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The formulations and methods of solution of the problems of diagnostics of thermoelasticity on a stationary boundary of a solid body are considered. A weak formulation of the boundary-value problem is presented for a nonlinear hyperbolic equation (a nonstationary wave equation with density, elasticity modulus, and thermal stress depending on temperature) with mixed boundary conditions. An approximate solution of this problem in the space W 2 1 has been obtained, and an example of solution is given. Based on the weak formulation of the boundary-value problem, problems of diagnostics have been formulated in the sense of the theory of function traces. According to the theorem about the uniqueness of the relationship between the function and its trace in the space W 2 1 , a conclusion has been drawn about the need to control the convergence of the calculated wave function to the experimental one only on boundaries with unstable conditions in solving the problem of diagnostics. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal Mathematics titled "Differential Equations of Mathematical Physics and Related Problems of Mechanics." The special issue covers a range of topics related to differential equations in mathematical physics and mechanics, including wave equations, spectral theory, scattering, and inverse problems. The editorial provides a summary of the published papers in the special issue, highlighting their contributions to the field. The document emphasizes the importance of the special issue in covering both applied and fundamental aspects of mathematics, physics, and their applications in various fields. The author expresses gratitude to the authors, reviewers, assistants, associate editors, and editors for their contributions to the special issue. The report does not provide specific details about the content of the papers or the nature of the special issue. [Extracted from the article]

This research article addresses a nonclassical initial boundary value problem characterized by a non-local constraint within the framework of a pseudo-hyperbolic equation. Employing rigorous analytical techniques, the paper establishes the existence, uniqueness, and continuous dependence of a strong solution to the problem at hand. With respect to the associated linear problem, the uniqueness of its solution is ascertained through an energy inequality, which provides an a priori bound for the solution. Moreover, the solvability of this linear problem is verified by proving that the operator range engendered by the problem is indeed dense. Extending the analysis to the nonlinear problem, an iterative methodology is utilized. This approach is predicated on the insights gained from the linear problem and facilitates the demonstration of both the existence and uniqueness of a solution for the nonlinear problem under study. Consequently, the paper contributes a robust mathematical framework for solving both linear and nonlinear variants of complex initial boundary value problems with non-local constraints. [ABSTRACT FROM AUTHOR]

The existence and uniqueness theorems of the solution to the boundary-value problem for a parabolic-hyperbolic fractional-order equation with the gluing condition are proved. [ABSTRACT FROM AUTHOR]

In this work, we investigate a generalization of the Tricomi problem for a loaded mixed‐type equation with the Riemann–Liouville fractional differential operator. By using the method of integral equations, a unique solvability of the formulated problem given in piecewise non‐parallel characteristics is proven. [ABSTRACT FROM AUTHOR]

We study the local-in-time well-posedness for an interface that separates an anisotropic plasma from a vacuum. The plasma flow is governed by the ideal Chew–Goldberger–Low (CGL) equations, which are the simplest collisionless fluid model with anisotropic pressure. The vacuum magnetic and electric fields are supposed to satisfy the pre-Maxwell equations. The plasma and vacuum magnetic fields are tangential to the interface. This represents a nonlinear hyperbolic-elliptic coupled problem with a characteristic free boundary. By a suitable symmetrization of the linearized CGL equations, we reduce the linearized free boundary problem to a problem analogous to that in isotropic magnetohydrodynamics (MHD). This enables us to prove the local existence and uniqueness of solutions to the nonlinear free boundary problem under the same non-collinearity condition for the plasma and vacuum magnetic fields on the initial interface required by Secchi and Trakhinin (Nonlinearity 27:105–169, 2014) in isotropic MHD. [ABSTRACT FROM AUTHOR]

In the 1950s, Protter proposed multi-dimensional analogues of the classical Guderley–Morawetz problem for mixed-type hyperbolic-elliptic equations on the plane that models transonic flows in fluid dynamics. The multi-dimensional variants turn out to be different from the two-dimensional case and the situation there is still not clear. Here, we study Protter problems in the hyperbolic part of the domain. Unlike the planar analogues, the four-dimensional variant is not well-posed for classical solutions. The problem is not Fredholm — there is an infinite number of necessary conditions for classical solvability. Alternatively, the notion of a generalized solution that may have singularities was introduced. It is known that for smooth right-hand sides, the uniquely determined generalized solution may have a power-type growth at one boundary point. The singularity is isolated at the vertex of the boundary characteristic light cone and does not propagate along the cone. Here, we construct a new singular solution with an exponential growth at the point where the singularity appears. [ABSTRACT FROM AUTHOR]

Solutions of boundary value problems for linear partial differential equations are known to exhibit singular behaviors near the boundary of nonsmooth domains. In this case, one is usually interested on the asymptotic behavior of the solutions near the geometric singularities. However, for both mathematical and engineering purposes, it is important to have extraction formulas for the coefficients in the asymptotic expansion. In this paper, we consider initial boundary value problems for the wave equation in two‐dimensional domains with corners and derive, by means of Fourier analysis, explicit computable expressions for the coefficients in the asymptotic expansion of the solution near the corners. The explicit structure of the singular solutions presented herein makes it fairly easy to compute numerically. Thus, the formulas can be used to construct postprocessing procedures for improving the accuracy of standard numerical techniques of approximating the solution and for improving the rates of convergence in the error estimates. The analysis presented herein would also apply to more general hyperbolic or parabolic equations with constant coefficients. [ABSTRACT FROM AUTHOR]

We study the inverse problem of recovering a coefficient at the nonlinearity in a second order hyperbolic equation with nonlinear damping. The unknown coefficient depends on one space variable . Also, we consider the process of wave propagation along the semiaxis given the derivative with respect to at . As additional information in the inverse problem we consider the trace of a solution to the initial boundary value problem on a finite segment of the axis and find the conditions for unique solvability of the direct problem. We also establish a local existence theorem and a global stability estimate for a solution to the inverse problem. [ABSTRACT FROM AUTHOR]

Asymptotic and spectral results on the initial boundary‐value problem for the coupled bending‐torsion vibration model (which is important in such areas of engineering sciences as bridge and tall building designs, aerospace and oil pipes modeling, etc.) are presented. The model is given by a system of two hyperbolic partial differential equations equipped with a three‐parameter family of non‐self‐adjoint (linear feedback type) boundary conditions modeling the actions of self‐straining actuators. The system is rewritten in the form of the first‐order evolution equation in a Hilbert space of a four‐component Cauchy data. It is shown that the dynamics generator is a matrix differential operator with compact resolvent, whose discrete spectrum splits asymptotically into two disjoint subsets called the α‐branch and the β‐branch, respectively. Precise spectral asymptotics for the eigenvalues from each branch as the number of an eigenvalue tends to ∞ have been derived. It is also shown that the leading asymptotical term of the α‐branch eigenvalue depends only on the torsion control parameter, while of the β‐branch eigenvalue depends on two bending control parameters. [ABSTRACT FROM AUTHOR]

In this paper we study an inverse boundary value problem with an unknown time-dependent coefficient for a third-order pseudo-hyperbolic equation with an additional integral condition. The definition of the classical solution of the problem is given. The essence of the problem is that it is required together with the solution to determine the unknown coefficient. The problem is considered in a rectangular area. When solving the original inverse boundary value problem, the transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to an auxiliary problem are proved with the help of contracted mappings. Then the transition to the original inverse problem is again made, as a result, a conclusion is made about the solvability of the original inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior (as t → ∞ ) of solutions as an initial boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis ( x > 0 ). The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum ( σ (H 0) ) of the one-dimensional Schrödinger operator H 0 with periodic coefficients p (x) and q (x) . [ABSTRACT FROM AUTHOR]

In this paper, an inverse boundary value problem for a two‐dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. We then use the Fourier method to reduce such an equivalent problem to a system of integral equations. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness theorem for the classical solution of the original inverse problem is proved. Some discussions on the numerical solutions for this inverse problem are presented including some numerical examples. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of finite rank perturbations of Toeplitz operators that have simple eigenvalues on the unit circle. Under a suitable assumption on the behavior of the essential spectrum, we show that such operators are power bounded. The problem originates in the approximation of hyperbolic partial differential equations with boundary conditions by means of finite difference schemes. Our result gives a positive answer to a conjecture by Trefethen, Kreiss and Wu that only a weak form of the so-called uniform Kreiss–Lopatinskii condition is sufficient to imply power boundedness. [ABSTRACT FROM AUTHOR]

We consider the mixed type equation with the nonlocal boundary condition and the initial condition. A uniqueness for the solvability of this boundary problem is shown. [ABSTRACT FROM AUTHOR]

This paper starts from several applications described by initial/boundary value problems for 1D (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties. [ABSTRACT FROM AUTHOR]

The theory of boundary value problems for hyperbolic equations is one of the most important sections of the theory of partial differential equations, which is explained both by the theoretical significance of the results and by the presence of their practical applications in gas dynamics, in the theory of infinitesimal bendings of surfaces, in the momentless theory of shells, in magnetic hydrodynamics, mathematical biology and other areas. In the XX century in connection with applied technical and economic needs, it became necessary to solve control and optimization problems. The most famous works in this area are J.I. S. Pontryagin and his students V. G. Boltyanskiy, R. V. Gamkrelidze, E. F. Mishchenko, studied issues of control of processes described by systems of ordinary differential equations [18,19]. Further development of applied research leads to the need to control objects, the behavior of which is described using methods with partial differential equations. The corresponding control problems were considered in the works of A.G Butkovsky and co-authors [3,4], A.I Egorov [6-9], J.-L. Lyons [12], K. A. Lurie [13], T. K. Sirazetdinov [20], as well as V. A. Ilyin and E. I. Moiseeva [21,22], [14,15], A.T.Ramazanova [26], Kuliyev Hamlet [27]. Optimal control problems for the processes described by hyperbolic differential equations are encountered in various applications [1,2]. Such problems have been most fully investigated in cases where the boundary conditions for the equations of state are classical. However, there are numerous problems in physics, technology, biology, etc., in which the processes are described by hyperbolic equations, where the boundary conditions are nonclassical or nonlocal [23,24]. Among the nonlocal boundary value problems for the hyperbolic equations, a special place is occupied by boundary value problems with integral boundary conditions [25]. Optimal control problems for hyperbolic-type equations with nonlocal boundary conditions, including those with integral boundary conditions, have been little studied. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation u tt - a ∫ Ø m e g a | ∇ u | 2 d x + b Δ u = λ u + | u | p - 1 u , where a, b > 0 , p > 1 , λ ∈ R and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of a, b, λ and the initial data for the following cases: (i) 1 < p < 3 , (ii) p = 3 and a > 1 / Λ , (iii) p = 3 , a ≤ 1 / Λ and λ < b λ 1 , (iv) p = 3 , a < 1 / Λ and λ > b λ 1 , (v) p > 3 and λ ≤ b λ 1 , (vi) p > 3 and λ > b λ 1 , where λ 1 = inf ‖ ∇ u ‖ 2 2 : u ∈ H 0 1 (Ø m e g a) and ‖ u ‖ 2 = 1 , and Λ = inf ‖ ∇ u ‖ 2 4 : u ∈ H 0 1 (Ø m e g a) and ‖ u ‖ 4 = 1 . Moreover, we prove the invariance of some stable and unstable sets of the solution for suitable a, b and λ , and give the sufficient conditions of initial data to generate a vacuum region of the solution. Due to the nonlocal effect caused by the nonlocal integro-differential term, we show many interesting differences between the blow-up phenomenon of the problem for a > 0 and a = 0 . [ABSTRACT FROM AUTHOR]

The problem of vibrations of objects with moving boundaries formulated as a differential equation with boundary and initial conditions is a nonclassical generalization of a hyperbolic problem. To facilitate the construction of the solution to this problem and to justify the choice of the form of the solution, equivalent integro-differential equations with symmetric and time-dependent kernels and time-varying integration limits are constructed. The advantages of the method of integro-differential equations are revealed in the transition to more complex dynamic systems carrying concentrated masses vibrating under the action of moving loads. The method is extended to a wider class of model boundary value problems that take into account bending stiffness, the resistance of the external environment, and the stiffness of the base of a vibrating object. The solution is given in dimensionless variables and is accurate up to second-order values with respect to small parameters characterizing the velocity of the boundary. An approximate solution is found for the problem of transverse vibrations of a hoisting rope having bending stiffness, one end of which is wound on a drum and a load is fixed on the other. [ABSTRACT FROM AUTHOR]

In this paper, we study a boundary-value problem with discontinuous conjugation conditions on the line of type changing for a model equation of mixed hyperbolic-parabolic type with degeneration of order in the hyperbolicity domain. In the parabolic domain, the equation is the fractional diffusion equation, whereas in the hyperbolic domain it is the loaded one-speed transfer equation. We prove the uniqueness and existence theorem and propose an explicit solution of the problem in the parabolic and hyperbolic domains. [ABSTRACT FROM AUTHOR]

We prove the well-posedness of the initial boundary value problem for the Einstein equations with sole boundary condition the requirement that the timelike boundary is totally geodesic. This provides the first well-posedness result for this specific geometric boundary condition and the first setting for which geometric uniqueness in the original sense of Friedrich holds for the initial boundary value problem. Our proof relies on the ADM system for the Einstein vacuum equations, formulated with respect to a parallelly propagated orthonormal frame along timelike geodesics. As an independent result, we first establish the well-posedness in this gauge of the Cauchy problem for the Einstein equations, including the propagation of constraints. More precisely, we show that by appropriately modifying the evolution equations, using the constraint equations, we can derive a first order symmetric hyperbolic system for the connection coefficients of the orthonormal frame. The propagation of the constraints then relies on the derivation of a hyperbolic system involving the connection, suitably modified Riemann and Ricci curvature tensors and the torsion of the connection. In particular, the connection is shown to agree with the Levi-Civita connection at the same time as the validity of the constraints. In the case of the initial boundary value problem with totally geodesic boundary, we then verify that the vanishing of the second fundamental form of the boundary leads to homogeneous boundary conditions for our modified ADM system, as well as for the hyperbolic system used in the propagation of the constraints. An additional analytical difficulty arises from a loss of control on the normal derivatives to the boundary of the solution. To resolve this issue, we work with an anisotropic scale of Sobolev spaces and exploit the specific structure of the equations. [ABSTRACT FROM AUTHOR]

We consider two-dimensional (2D) pseudosteady flows around a sharp corner. This problem can be seen as a 2D Riemann initial and boundary value problem (IBVP) for the compressible Euler system. The initial state is a combination of a uniform flow in one quadrant and vacuum in the remaining domain. The boundary condition on the wall of the sharp corner is a slip boundary condition. By a self-similar transformation, the 2D Riemann IBVP is converted into a boundary value problem (BVP) for the 2D self-similar Euler system. Existence of global piecewise smooth (or Lipshitz-continuous) solutions to the BVP are obtained. One of the main difficulties for the global existence is that the type of the 2D self-similar Euler system is a priori unknown. In order to use the method of characteristic analysis, we establish some a priori estimates for the hyperboliciy of the system. The other main difficulty is that when the uniform flow is sonic or subsonic, the hyperbolic system becomes degenerate at the origin. Moreover, there is a multi-valued singularity at the origin. To solve this degenerate hyperbolic boundary value problem, we establish some uniform interior C 0 , 1 norm estimates for the solutions of a sequence of regularized hyperbolic boundary value problems, and then use the Arzela–Ascoli theorem and a standard diagonal procedure to construct a global Lipschitz continuous solution. The method used here may also be used to construct continuous solutions of some other degenerate hyperbolic boundary value problems and sonic-supersonic flow problems. [ABSTRACT FROM AUTHOR]

Two types of existing iterative methods for solving the nonlinear balance equation (NBE) are revisited. In the first type, the NBE is rearranged into a linearized equation for a presumably small correction to the initial guess or the subsequent updated solution. In the second type, the NBE is rearranged into a quadratic form of the absolute vorticity with the positive root of this quadratic form used in the form of a Poisson equation to solve NBE iteratively. The two methods are rederived by expanding the solution asymptotically upon a small Rossby number, and a criterion for optimally truncating the asymptotic expansion is proposed to obtain the super-asymptotic approximation of the solution. For each rederived method, two iterative procedures are designed using the integral-form Poisson solver versus the over-relaxation scheme to solve the boundary value problem in each iteration. Upon testing with analytically formulated wavering jet flows on the synoptic, sub-synoptic and meso-α scales, the iterative procedure designed for the first method with the Poisson solver, named M1a, is found to be the most accurate and efficient. For the synoptic wavering jet flow in which the NBE is entirely elliptic, M1a is extremely accurate. For the sub-synoptic wavering jet flow in which the NBE is mostly elliptic, M1a is sufficiently accurate. For the meso-α wavering jet flow in which the NBE is partially hyperbolic so its boundary value problem becomes seriously ill-posed, M1a can effectively reduce the solution error for the cyclonically curved part of the wavering jet flow, but not for the anti-cyclonically curved part. [ABSTRACT FROM AUTHOR]

The goal of this paper is to accurately describe the metastable dynamics of the solutions to the hyperbolic relaxation of the Cahn–Hilliard equation in a bounded interval of the real line, subject to homogeneous Neumann boundary conditions. We prove the existence of an approximately invariant manifold M 0 for such boundary value problem, that is we construct a narrow channel containing M 0 and satisfying the following property: a solution starting from the channel evolves very slowly and leaves the channel only after an exponentially long time. Moreover, in the channel the solution has a transition layer structure and we derive a system of ODEs, which accurately describes the slow dynamics of the layers. A comparison with the layer dynamics of the classic Cahn–Hilliard equation is also performed. [ABSTRACT FROM AUTHOR]

We establish the well-posedness of an initial-boundary value problem of mixed type for a stochastic nonlinear parabolic-hyperbolic equation on a space domain O = O ′ × O ″ where a Neumann boundary condition is imposed on ∂ O ′ × O ″ , the hyperbolic boundary, and a Dirichlet condition is imposed on O ′ × ∂ O ″ , the parabolic boundary. Among other points to be highlighted in our analysis of this problem we mention the new strong trace theorem for the special class of stochastic nonlinear parabolic-hyperbolic equations studied here, which is decisive for the uniqueness of the kinetic solution, and the new averaging lemma for the referred class of equations which is a vital part of the proof of the strong trace property. We also provide a detailed analysis of the approximate nondegenerate problems, which is also made here for the first time, as far as the authors know, whose solutions we prove to converge to the solution of our initial-boundary value problem. [ABSTRACT FROM AUTHOR]

This article is concerned with the study of the unique solvability of a timenonlocal inverse boundary value problem for second-order hyperbolic equation with an integral overdetermination condition. To study the solvability of the inverse problem, we first reduce the considered problem to an auxiliary system with trivial data and prove its equivalence (in a certain sense) to the original problem. Then using the Banach fixed point principle, the existence and uniqueness of a solution to this system is shown. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the inverse coefficient problem is proved for the smaller value of time. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the stability of the steady normal shock structure in potential flows under an unsteady perturbation. The dynamic stability problem is formulated as the well-posedness problem of an initial boundary value problem of a nonlinear wave equation in a cornered space domain with a free boundary. The corner singularity is the essential difficulty and there is no result available even for the linear problem without the symmetry assumptions, i.e., \even" or \odd" traces vanish on the solid boundary, which allows extension from the cornered space domain to the half-space domain, as in the previous works. In this paper, we first obtain an existence result for the initial boundary value problem of linear hyperbolic equations of second order in a cornered space domain without such symmetry assumptions. The key idea is based on the construction of a new auxilliary problem, which allows us to reduce the linear problem to a new one that can be even extended to a half-space domain such that the existence of H² η-solutions can be established. However, due to the lack of the symmetry assumptions, the low regularity of the extended coefficients block us from obtaining the higher regularity of the solutions in the extended domain, which is necessary for the iteration to the nonlinear problem. In order to deal with it, new hyperbolic type and elliptic type estimates in the cornered space domain are established carefully. The results on the general linear problems can be applied to the linearized problem that we are concerned with in this paper. Due to the loss of regularity in the estimates of the linearized problem, a modified Nash-Moser iteration is developed. [ABSTRACT FROM AUTHOR]

Consider a single hyperbolic PDE u x y = f (x , y , u , u x , u y) , with locally prescribed data: u along a non-characteristic curve M and u x along a non-characteristic curve N. We assume that M and N are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the (x , y) -plane. It is known that if M is located above N , then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when M lies below N , the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function f). The construction, via Picard iteration, makes use of a careful choice of additional u -data which are updated in each iteration step. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to establish the weak existence theorem and find under suitable assumptions sufficient conditions on m; p and the initial data for which the blow up takes place for the following boundary value problem: ... This paper extends some of the results obtained by the authors and it is focused on new results which are consequence of the presence of variable exponents. [ABSTRACT FROM AUTHOR]

This paper is concerned with studying the numerical solution for the discrete classical optimal control problem (NSDCOCP) governed by a variable coefficients nonlinear hyperbolic boundary value problem (VCNLHBVP). The DSCOCP is solved by using the Galerkin finite element method (GFEM) for the space variable and implicit finite difference scheme (GFEM-IFDS) for the time variable to get the NS for the discrete weak form (DWF) and for the discrete adjoint weak form (DSAWF) While, the gradient projection method (GRPM), also called the gradient method (GRM), or the Frank Wolfe method (FRM) are used to minimize the discrete cost function (DCF) to find the DSCOC. Within these three methods, the Armijo step option (ARMSO) or the optimal step option (OPSO) are used to improve the discrete classical control (DSCC). Finally, some illustrative examples for the problem are given to show the accuracy and efficiency of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear hyperbolic equation. We establish a blow up result for the initial and boundary value problem. Furthermore, the upper bound of the blow up time is derived. [ABSTRACT FROM AUTHOR]

We study a boundary value problem with periodicity conditions and with a nonlocal Dezin condition for a mixed elliptic-hyperbolic equation in a rectangular domain with power-law degeneracy on the transition line. Necessary and sufficient conditions for the uniqueness of the solution are established, the uniqueness of the solution being proved based on the completeness of the system of eigenfunctions of a one-dimensional eigenvalue problem. The solution is constructed in the form of a series. The problem of small denominators occurs when justifying the convergence of the series. Under some conditions imposed on the given parameters and functions, the convergence of the series is proved in the class of regular solutions. [ABSTRACT FROM AUTHOR]

A non-stationary model is shown, formed by the 1D hyperbolic differential equation, which is commonly used in practice for wave propagation of rivers and channels. As an upstream boundary condition, a function that depends on time has been taken, representing the variation of the water levels in the channel. Downstream, the wave equation with first derivatives in space and time is taken as the boundary condition, thus ensuring that there is no reflection of the wave in that boundary. Numerical algorithms obtained from the application of the Finite Differences Method and the Finite Element Method are formulated and the calculation algorithms are compared with an analytical solution showing good results. [ABSTRACT FROM AUTHOR]

We investigate global existence and asymptotic behavior of the 3D quasilinear hyperbolic equations with nonlinear damping on a bounded domain with slip boundary condition, which describes the propagation of heat waves for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of classical solutions are obtained when the initial data are near its equilibrium. Time asymptotically, the internal energy is conjectured to satisfy the porous medium equation and the heat flux obeys the classical Darcy's-type law. Based on energy estimates, we show that the classical solution converges to steady state exponentially fast in time. Moreover, we also verify that the same is true for the corresponding initial boundary value problem of porous medium equation and thus justifies the validity of Darcy's-type law in large time. [ABSTRACT FROM AUTHOR]

In this study, a mathematical model describing two-dimensional bio-heat transfer during cryosurgery of lung cancer is developed. The lung tissue is cooled by a cryoprobe by imposing its surface at a constant temperature or a constant heat flux or a constant heat transfer coefficient. The freezing starts and the domain is distributed into three stages namely: unfrozen, mushy and frozen regions. In stage I where the only unfrozen region is formed, our problem is an initial-boundary value problem of the hyperbolic partial differential equation. In stage II where mushy and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations and in stage III where frozen, mushy, and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations. The solution consists of the three-step procedure: (i) transformation of problem in non-dimensional form, (ii) by using finite differences, the problem converted into ordinary matrix differential equation and moving boundary problem of ordinary matrix differential equations, (iii) applying Legendre wavelet Galerkin method the problem is transferred into the generalized system of Sylvester equations which are solved by applying Bartels-Stewart algorithm of generalized inverse. The complete analysis is presented in the non-dimensional form. The consequence of the imposition of boundary conditions on moving layer thickness and temperature distribution are studied in detail. The consequence of Stefan number, Kirchoff number and Biot number on moving layer thickness are also studied in specific. • In this study, we have developed two-dimensional mathematical model describing bio-heat transfer during cryosurgery of lung cancer. • We see the effect of imposition of different type of boundary condition on temperature distribution and moving layer thickness. • We find the temperature distribution in all the three stages and moving layer thickness in stage 2 & 3. • Our model will be very useful for the experimental analysis of freezing process of cryosurgical treatment in lung cancer. • We will use this novel method in various type of moving boundary problem. [ABSTRACT FROM AUTHOR]

The Escalator Boxcar Train (EBT) numerical method has been designed and widely used by theoretical biologists to compute solutions of one-dimensional structured population models of McKendrick--von Foerster type. Recently the method has been derived for an age-structured twosex population model (Fredrickson--Hoppenstaedt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. The convergence of the EBT method for the Fredrickson--Hoppenstaedt model has not been analyzed, and relevant numerical examples are still missing. In this paper, we derive a simplified EBT method for the "two-sex model" and prove its convergence. However, due to the interest in tracking specified cohorts of individuals, the analytical results cannot be analyzed in the L1 norm. Instead, we embedded the problem in the space of nonnegative Radon measures equipped with the bounded Lipschitz distance (the flat metric). We also present numerical examples to illustrate the results, compute the error in bounded Lipschitz distance, and compare it against the total variation (TV) distance. [ABSTRACT FROM AUTHOR]

In this paper, we consider Goursat boundary value problems for fuzzy delay fractional partial differential equations under Caputo gH-derivatives. Firstly, the unique solvability of the problems in finite domain is considered. Secondly, by restricting the force function and boundary conditions by exponential growth and using fixed point approach, the existence and the Ulam-Hyers stability of fuzzy mild solutions of the problem in infinite domain are investigated in two types with different geometric behaviors. It is necessary to implement that in many engineering applications, e.g. numerical solutions of a fuzzy dynamical system, the object of optimization problems or the trajectory of a economical dynamics etc., where finding the exact solution is more difficult than solving its approximate solutions, Ulam-Hyers stability is quite useful. Finally, as usual some examples are given to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]