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Статті в журналах з теми "First-order hyperbolic partial differential equations"

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Автор: Grafiati

Опубліковано: 7 липня 2024

Оновлено: 7 липня 2024

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1

Cheema, T. A., M. S. A. Taj, and E. H. Twizell. "Third-order methods for first-order hyperbolic partial differential equations." Communications in Numerical Methods in Engineering 20, no. 1 (November 4, 2003): 31–41. http://dx.doi.org/10.1002/cnm.650.

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2

Turo, Jan. "On some class of quasilinear hyperbolic systems of partial differential-functional equations of the first order." Czechoslovak Mathematical Journal 36, no. 2 (1986): 185–97. http://dx.doi.org/10.21136/cmj.1986.102083.

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3

Tokibetov, Zh A., N. E. Bashar, and А. К. Pirmanova. "THE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS." BULLETIN Series of Physics & Mathematical Sciences 72, no. 4 (December 29, 2020): 68–72. http://dx.doi.org/10.51889/2020-4.1728-7901.10.

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For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Dirichlet problem. This work dedicated to systems of first-order partial differential equations of elliptic and hyperbolic types consisting of four equations with three unknown variables. An explicit solution of the CauchyDirichlet problem is constructed using the method of an exponential – differential operator. Giving a very simple example of the co-solution of the Cauchy problem for a second-order differential equation and the Cauchy problem for systems of first-order hyperbolic differential equations.

4

Kamont, Z., and S. Kozieł. "First Order Partial Functional Differential Equations with Unbounded Delay." gmj 10, no. 3 (September 2003): 509–30. http://dx.doi.org/10.1515/gmj.2003.509.

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Abstract The phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed. The set of axioms for generalized solutions of initial problems is presented. A theorem on the existence and continuous dependence upon initial data is given. The Cauchy problem is transformed into a system of integral functional equations. The existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities. Examples of phase spaces are given.

5

Karafyllis, Iasson, and Miroslav Krstic. "On the relation of delay equations to first-order hyperbolic partial differential equations." ESAIM: Control, Optimisation and Calculus of Variations 20, no. 3 (June 13, 2014): 894–923. http://dx.doi.org/10.1051/cocv/2014001.

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6

Verma, Anjali, and Ram Jiwari. "Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 7 (September 7, 2015): 1574–89. http://dx.doi.org/10.1108/hff-08-2014-0240.

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Purpose – The purpose of this paper is to present the computational modeling of second-order two-dimensional nonlinear hyperbolic equations by using cosine expansion-based differential quadrature method (CDQM). Design/methodology/approach – The CDQM reduced the equations into a system of second-order differential equations. The obtained system is solved by RK4 method by converting into a system of first ordinary differential equations. Findings – The computed numerical results are compared with the results presented by other workers (Mohanty et al., 1996; Mohanty, 2004) and it is found that the present numerical technique gives better results than the others. Second, the proposed algorithm gives good accuracy by using very less grid point and less computation cost as comparison to other numerical methods such as finite difference methods, finite elements methods, etc. Originality/value – The author extends CDQM proposed in (Korkmaz and Dağ, 2009b) for two-dimensional nonlinear hyperbolic partial differential equations. This work is new for two-dimensional nonlinear hyperbolic partial differential equations.

7

Hou, Lei, Pan Sun, Jun Jie Zhao, Lin Qiu, and Han Lin Li. "Evaluation of Coupled Rheological Equations." Applied Mechanics and Materials 433-435 (October 2013): 1943–46. http://dx.doi.org/10.4028/www.scientific.net/amm.433-435.1943.

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Apart from the convergence of the first-order hyperbolic partial differential equations in rheological flow,this paper estimate the general behavior of the solution. By analyzing the coupled partial differential equations on a macroscopic scale the solution of free surface flow has been obtained. Its asymptotic estimate of the solution and super convergence are proposed in the internal boundary layer.

8

Ashyralyev, A., A. Ashyralyyev, and B. Abdalmohammed. "On the hyperbolic type differential equation with time involution." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 109, no. 1 (March 30, 2023): 38–47. http://dx.doi.org/10.31489/2023m1/38-47.

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In the present paper, the initial value problem for the hyperbolic type involutory in t second order linear partial differential equation is studied. The initial value problem for the fourth order partial differential equations equivalent to this problem is obtained. The stability estimates for the solution and its first and second order derivatives of this problem are established.

9

Hou, Lei, Jun Jie Zhao, and Han Ling Li. "Finite Element Convergence Analysis of Two-Scale Non-Newtonian Flow Problems." Advanced Materials Research 718-720 (July 2013): 1723–28. http://dx.doi.org/10.4028/www.scientific.net/amr.718-720.1723.

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The convergence of the first-order hyperbolic partial differential equations in non-Newton fluid is analyzed. This paper uses coupled partial differential equations (Cauchy fluid equations, P-T/T stress equation) on a macroscopic scale to simulate the free surface elements. It generates watershed by excessive tensile elements. The semi-discrete finite element method is used to solve these equations. These coupled nonlinear equations are approximated by linear equations. Its super convergence is proposed.

10

Bainov, Drumi, Zdzisław Kamont, and Emil Minchev. "Periodic boundary value problem for impulsive hyperbolic partial differential equations of first order." Applied Mathematics and Computation 68, no. 2-3 (March 1995): 95–104. http://dx.doi.org/10.1016/0096-3003(94)00083-g.

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11

Vassiliou, Peter J. "An integrable system of partial differential equations on the special linear group." ANZIAM Journal 44, no. 1 (July 2002): 83–93. http://dx.doi.org/10.1017/s1446181100007938.

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AbstractWe give an intrinsic construction of a coupled nonlinear system consisting of two first-order partial differential equations in two dependent and two independent variables which is determined by a hyperbolic structure on the complex special linear group regarded as a real Lie groupG. Despite the fact that the system is not Darboux semi-integrable at first order, the construction of a family of solutions depending.upon two arbitrary functions, each of one variable, is reduced to a system of ordinary differential equations on the 1-jets. The ordinary differential equations in question are of Lie type and associated withG.

12

Ashyralyev, Allaberen, and Fadime Dal. "Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition." Discrete Dynamics in Nature and Society 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/434976.

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The numerical and analytic solutions of the mixed problem for multidimensional fractional hyperbolic partial differential equations with the Neumann condition are presented. The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation with the Neumann condition is presented. Stability estimates for the solution of this difference scheme and for the first- and second-order difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations. He's variational iteration method is applied. The comparison of these methods is presented.

13

Jumayel, Josef Al. "A Note on the Uniqueness and Existence of the Solution to the Problem of Boundary Values for Some Partial Differential Equations of Second Order." Galoitica: Journal of Mathematical Structures and Applications 3, no. 2 (2023): 08–16. http://dx.doi.org/10.54216/gjmsa.030201.

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Анотація:

Most physical mathematical problems, when solved, turn into one or more partial differential equations with imposed initial conditions or boundary conditions . This is known as the boundary value problems of differential equations .This research studies the solution of the set of Partial Differential Equations of parabolic and hyperbolic-hyperbolic type with boundary conditions imposed in different regions of the plane x o y This research has been proving the theorem of uniqueness and the existence of the solution.

14

Assanova, A. T., and Zh S. Tokmurzin. "Method of functional parametrization for solving a semi-periodic initial problem for fourth-order partial differential equations." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 100, no. 4 (December 30, 2020): 5–16. http://dx.doi.org/10.31489/2020m4/5-16.

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A semi-periodic initial boundary-value problem for a fourth-order system of partial differential equations is considered. Using the method of functional parametrization, an additional parameter is carried out and the studied problem is reduced to the equivalent semi-periodic problem for a system of integro-differential equations of hyperbolic type second order with functional parameters and integral relations. An interrelation between the semi-periodic problem for the system of integro-differential equations of hyperbolic type and a family of Cauchy problems for a system of ordinary differential equations is established. Algorithms for finding of solutions to an equivalent problem are constructed and their convergence is proved. Sufficient conditions of a unique solvability to the semi-periodic initial boundary value problem for the fourth-order system of partial differential equations are obtained.

15

Potts, Renfrey B. "Ordinary and partial difference equations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 27, no. 4 (April 1986): 488–501. http://dx.doi.org/10.1017/s0334270000005099.

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AbstractOrdinary difference equations (OΔE's), mostly of order two and three, are derived for the trigonometric, Jacobian elliptic, and hyperbolic functions. The results are used to derive partial difference equations (PΔE's) for simple solutions of the wave equation and three nonlinear evolutionary partial differential equations.

16

Syazana Saharizan, Nur, and Nurnadiah Zamri. "Numerical solution for a new fuzzy transform of hyperbolic goursat partial differential equation." Indonesian Journal of Electrical Engineering and Computer Science 16, no. 1 (October 1, 2019): 292. http://dx.doi.org/10.11591/ijeecs.v16.i1.pp292-298.

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<p>The main objective of this paper is to present a new numerical method with utilization of fuzzy transform in order to solve various engineering problems that represented by hyperbolic Goursat partial differentical equation (PDE). The application of differential equations are widely used for modelling physical phenomena. There are many complicated and dynamic physical problems involved in developing a differential equation with high accuracy. Some problems requires a complex and time consuming algorithms. Therefore, the application of fuzzy mathematics seems to be appropriate for solving differential equations due to the transformation of differential equations to the algebraic equation which is solvable. Furthermore, development of a numerical method for solving hyperbolic Goursat PDE is presented in this paper. The method are supported by numerical experiment and computation using MATLAB. This will provide a clear picture to the researcher to understand the utilization of fuzzy transform to the hyperbolic Goursat PDE.</p>

17

Mokhtar, Mahmoud M., and M. H. El Dewaik. "New Fifth-Kind Chebyshev Collocation Scheme for First-Order Hyperbolic and Second-Order Convection-Diffusion Partial Differential Equations." Mathematical Problems in Engineering 2022 (October 4, 2022): 1–11. http://dx.doi.org/10.1155/2022/3789611.

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The fifth type of Chebyshev polynomials was used in tandem with the spectral tau method to achieve a semianalytical solution for the partial differential equation of the hyperbolic first order. For this purpose, the problem was diminished to the solution of a set of algebraic equations in unspecified expansion coefficients. The convergence and error analysis of the proposed expansion were studied in-depth. Numerical trials have confirmed the applicability and the accuracy.

18

Atta, A. G., W. M. Abd-Elhameed, G. M. Moatimid, and Y. H. Youssri. "Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations." Applied Numerical Mathematics 167 (September 2021): 237–56. http://dx.doi.org/10.1016/j.apnum.2021.05.010.

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19

Deo, S. G., and S. Sivasundaram. "Extension of the method of quasilinearization to hyperbolic partial differential equations of first order." Applicable Analysis 59, no. 1-4 (December 1995): 153–62. http://dx.doi.org/10.1080/00036819508840396.

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20

Człapiński, Tomasz. "On the mixed problem for hyperbolic partial differential-functional equations of the first order." Czechoslovak Mathematical Journal 49, no. 4 (December 1999): 791–809. http://dx.doi.org/10.1023/a:1022453117846.

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21

Watanabe, Yoichi. "A nondispersive and nondissipative numerical method for first-order linear hyperbolic partial differential equations." Numerical Methods for Partial Differential Equations 3, no. 1 (1987): 1–8. http://dx.doi.org/10.1002/num.1690030102.

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22

Xue, Xiaomin, Juanjuan Xu, and Huanshui Zhang. "Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations." Journal of Systems Science and Complexity 37, no. 1 (February 2024): 230–52. http://dx.doi.org/10.1007/s11424-024-3324-8.

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23

Kiguradze, Tariel, Takasi Kusano, and Norio Yoshida. "Oscillation criteria for a class of partial functional-differential equations of higher order." Journal of Applied Mathematics and Stochastic Analysis 15, no. 3 (January 1, 2002): 255–67. http://dx.doi.org/10.1155/s1048953302000229.

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Higher order partial differential equations with functional arguments including hyperbolic equations and beam equations are studied. Sufficient conditions are derived for every solution of certain boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multi-dimensional oscillation problem to a one-dimensional problem for higher order functional differential inequalities.

24

Khan, Hassan, Rasool Shah, Dumitru Baleanu, Poom Kumam, and Muhammad Arif. "Analytical Solution of Fractional-Order Hyperbolic Telegraph Equation, Using Natural Transform Decomposition Method." Electronics 8, no. 9 (September 11, 2019): 1015. http://dx.doi.org/10.3390/electronics8091015.

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In the current paper, fractional-order hyperbolic telegraph equations are considered for analytical solutions, using the decomposition method based on natural transformation. The fractional derivative is defined by the Caputo operator. The present technique is implemented for both fractional- and integer-order equations, showing that the current technique is an accurate analytical instrument for the solution of partial differential equations of fractional-order arising in all branches of applied sciences. For this purpose, several examples related to hyperbolic telegraph models are presented to explain the procedure of the suggested method. It is noted that the procedure of the present technique is simple, straightforward, accurate, and found to be a better mathematical technique to solve non-linear fractional partial differential equations.

25

Avetisyan, Zhirayr, and Matteo Capoferri. "Partial Differential Equations and Quantum States in Curved Spacetimes." Mathematics 9, no. 16 (August 13, 2021): 1936. http://dx.doi.org/10.3390/math9161936.

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In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

26

Grazhdantseva, Elena Yu. "On exact solution of a hyperbolic system of differential equations." Russian Universities Reports. Mathematics, no. 140 (2022): 328–38. http://dx.doi.org/10.20310/2686-9667-2022-27-140-328-338.

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The paper considers a hyperbolic system of two first-order partial differential equations with constant coefficients, one of which is nonlinear and contains the square of one of the unknown functions. Moreover, each equation contains two unknown functions which in turn depend on two variables. Exact solutions are found for this system: a traveling wave solution and a self-similar solution. There is also defined the type of initial-boundary conditions which allow to use the constructed general solutions in order to write out a solution of the initial-boundary value problem for the system of differential equations under consideration.

27

Agarwal, Ravi P. "The method of upper, lower solutions and monotone iterative scheme for higher order hyperbolic partial differential equations." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 47, no. 1 (August 1989): 153–70. http://dx.doi.org/10.1017/s1446788700031293.

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AbstractUniformly monotone convergent iterative methods for obtaining multiple solutions of (n + m)th order hyperbolic partial differential equations together with initial conditions are discussed. Appropriate partial differential inequalities which connect upper and lower solutions, and variation of parameters formula is developed.

28

Freistühler, Heinrich, and Blake Temple. "Causal dissipation for the relativistic dynamics of ideal gases." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2201 (May 2017): 20160729. http://dx.doi.org/10.1098/rspa.2016.0729.

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We derive a general class of relativistic dissipation tensors by requiring that, combined with the relativistic Euler equations, they form a second-order system of partial differential equations which is symmetric hyperbolic in a second-order sense when written in the natural Godunov variables that make the Euler equations symmetric hyperbolic in the first-order sense. We show that this class contains a unique element representing a causal formulation of relativistic dissipative fluid dynamics which (i) is equivalent to the classical descriptions by Eckart and Landau to first order in the coefficients of viscosity and heat conduction and (ii) has its signal speeds bounded sharply by the speed of light. Based on these properties, we propose this system as a natural candidate for the relativistic counterpart of the classical Navier–Stokes equations.

29

Yu, Xin, Chao Xu, Huacheng Jiang, Arthi Ganesan, and Guojie Zheng. "Backstepping Synthesis for Feedback Control of First-Order Hyperbolic PDEs with Spatial-Temporal Actuation." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/643640.

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This paper deals with the stabilization problem of first-order hyperbolic partial differential equations (PDEs) with spatial-temporal actuation over the full physical domains. We assume that the interior actuator can be decomposed into a product of spatial and temporal components, where the spatial component satisfies a specific ordinary differential equation (ODE). A Volterra integral transformation is used to convert the original system into a simple target system using the backstepping-like procedure. Unlike the classical backstepping techniques for boundary control problems of PDEs, the internal actuation can not eliminate the residual term that causes the instability of the open-loop system. Thus, an additional differential transformation is introduced to transfer the input from the interior of the domain onto the boundary. Then, a feedback control law is designed using the classic backstepping technique which can stabilize the first-order hyperbolic PDE system in a finite time, which can be proved by using the semigroup arguments. The effectiveness of the design is illustrated with some numerical simulations.

30

Jokhadze, O. "On the Boundary Value Problem in A Dihedral Angle for Normally Hyperbolic Systems of First Order." gmj 5, no. 2 (April 1998): 121–38. http://dx.doi.org/10.1515/gmj.1998.121.

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Abstract Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

31

Ashyralyev, Allaberen, Fadime Dal, and Zehra Pinar. "On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations." Mathematical Problems in Engineering 2009 (2009): 1–11. http://dx.doi.org/10.1155/2009/730465.

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The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this difference scheme and for the first and second orders difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.

32

Abalos, Fernando. "A necessary condition ensuring the strong hyperbolicity of first-order systems." Journal of Hyperbolic Differential Equations 16, no. 01 (March 2019): 193–221. http://dx.doi.org/10.1142/s0219891619500073.

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We study strong hyperbolicity of first-order partial differential equations for systems with differential constraints. In these cases, the number of equations is larger than the unknown fields, therefore, the standard Kreiss necessary and sufficient conditions of strong hyperbolicity do not directly apply. To deal with this problem, one introduces a new tensor, called a reduction, which selects a subset of equations with the aim of using them as evolution equations for the unknown. If that tensor leads to a strongly hyperbolic system we call it a hyperbolizer. There might exist many of them or none. A question arises on whether a given system admits any hyperbolization at all. To sort-out this issue, we look for a condition on the system, such that, if it is satisfied, there is no hyperbolic reduction. To that purpose we look at the singular value decomposition of the whole system and study certain one parameter families ([Formula: see text]) of perturbations of the principal symbol. We look for the perturbed singular values around the vanishing ones and show that if they behave as [Formula: see text], with [Formula: see text], then there does not exist any hyperbolizer. In addition, we further notice that the validity or failure of this condition can be established in a simple and invariant way. Finally, we apply the theory to examples in physics, such as Force-Free Electrodynamics in Euler potentials form and charged fluids with finite conductivity. We find that they do not admit any hyperbolization.

33

Abbas, Said, Mouffak Benchohra, and Yong Zhou. "Darboux problem for fractional order neutral functional partial hyperbolic differential equations." International Journal of Dynamical Systems and Differential Equations 2, no. 3/4 (2009): 301. http://dx.doi.org/10.1504/ijdsde.2009.031110.

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34

Abbas, Said, Mouffak Benchohra, and Yong Zhou. "Fractional order partial hyperbolic functional differential equations with state-dependent delay." International Journal of Dynamical Systems and Differential Equations 3, no. 4 (2011): 459. http://dx.doi.org/10.1504/ijdsde.2011.042941.

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35

Abbas, Saïd, Mouffak Benchohra, and JuanJ Nieto. "Global Uniqueness Results for Fractional Order Partial Hyperbolic Functional Differential Equations." Advances in Difference Equations 2011, no. 1 (2011): 379876. http://dx.doi.org/10.1155/2011/379876.

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36

Capdeville, G. "Compact high-order numerical schemes for scalar hyperbolic partial differential equations." Journal of Computational and Applied Mathematics 363 (January 2020): 171–210. http://dx.doi.org/10.1016/j.cam.2019.05.029.

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37

Пономаренко, Артем. "METHOD OF SOLVING THE CAUCHY PROBLEM FOR THE WAVE EQUATION IN ONE SPACE DIMENSION BY REDUCING THE ORDER OF DIFFERENTIATIONS." Молодий вчений, no. 9 (121) (September 30, 2023): 1–6. http://dx.doi.org/10.32839/2304-5809/2023-9-121-1.

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In this article, we present method of solving the Cauchy problem for the wave equation in one space dimension by reducing the order of differentiations. To obtain its solution, we use the solution of the Cauchy problem for a linear partial differential equation of the first order with constant coefficients. These methods for solving hyperbolic partial differential equations can be computationally cumbersome and involve much more elementary transformations, but with their help we can quite simply find the class of functions for which we obtain a given solution. In this way, it is possible to transform many types of partial differential equations with constant coefficients, in which differentiation of some given order is carried out in both variables, to an equation whose partial derivative order is one less. The main point of the method for obtaining a solution is the choice of the correct sequence of application of elementary transformations and the simplest operations of differential and integral calculus on the initial given equation.

38

Mansimov, K. B., and R. O. Mastaliyev. "On the Representation of the Goursat Boundary Problem Solution for the First Order Partial Derivatives Stochastic Hyperbolic Equations." Bulletin of Irkutsk State University. Series Mathematics 45 (2023): 145–51. http://dx.doi.org/10.26516/1997-7670.2023.45.145.

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We study the standard canonical form of a stochastic analog of a system of linear partial differential equations of first order hyperbolic type with Goursat boundary conditions. The stochastic analogue of the Riemann matrix in block form is introduced, an integral representation of the solution of the boundary value problem under consideration is obtained in an explicit integral form in terms of boundary conditions.

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Shoimkulov, B. M. "Of one over determined system differential equation at private derivative second order with one singular point and one singular line." Вестник Пермского университета. Математика. Механика. Информатика, no. 4(55) (2021): 14–18. http://dx.doi.org/10.17072/1993-0550-2021-4-14-18.

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In this paper, a over determined system of second-order partial differential equations with one singular point and one singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular point and one singular line. If the compatibility condition is met, integral representations of the variety of solutions are found explicitly in terms of three arbitrary constants, when the singular line is in the boundaries of the domain for which initial data problems (Cauchy-type Problems) can be set. In this paper considers a redefined system of second-order partial differential equations, when the coefficients and right parts have one singular point and one singular line. Obtaining a variety of solutions and studying boundary value problems for linear differential equations of the hyperbolic type of the second order, some linear redefined systems of the first and second order with one and two supersingular lines and supersingular points is devoted to the monograph of academician of the National Academy of Sciences of the Republic of Tatarstan Rajabov N. - 1992 "Introduction to the theory of partial differential equations with supersingular coefficients" [6, p.126]. Using the obtained results of The monograph of Rajabov N., a variety of solutions of redefined systems of partial differential equations of the second order with one singular point and one singular line in an explicit form, through three arbitrary constants, was found.

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Nguyen, Nhan, and Mark Ardema. "Optimality of Hyperbolic Partial Differential Equations With Dynamically Constrained Periodic Boundary Control—A Flow Control Application." Journal of Dynamic Systems, Measurement, and Control 128, no. 4 (April 26, 2006): 946–59. http://dx.doi.org/10.1115/1.2362814.

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This paper is concerned with optimal control of a class of distributed-parameter systems governed by first-order, quasilinear hyperbolic partial differential equations that arise in optimal control problems of many physical systems such as fluids dynamics and elastodynamics. The distributed system is controlled via a forced nonlinear periodic boundary condition that describes a boundary control action. Further, the periodic boundary control is subject to a dynamic constraint imposed by a lumped-parameter system governed by ordinary differential equations that model actuator dynamics. The partial differential equations are thus coupled with the ordinary differential equations via the periodic boundary condition. Optimality of this coupled system is investigated using variational principles to seek an adjoint formulation of the optimal control problem. The results are then applied to solve a feedback control problem of the Mach number in a wind tunnel.

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Mansimov, K. B., and R. O. Mastaliyev. "Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations." Bulletin of Irkutsk State University. Series Mathematics 36 (2021): 29–43. http://dx.doi.org/10.26516/1997-7670.2021.36.29.

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The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

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Wei, Lin. "Some second-order systems of partial differential equations of composite type." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 106, no. 1-2 (1987): 73–88. http://dx.doi.org/10.1017/s0308210500018217.

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SynopsisThe Cauchy problem and the Dirichlet-Cauchy type problem of some second-order systems of partial differential equations of composite type of two unknown functions are investigated. Such systems possess some of the characteristics not only of elliptic but also of hyperbolic systems in the same domain. Representations of the solutions are found for the upper half plane. To this end, the composite systems are reduced to the canonical form by means of successive applications of three kinds of linear transformations. Function theoretic methods are used to obtain representation formulae. Furthermore, some composite systems of 2m-unknown function are also considered.

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Adomian, G. "Application of decomposition to hyperbolic, parabolic, and elliptic partial differential equations." International Journal of Mathematics and Mathematical Sciences 12, no. 1 (1989): 137–43. http://dx.doi.org/10.1155/s0161171289000190.

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The decomposition method is applied to examples of hyperbolic, parabolic, and elliptic partial differential equations without use of linearizatlon techniques. We consider first a nonlinear dissipative wave equation; second, a nonlinear equation modeling convectlon-diffusion processes; and finally, an elliptic partial differential equation.

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El-Ganaini, Shoukry Ibrahim Atia. "The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/349173.

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The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.

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de Vries, H. B. "A Survey of One-Step Splitting Methods for Semi-Discrete First Order Hyperbolic Partial Differential Equations." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 65, no. 2 (1985): 109–18. http://dx.doi.org/10.1002/zamm.19850650211.

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Paliathanasis, Andronikos. "Similarity Transformations and Linearization for a Family of Dispersionless Integrable PDEs." Symmetry 14, no. 8 (August 4, 2022): 1603. http://dx.doi.org/10.3390/sym14081603.

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We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlevé transcendents. The main results of this study are that from the application of the similarity transformations provided by the Lie point symmetries, all the members of the family of the partial differential equations are reduced to second-order differential equations, which are maximal symmetric and can be linearized.

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Heikkilä, S., and S. Leela. "On second order discontinuous differential equations in Banach spaces." Journal of Applied Mathematics and Stochastic Analysis 6, no. 4 (January 1, 1993): 303–23. http://dx.doi.org/10.1155/s1048953393000279.

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In this paper we study a second order semilinear initial value problem (IVP), where the linear operator in the differential equation is the infinitesimal generator of a strongly continuous cosine family in a Banach space E. We shall first prove existence, uniqueness and estimation results for weak solutions of the IVP with Carathéodory type of nonlinearity, by using a comparison method. The existence of the extremal mild solutions of the IVP is then studied when E is an ordered Banach space. We shall also discuss the dependence of these solutions on the data. A characteristic feature of the results concerning extremal solutions is that the nonlinearity is not assumed to be continuous in any of its arguments. Moreover, no compactness conditions are assumed. The obtained results are then applied to a second order partial differential equation of hyperbolic type.

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Helzel, Christiane. "A Third-Order Accurate Wave Propagation Algorithm for Hyperbolic Partial Differential Equations." Communications on Applied Mathematics and Computation 2, no. 3 (January 16, 2020): 403–27. http://dx.doi.org/10.1007/s42967-019-00056-3.

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Ashyralyev, Allaberen, and Deniz Agirseven. "Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations." Mathematics 7, no. 12 (December 2, 2019): 1163. http://dx.doi.org/10.3390/math7121163.

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In this paper, we study the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive definite operator. The mean theorem on the existence and uniqueness of a bounded solution of this differential problem for a semilinear hyperbolic equation with unbounded time delay term is established. In applications, the existence and uniqueness of bounded solutions of four problems for semilinear hyperbolic equations with time delay in unbounded term are obtained. For the approximate solution of this abstract differential problem, the two-step difference scheme of a first order of accuracy is presented. The mean theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme with respect to time stepsize is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay in unbounded term are obtained. In general, it is not possible to get the exact solution of semilinear hyperbolic equations with unbounded time delay term. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equation with time delay are presented. Finally, some numerical examples are given to confirm the theoretical analysis.

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Otarova, J. A. "A boundary value problem for the fourth-order degenerate equation of the mixed type." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 113, no. 1 (March 29, 2024): 140–48. http://dx.doi.org/10.31489/2024m1/140-148.

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Many problems in mechanics, physics, and geophysics lead to solving partial differential equations that are not included in the known classes of elliptic, parabolic or hyperbolic equations. Such equations, as a rule, began to be called non-classical equations of mathematical physics. The theory of degenerate equations is one of the central branches of the modern theory of partial differential equations. This is primarily due to the identification of a variety of applied problems, the mathematical modeling of which serves the study of various types of degenerate equations. The study of boundary value problems for mixed type’s equations of the fourth-order with power-law degeneration remains relevant. In this work, a boundary value problem in a rectangular domain for a degenerate equation of the fourth-order mixed-type is posed and investigated. Well-posedness of the boundary value problem for a fourth-order partial differential equation is established by proving the existence and uniqueness of the solution. Under sufficient conditions, a solution to the problem under consideration was explicitly found by the variable separation method.