed point. The bifurcation function for the existence of a quasi-periodic solution of the perturbed system is obtained by functional analysis methods. The zeros of the bifurcation function correspond to the existence of the quasi-periodic solution at the non-zero parameter values. Some solvable conditions of the bifurcation equations are investigated. Two examples are given to illustrate the results.

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8

Kara, Merve, and Yasin Yazlik. "Solvable three-dimensional system of higher-order nonlinear difference equations." Filomat 36, no. 10 (2022): 3449–69. http://dx.doi.org/10.2298/fil2210449k.

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In this work, we indicate three-dimensional system of difference equations xn = ayn?k + dyn?kxn?k?l/?bxn?k?l+?czn?l, yn = ?zn?k + ?zn?kyn?k?l/??yn?k?l + ??xn?l, zn = exn?k + hxn?kzn?k?l/?fzn?k?l +?g?yn?l , n ? N0, where k and l are positive integers, the parameters a,?b, ?c, d, ?, ??, ??, ?, e, ?f ,?g, h and the initial values x?j, y?j, z?j j = ?1,k+l, are non-zero real numbers, can be solved in closed form. In addition, we obtain explicit formulas for the well-defined solutions of the aforementioned system for the case l = 1. Also, the set of undefinable solutions of the system is found. Finally, an application about a three-dimensional system of difference equations is given.

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9

Pinar, Zehra. "The symmetry analysis of electrostatic micro-electromechanical system (MEMS)." Modern Physics Letters B 34, no. 18 (2020): 2050199. http://dx.doi.org/10.1142/s0217984920501997.

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The model of electrostatic Micro-Electromechanical System (MEMS) is considered without/with an external pressure. The model represents nonlinear elliptic or parabolic problem due to the steady or non-steady state problem, respectively. To obtain exact solutions in an explicit form, the symmetry analysis is considered. With symmetries, the transformations are obtained and by means of these transformations, solvable equations are hold. The obtained results have a major role in the literature so that the considered equation is seen in a large-scale applications.

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10

Tarasov, Vasily E. "General Non-Markovian Quantum Dynamics." Entropy 23, no. 8 (2021): 1006. http://dx.doi.org/10.3390/e23081006.

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A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.

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11

Kara, Merve, and Yasin Yazlik. "On a solvable three-dimensional system of difference equations." Filomat 34, no. 4 (2020): 1167–86. http://dx.doi.org/10.2298/fil2004167k.

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In this paper, we show that the following three-dimensional system of difference equations xn = zn-2xn-3/axn-3 + byn-1, yn = xn-2yn-3/cyn-3 + dzn-1, zn = yn-2zn-3/ezn-3+ fxn-1, n ? N0, where the parameters a, b, c, d, e, f and the initial values x-i, y-i, z-i, i ? {1, 2, 3}, are real numbers, can be solved, extending further some results in literature. Also, we determine the asymptotic behavior of solutions and the forbidden set of the initial values by using the obtained formulas.

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12

HALIM, Yacine, Massaoud BERKAL, and Amira KHELIFA. "On a three-dimensional solvable system of difference equations." TURKISH JOURNAL OF MATHEMATICS 44, no. 4 (2020): 1263–88. http://dx.doi.org/10.3906/mat-2001-40.

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13

Gora, P. "Approximate master equations for an exactly solvable quantum system." Journal of Physics A: Mathematical and General 21, no. 9 (1988): 2061–73. http://dx.doi.org/10.1088/0305-4470/21/9/020.

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14

Erdem, İbrahim, and Yasin Yazlik. "On a solvable four-dimensional system of difference equations." Mathematica Slovaca 74, no. 4 (2024): 929–46. http://dx.doi.org/10.1515/ms-2024-0069.

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Abstract In this paper we show that the following four-dimensional system of difference equations x n + 1 = y n α z n − 1 β , y n + 1 = z n γ t n − 1 δ , z n + 1 = t n ϵ x n − 1 μ , t n + 1 = x n ξ y n − 1 ρ , n ∈ N 0 , $$\begin{array}{} \displaystyle x_{n+1}=y_{n}^{\alpha}z_{n-1}^{\beta}, \quad y_{n+1}=z_{n}^{\gamma}t_{n-1}^{\delta}, \quad z_{n+1}=t_{n}^{\epsilon}x_{n-1}^{\mu}, \quad t_{n+1}=x_{n}^{\xi}y_{n-1}^{\rho}, \qquad n\in \mathbb{N}_{0}, \end{array}$$ where the parameters α, β, γ, δ, ϵ, μ, ξ, ρ ∈ ℤ and the initial values x –i , y –i , z –i , t –i , i ∈ {0, 1}, are real numbers, can be solved in closed forms, extending further some results in literature.

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15

Stevic, Stevo. "On a two-dimensional solvable system of difference equations." Electronic Journal of Qualitative Theory of Differential Equations, no. 104 (2018): 1–18. http://dx.doi.org/10.14232/ejqtde.2018.1.104.

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16

Kara, Merve, and Yasin Yazlik. "On a solvable three-dimensional system of difference equations." Filomat 34, no. 4 (2020): 1167–86. http://dx.doi.org/10.2298/fil2004167k.

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In this paper, we show that the following three-dimensional system of difference equations xn = zn-2xn-3/axn-3 + byn-1, yn = xn-2yn-3/cyn-3 + dzn-1, zn = yn-2zn-3/ezn-3+ fxn-1, n ? N0, where the parameters a, b, c, d, e, f and the initial values x-i, y-i, z-i, i ? {1, 2, 3}, are real numbers, can be solved, extending further some results in literature. Also, we determine the asymptotic behavior of solutions and the forbidden set of the initial values by using the obtained formulas.

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17

Akhiezer, A., A. Borovick, and V. Popkov. "An exactly solvable system of coupled nonlinear Schrödinger equations." Physics Letters A 182, no. 1 (1993): 44–48. http://dx.doi.org/10.1016/0375-9601(93)90050-a.

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18

Ignatyuk, Vasyl’. "Dynamic Correlations in Open Quantum Systems: The Dephasing Model." Open Systems & Information Dynamics 27, no. 02 (2020): 2050007. http://dx.doi.org/10.1142/s1230161220500079.

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We study the dynamical correlations in open quantum systems on an example of an exactly solvable dephasing model. The system of non-Markovian kinetic equations for the generalized coherence and quasi-temperature is derived up to the 2-nd order in the coupling constant using the generalized quantum master equation [20]. Numerical and analytical solutions of the kinetic equations are obtained. The analytical result shows a redistribution of the real and imaginary parts of the coherence as compared to the exact one. We give a physical interpretation to the quasi-temperature, relating it to the nonequilibrium kinetic energy of the environment.

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19

De Nittis, Giuseppe, and Antonio Moro. "Thermodynamic phase transitions and shock singularities." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2139 (2011): 701–19. http://dx.doi.org/10.1098/rspa.2011.0459.

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We show that, under rather general assumptions on the form of the entropy function, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase transition. A family of explicitly solvable models of non-hydrodynamic type such as the classical plasma and the ideal Bose gas is also discussed.

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20

Zhuravlov, V., N. Gongalo, and I. Slusarenko. "CONTROLLABILITY OF FREDHOLM’S INTEGRO-DIFFERENTIAL EQUATIONS WITH BY A DEGENERATE KERNEL IN HILBERT SPACES." Bukovinian Mathematical Journal 10, no. 1 (2022): 51–60. http://dx.doi.org/10.31861/bmj2022.01.05.

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The work examines integro-diﬀerential equations Fredholm with a degenerate kernel with Hilbert control spaces. The need to study these equations is related to numerous ones applications of integro- diﬀerential equations in mathematics, physics, technology, economy and other ﬁelds. Complexity the study of integro-diﬀerential equations is connected with the fact that the integral-diﬀerential operator is not solvable everywhere. There are diﬀerent approaches to the solution of not everywhere solvable linear operator equations: weak perturbation of the right-hand side of this equation with further application of the Vishyk-Lyusternyk method, introduction to system of impulse action, control, etc. The problem of obtaining coeﬃcient conditions of solvability and analytical presentation of general solutions of integro-diﬀerential equations is a rather diﬃcult problem, so frequent solutions will suﬃce are obtained by numerical methods. In this connection, Fredholm’s integro-diﬀerential equations with degenerate kernel and control in Hilbert spaces no were investigated. Therefore, the task of establishing conditions is urgent controllability, construction of general solutions in an analytical form and corresponding general controls of integro-diﬀerential equations with a degenerate kernel in abstract Hilbert spaces. As an intermediate result in the work using the results of pseudoinversion of integral operators in Hilbert spaces the solvability criterion and the form of general solutions are established integro-diﬀerential equations without control in the abstract Hilbert spaces. To establish the controllability criterion is not solvable everywhere integro-diﬀerential equations with Hilbert control spaces, the general theory of research is not applied everywhere solvable operator equations. At the same time, they are used signiﬁcantly orthoprojectors, pseudo-inverse operators to normally solvable ones operators in Hilbert spaces. With the use of orthoprojectors, pseudo-inverse operators and pseudoinversion of integraloperators, a criterion is obtained solutions and the general form of solutions of integro-diﬀerential equations with a degenerate kernel with control y Hilbert spaces. An image of the general appearance is obtained control under which these solutions exist.

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21

ALMEIDA, JORGE, and MANUEL DELGADO. "TAMENESS OF THE PSEUDOVARIETY OF ABELIAN GROUPS." International Journal of Algebra and Computation 15, no. 02 (2005): 327–38. http://dx.doi.org/10.1142/s0218196705002311.

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In this paper we prove that the pseudovariety of Abelian groups is hyperdecidable and moreover that it is completely tame. This is a consequence of the fact that a system of group equations on a free Abelian group with certain rational constraints is solvable if and only if it is solvable in every finite quotient.

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22

Mikheenko, Mikhail Alexandrovich. "$p$-Nonsingular systems of equations over solvable groups." Sbornik: Mathematics 215, no. 6 (2024): 775–89. http://dx.doi.org/10.4213/sm10009e.

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Any group that has a subnormal series all factors in which are abelian and all factors except the last one are $p'$-torsion free, can be embedded into a group with a subnormal series of the same length, with the same properties and such that any $p$-nonsingular system of equations over this group is solvable in this group itself. Using this we prove that the minimal order of a metabelian group over which there exists a unimodular equation that is unsolvable in metabelian groups is $42$. Bibliography: 14 titles.

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23

Stevic, Stevo, Bratislav Iricanin, and Zdenk Smarda. "On a solvable class of product-type systems of difference equations." Filomat 31, no. 19 (2017): 6113–29. http://dx.doi.org/10.2298/fil1719113s.

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It is shown that the following class of systems of difference equations zn+1 = ?zanwbn, wn+1 = ?wcnzdn-2, n ? N0, where a,b,c,d ? Z, ?, ?, z-2, z-1, z0,w0 ? C \ {0}, is solvable, continuing our investigation of classification of solvable product-type systems with two dependent variables. We present closed form formulas for solutions to the systems in all the cases. In the main case, when bd ? 0, a detailed investigation of the form of the solutions is presented in terms of the zeros of an associated polynomial whose coefficients depend on some of the parameters of the system.

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24

Stević, Stevo. "On a solvable system of difference equations of fourth order." Applied Mathematics and Computation 219, no. 10 (2013): 5706–16. http://dx.doi.org/10.1016/j.amc.2012.11.024.

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25

Stević, Stevo. "On a solvable system of difference equations of kth order." Applied Mathematics and Computation 219, no. 14 (2013): 7765–71. http://dx.doi.org/10.1016/j.amc.2013.01.064.

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26

Stević, Stevo. "Two‐dimensional solvable system of difference equations with periodic coefficients." Mathematical Methods in the Applied Sciences 42, no. 18 (2019): 6757–74. http://dx.doi.org/10.1002/mma.5780.

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27

Karakaya, Dilek, Yasin Yazlik, and Merve Kara. "On a solvable system of difference equations of sixth-order." Miskolc Mathematical Notes 24, no. 3 (2023): 1405. http://dx.doi.org/10.18514/mmn.2023.4248.

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28

Ghosh, Pijush K. "Classical Hamiltonian Systems with balanced loss and gain." Journal of Physics: Conference Series 2038, no. 1 (2021): 012012. http://dx.doi.org/10.1088/1742-6596/2038/1/012012.

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Abstract Classical Hamiltonian systems with balanced loss and gain are considered in this review. A generic Hamiltonian formulation for systems with space-dependent balanced loss and gain is discussed. It is shown that the loss-gain terms may be removed completely through appropriate co-ordinate transformations with its effect manifested in modifying the strength of the velocity-mediated coupling. The effect of the Lorentz interaction in improving the stability of classical solutions as well as allowing a possibility of defining the corresponding quantum problem consistently on the real line, instead of within Stokes wedges, is also discussed. Several exactly solvable models based on translational and rotational symmetry are discussed which include coupled cubic oscillators, Landau Hamiltonian etc. The role of PT -symmetry on the existence of periodic solution in systems with balanced loss and gain is critically analyzed. A few non- PT -symmetric Hamiltonian as well as non-Hamiltonian systems with balanced loss and gain, which include mechanical as well as extended system, are shown to admit periodic solutions. An example of Hamiltonian chaos within the framework of a non- PT -symmetric system of coupled Duffing oscillator with balanced loss-gain and/or positional non-conservative forces is discussed. It is conjectured that a non- PT -symmetric system with balanced loss-gain and without any velocity mediated interaction may admit periodic solution if the linear part of the equations of motion is necessarily PT symmetric —the nonlinear interaction may or may not be PT -symmetric. Further, systems with velocity mediated interaction need not be PT -symmetric at all in order to admit periodic solutions. Results related to nonlinear Schrödinger and Dirac equations with balanced loss and gain are mentioned briefly. A class of solvable models of oligomers with balanced loss and gain is presented for the first time along with the previously known results.

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29

Xie, Lv-Ming, and Qing-Wen Wang. "A system of matrix equations over the commutative quaternion ring." Filomat 37, no. 1 (2023): 97–106. http://dx.doi.org/10.2298/fil2301097x.

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In this paper, we propose a necessary and sufficient condition for the solvability to a system of matrix equations over the commutative quaternion ring, and establish an expression of its general solution when it is solvable. We also present an algorithm for finding an approximate solution to the system when it is inconsistent. Finally, we give an example to illustrate the main results of this paper.

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30

Zhang, Xin, Xun Li, and Jie Xiong. "Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markovian regime switching system." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 69. http://dx.doi.org/10.1051/cocv/2021066.

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This paper investigates the stochastic linear quadratic (LQ, for short) optimal control problem of Markovian regime switching system. The representation of the cost functional for the stochastic LQ optimal control problem of Markovian regime switching system is derived by the technique of Itô’s formula with jumps. For the stochastic LQ optimal control problem of Markovian regime switching system, we establish the equivalence between the open-loop (closed-loop, resp.) solvability and the existence of an adapted solution to the corresponding forward-backward stochastic differential equation with constraint. (i.e., the existence of a regular solution to Riccati equations). Also, we analyze the interrelationship between the strongly regular solvability of Riccati equations and the uniform convexity of the cost functional. Finally, we present an example which is open-loop solvable but not closed-loop solvable.

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31

Wang, Ruo-Nan, Qing-Wen Wang, and Long-Sheng Liu. "Solving a System of Sylvester-like Quaternion Matrix Equations." Symmetry 14, no. 5 (2022): 1056. http://dx.doi.org/10.3390/sym14051056.

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Using the ranks and Moore-Penrose inverses of involved matrices, in this paper we establish some necessary and sufficient solvability conditions for a system of Sylvester-type quaternion matrix equations, and give an expression of the general solution to the system when it is solvable. As an application of the system, we consider a special symmetry solution, named the η-Hermitian solution, for a system of quaternion matrix equations. Moreover, we present an algorithm and a numerical example to verify the main results of this paper.

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32

Zhang, Yue, Qing-Wen Wang, and Lv-Ming Xie. "The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations." Symmetry 16, no. 3 (2024): 361. http://dx.doi.org/10.3390/sym16030361.

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This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples.

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33

Bogatov, Andrei V., Anton V. Gilev, and Ludmila S. Pulkina. "A problem with a non-local condition for a fourth-order equation with multiple characteristics." Russian Universities Reports. Mathematics, no. 139 (2022): 214–30. http://dx.doi.org/10.20310/2686-9667-2022-27-139-214-230.

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In this article, we consider a non-local problem with an integral condition for a fourth-order equation. The unique solvability of the problem is proved. The proof of the uniqueness of a solution is based on the a priori estimates derived in the paper. To prove the existence of a solution, the problem is reduced to two Goursat problems for second-order equations, and the equivalence of the stated problem and the resulting system of Goursat problems is proved. One of the problems of the system is the classical Goursat problem. The second problem is a characteristic problem for an integro-differential equation with a non-local integral condition on one of the characteristics. It is impossible to apply the well-known methods of substantiating the solvability of problems with conditions on characteristics to the study of this problem. The introduction of a new unknown function made it possible to reduce the second problem to an equation with a completely continuous operator, to verify, on the basis of the uniqueness theorem, that it is solvable and, by virtue of the proven equivalence of the problems, that the problem posed is solvable.

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34

Yu, Shao-Wen, Wei-Lu Qin, and Zhuo-Heng He. "Some systems of tensor equations under t-product and their applications." Filomat 35, no. 11 (2021): 3663–77. http://dx.doi.org/10.2298/fil2111663y.

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In this paper, some systems of tensor equations under t-product are considered. Some practical necessary and sufficient conditions for the existence of a solution to two systems of tensor equations in terms of the Moore-Penrose inverses are given. The general solutions to the systems of tensor equations are presented when they are solvable. An application of the tensor equations in the solvability conditions and general symmetric solution to a system of tensor equations. Some algorithms and numerical examples are provided to illustrate the main results.

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35

Allakov, I., and B. Kh Erdonov. "On the solution of a system of linear Diophantine equations in prime numbers." UZBEK MATHEMATICAL JOURNAL 69, no. 1 (2025): 14–26. https://doi.org/10.29229/uzmj.2025-1-2.

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In the paper, it is proved that the system of linear Diophantine equations consisting of s equations with m unknowns (where s < m ≤ 2s) is solvable in prime numbers with some exceptions. A lower bound for the number of solutions of the system under consideration is also obtained for the exceptional set. The obtained results complement the corresponding results of Wu Fang, M. C. Liu, K. M. Tsang and I. Allakov

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36

Stevic, Stevo, Bratislav Iricanin, and Zdenĕk Smarda. "On a solvable symmetric and a cyclic system of partial difference equations." Filomat 32, no. 6 (2018): 2043–65. http://dx.doi.org/10.2298/fil1806043s.

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It is shown that the following symmetric system of partial difference equations cm,n = dm-1,n + cm-1, n-1, dm,n = cm-1,n + dm-1,n-1, is solvable on the combinatorial domain C = n {(m,n) ? N20 : 0 ? n ? m}\{(0,0)}, by presenting some formulas for the general solution to the system on the domain in terms of the boundary values cj,j, cj,0, dj,j, dj,0, j ? N, and the indices m and n. The corresponding result for a related three-dimensional cyclic system of partial difference equations is also proved. These results can serve as a motivation for further studies of the solvability of symmetric, close-to-symmetric, cyclic, close-to-cyclic and other related systems of partial difference equations.

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Burmasheva, Natalya V., and Evgeniy Yu Prosviryakov. "Influence of the Dufour Effect on Shear Thermal Diffusion Flows." Dynamics 2, no. 4 (2022): 367–79. http://dx.doi.org/10.3390/dynamics2040021.

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The article considers thermal diffusion shear flows of a viscous incompressible fluid with spatial acceleration. The simulation uses a system of thermal diffusion equations (in the Boussinesq approximation), taking into account the Dufour effect. This system makes it possible to describe incompressible gases, for which this effect prevails, from a unified standpoint. It is shown that for shear flows, the system of equations under study is nonlinear and overdetermined. In view of the absence of a theorem on the existence and smoothness of the solution of the Navier–Stokes equation, the integration of the existing system seems to be an extremely difficult task. The article studies the question of the existence of a solution in the class of functions represented as complete linear forms in two Cartesian coordinates with non-linear (with respect to the third Cartesian coordinate) coefficients. It is shown that the system is non-trivially solvable under a certain condition (compatibility condition) constructed by the authors. The corresponding theorem is formulated and proven. These conclusions are illustrated by a comparison with the previously obtained results.

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Ghezal, Ahmed, and Imane Zemmouri. "Representation of Solutions of a Second-Order System of Two Difference Equations With Variable Coefficients." Pan-American Journal of Mathematics 2 (January 15, 2023): 2. http://dx.doi.org/10.28919/cpr-pajm/2-2.

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A definition of system of two nonlinear difference equations with variable coefficients is given. Our main result shows that the difference equation is solvable in closed form and thus for the constant coefficients. Some applications of the main result are also given.

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He, Zhuoheng, and Qingwen Wang. "A System of Periodic Discrete-time Coupled Sylvester Quaternion Matrix Equations." Algebra Colloquium 24, no. 01 (2017): 169–80. http://dx.doi.org/10.1142/s1005386717000104.

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We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations [Formula: see text] over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

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Baleanu, Dumitru, Vladimir E. Fedorov, Dmitriy M. Gordievskikh, and Kenan Taş. "Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case." Mathematics 7, no. 8 (2019): 735. http://dx.doi.org/10.3390/math7080735.

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We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter—Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input.

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Šeda, V., and J. Eliaš. "On the Initial Value Problem for Functional Differential Systems." gmj 1, no. 4 (1994): 419–27. http://dx.doi.org/10.1515/gmj.1994.419.

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Abstract For a system of functional differential equations of an arbitrary order the conditions are established for the initial value problem to be solvable on an infinite interval, and the structure of the set of solutions to this problem is studied.

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Stević, Stevo. "Solvable Three-Dimensional Product-Type System of Difference Equations with Multipliers." Symmetry 9, no. 9 (2017): 195. http://dx.doi.org/10.3390/sym9090195.

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Mao, Linfan. "Cauchy problem on non-solvable systems of first-order partial differential equations with applications." Methods and Applications of Analysis 22, no. 2 (2015): 171–200. http://dx.doi.org/10.4310/maa.2015.v22.n2.a3.

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SPARENBERG, JEAN-MARC, ANDREY M. PUPASOV, BORIS F. SAMSONOV, and DANIEL BAYE. "EXACTLY-SOLVABLE COUPLED-CHANNEL MODELS FROM SUPERSYMMETRIC QUANTUM MECHANICS." Modern Physics Letters B 22, no. 23 (2008): 2277–86. http://dx.doi.org/10.1142/s0217984908017023.

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Starting from a system of N radial Schrödinger equations with a vanishing potential and finite threshold differences between the channels, a coupled N × N exactly-solvable potential model is obtained with the help of a single non-conservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as well as its Jost matrix. It depends on N(N + 1)/2 unconstrained parameters and on one upper-bounded parameter, the factorization energy. For N = 2, previous results are reviewed, in particular regarding the number of bound states and resonances of the potential. A schematic inverse problem with one resonance is considered.

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Fauzi, F., A. R. M. Kasim, and S. F. H. M. Kanafiah. "Mixed convection of nanofluid at the lower stagnation point of a horizontal circular cylinder: Brinkman–viscoelastic." Mathematical Modeling and Computing 12, no. 1 (2025): 187–96. https://doi.org/10.23939/mmc2025.01.187.

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Researchers have formulated the Brinkman–viscoelastic model to study the convective heat transfer of viscoelastic fluids traversing porous media recently. Nevertheless, they did not employ nanofluids. Currently, scientists and researchers are employing nanofluids and hybrid nanofluids in their studies, products, and technologies due to their ability to enhance the transfer of heat. Therefore, the aim of this research is to study the heat transfer of a viscoelastic fluid with nanoparticles as it moves over a horizontal circular cylinder in a saturated porous region at the lower stagnation point. The model, described by a collection of partial differential equations, is simplified into solvable equations by utilizing non-dimensional and non-similarity variables. The simplified system of equations is then solved using the Runge–Kutta–Fehlberg technique and the outcomes are verified through comparative study. It has been found that the temperature goes up when the Brinkman parameter, the viscoelastic parameter, and the volume fraction of nanoparticles are all raised. This temperature rise indicates convective heat transfer.

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Khasanov, Aknazar Bekdurdievich, Raykhonbek Khubaydullo ugli Eshbekov, and Temur Gafurjonovich Hasanov. "Integration of a non-linear Hirota type equation with additional terms." Izvestiya: Mathematics 89, no. 1 (2025): 196–219. https://doi.org/10.4213/im9559e.

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In this paper, the inverse spectral problem method is used to integrate a Hirota type equation with additional terms in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-gap functions is proved. It is also shown that the Cauchy problem is solvable at all times for sufficiently smooth initial conditions.

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Hassani, Murad Khan, Nouressadat Touafek, and Yasin Yazlik. "On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence." Mathematica Slovaca 74, no. 3 (2024): 703–16. http://dx.doi.org/10.1515/ms-2024-0053.

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Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

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Omonov, Abbos Ulug'bekоvich. "ANALYSIS OF THE EFFECT OF DISPLACEMENT OF THE CORE FROM THE GEOMETRIC CENTER OF SPIRAL GALAXIES." physical and mathematical science 3, no. 1 (2022): 4. https://doi.org/10.5281/zenodo.7198968.

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To analyze the physics of the formation of galaxies and other gravitational systems, we first need to create accurate analytically solvable models for them. Taking this into account, new analytical models were created, and critical diagrams were found, forming analogs of nonlinear non-stationary equations for the (1;3) mode, which corresponds to the displacement of the nucleus from the geometric center, which occurs in spiral galaxies  

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Polin, Sergey V. "Families of quasigroup operations satisfying the generalized distributive law." Discrete Mathematics and Applications 30, no. 3 (2020): 187–202. http://dx.doi.org/10.1515/dma-2020-0018.

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AbstractThe previous paper was concerned with systems of equations over a certain family 𝓢 of quasigroups. In that work a method of elimination of an outermost variable from the system of equations was suggested and it was shown that further elimination of variables requires that the family 𝓢 of quasigroups satisfy the generalized distributive law (GDL). In this paper we describe families 𝓢 that satisfy GDL. The results are applied to construct classes of easily solvable systems of equations.

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JANA, T. K., and P. ROY. "EXACTLY SOLVABLE INTERPOLATING HAMILTONIANS WITH POSITION-DEPENDENT MASS." Modern Physics Letters A 25, no. 34 (2010): 2915–22. http://dx.doi.org/10.1142/s0217732310033645.

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It is shown that Hamiltonians of the form H(s) = (1 - s)H- + sH+, 0 ≤ s ≤ 1 where H± are supersymmetric partner Hamiltonians corresponding to position-dependent mass Schrödinger equations are exactly solvable for a number of deformed shape-invariant potentials. The method has also been extended to a system with broken supersymmetry.

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