Multidimensional Hyperbolic Equations Research Articles

High-frequency homogenization of multidimensional hyperbolic equations

In L 2 ( R d ) , we consider an elliptic differential operator A ϵ , ϵ > 0 , of the form A ϵ = − div ⁡ g ( x / ϵ ) ∇ + ϵ − 2 V ( x / ϵ ) with periodic coefficients. For hyperbolic equations with the operator A ϵ , analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator A 1 are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L 2 ( R d ) -norm for small ε are obtained.

Identification of an unknown spatial source function in a multidimensional hyperbolic partial differential equation with interior degeneracy

A reconstruction of an unknown source function is considered for hyperbolic partial differential equations with interior degeneracy. We identify the spatial element of the source term of a degenerate wave equation using the final observation data. The existence and uniqueness of the direct problem with interior degeneracy within the spatial domain are stated and proved. The inverse problem can be formulated as a nonlinear optimization problem and the unknown source term can be characterized as the solution to a minimization problem. The Tikhonov regularization technique is employed to accomplish the inclusion of noise in the input data, based on the insertion of the regularization term into the cost functional. The conjugate gradient algorithm in conjunction with Morozov's discrepancy principle as a stopping criterion is then utilized to develop an iterative reconstruction procedure. Finally, some numerical simulation results are provided to show the performance of the proposed scheme in one and two dimensions.

A criterion for the unique solvability of the spectral Poincare problem for a class of multidimensional hyperbolic equations

A criterion for the unique solvability of the spectral Poincare problem for a class of multidimensional hyperbolic equations

Gelfand-Levitan integral equation for solving coefficient inverse problem

In this paper, numerical methods for solving multidimensional equations of hyperbolic type by the Gelfand-Levitan method are proposed and implemented. The Gelfand-Levitan method is one of the most widely used in the theory of inverse problems and consists in reducing a nonlinear inverse problem to a one-parameter family of linear Fredholm integral equations of the first and second kind. In the class of generalized functions, the initial-boundary value problem for a multidimensional hyperbolic equation is reduced to the Goursat problem. Discretization and numerical implementation of the direct Goursat problem are obtained to obtain additional information for solving a multidimensional inverse problem of hyperbolic type. For the numerical solution, a sequence of Goursat problems is used for each giveny. A comparative analysis of numerical experiments of the two-dimensional Gelfand-Levitan equation is performed. Numerical experiments are presented in the form of tables and figures for various continuous functions q(x, y).

Задача Трикоми для одного класса многомерных смешанно гиперболо-параболических уравнений

It is known that in the mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate multidimensional hyperbolic equations. If the medium has a high conductivity, then we come to degenerate multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) is reduced to degenerate multidimensional hyperbolic-parabolic equations. It is also known that the oscillations of elastic membranes in space can be modeled according to the Hamilton principle by degenerate multidimensional hyperbolic equations. The study of the process of heat propagation in a medium filled with mass leads to degenerate multidimensional parabolic equations. Therefore, by studying the mathematical modeling of the heat propagation process in oscillating elastic membranes, we also arrive at degenerate multidimensional hyperbolic-parabolic equations. When studying these applications, it becomes necessary to obtain an explicit representation of the solutions to the problems under study. Boundary value problems for hyperbolicparabolic equations on the plane are well studied, and their multidimensional analogues are little studied. The Tricomi problem for these equations was previously investigated. As far as we know, this problem has not been studied in space. In this paper, the Tricomi problem is shown to be ambiguously solvable for a class of multidimensional mixed hyperbolic-parabolic equations.

Well-posedness of the main mixed problem for the multidimensional Lavrentiev — Bitsadze equation

It is known that the oscillations of elastic membranes in space are modelled with partial differential equations. If the deflection of the membrane is considered as a function of u(x; t); x = (x1; :::; xm);m 2; then, according to the Hamilton principle, we arrive to a multidimensional wave equation.Assuming that the membrane is in equilibrium in the bending position, we also obtain the multidimensional Laplace equation from the Hamiltons principle.Consequently, the oscillations of elastic membranes in space can be modelled with a multidimensional Lavrentiev Bitsadze equation.The main mixed problem in the cylindrical domain for multidimensional hyperbolic equations in the space of generalized functions is well studied. In the works of the author, the well-posedness of this problem for multidimensional hyperbolic and elliptic equations is proved, and the explicit forms of classical solutions are obtained.As far as we know, these questions for multidimensional hyperbolic-elliptic equations have not been studied.The mixed problem with boundary-value conditions for the multidimensional Lavrentiev Bitsazde equation is ill-posed.In this paper, we prove the unique solvability and obtain an explicit form of classical solution of themain mixed problem with boundary and initial conditions for the multidimensional Lavrentiev Bitsadze equation.

КОМПОЗИЦИЯЛЫҚ МАТЕРИАЛДАР ҚҰРЫЛЫМЫ ЖӘНЕ КОМПОЗИТТЕР МЕХАНИКАСЫНЫҢ ЕСЕПТЕРІ

It is known that in space during mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conductive, then we get degenerating multidimensional hyperbolic equations. Therefore, the analysis of electromagnetic fields in complex environments (for example, if the conductivity of the medium changes) is reduced to degenerating multidimensional hyperbolic equations. It is also known that oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Therefore, by studying mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerating multidimensional hyperbolic equations. When studying these applications, it becomes necessary to obtain a clear representation of the solutions to the investigated problems. The mixed problem for degenerating multidimensional hyperbolic equations in generalized spaces is well researched. This task is also studied in the works of S. A. Aldashev, where it is shown that its correctness significantly depends on the height of the cylindrical region under consideration. A.V. Bitsadze drew attention to the importance of studies of multidimensional hyperbolic equations with degeneration of type and order. Mixed problems for these equations have not previously been studied. In this work, the solvability of a mixed problem is shown and a clear form of a classical solution for three-dimensional hyperbolic equations with degeneration of type and order is obtained.

MODERN METHODS FOR EVALUATING BUSINESS PROCESSES OF AN ENTERPRISE USING A BALANCED SCORECARD

It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conductive, we get degenerate multi-dimensional hyperbolic equations. If the medium has a high conductivity, then we go to degenerate multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to degenerate multidimensional hyperbolic-parabolic equations. Also, it is known that the oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Studying the process of heat propagation in a medium filled with mass leads to degenerate multidimensional parabolic equations. Consequently, by studying the mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerate multidimensional hyperbolic-parabolic equations. When studying these applications, it is necessary to obtain an explicit representation of the solutions of the studied problems. The mixed problem for degenerate multidimensional hyperbolic equations was previously considered. As far as is known, these questions for degenerate multidimensional hyperbolic-parabolic equations have not been studied. In this paper, unique solvability is shown and an explicit form of the classical solution of the mixed problem for one class of degenerate multidimensional hyperbolic-parabolic equations is obtained.

TRICOMI PROBLEM FOR MULTIDIMENSIONAL MIXED HYPERBOLIC-PARABOLIC EQUATION

It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the media. If the medium is non-conducting, then we obtain multidimensional hyperbolic equations. If the mediums conductivity is higher, then we arrive at multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to multidimensional hyperbolic-parabolic equations. When studying these applications, one needs to obtain an explicit representation of solutions to the problems under study. Boundary-value problems for hyperbolic-parabolic equations on a plane are well studied; however, their multidimensional analogs have been analyzed very little. The Tricomi problem for the above equations has been previously investigated, but this problem in space has not been studied earlier. This article shows that the Tricomi problem is not uniquely solvable for a multidimensional mixed hyperbolic-parabolic equation. An explicit form of these solutions is given.

Parameter identification in multidimensional hyperbolic partial differential equations using wavelet collocation method

In this article, we have proposed a highly efficient and accurate collocation method based on Haar wavelet for the parameter identification in multidimensional hyperbolic partial differential equations (PDEs). In the proposed method, highest order derivative is represented in terms of Haar wavelet and required term of the PDE is obtained using successive integration. Due to compact support and orthogonality of Haar wavelet, this method leads to significant reduction in the computational cost. Cubic spline interpolation and Taylor series approximation are used for the parameter identification. Error analysis is carried out in order to prove the convergence of the method. The obtained numerical results and central processing unit (CPU) time ensure the efficiency and accuracy of the method.

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method

Abstract This research work is to study the numerical solution of three-dimensional second-order hyperbolic telegraph equations using an efficient local meshless method based on radial basis function (RBF). The model equations are used in nuclear material science and in the modeling of vibrations of structures. The explicit time integration technique is utilized to semi-discretize the model in the time direction whereas the space derivatives of the model are discretized by the proposed local meshless procedure based on multiquadric RBF. Numerical experiments are performed with the proposed numerical scheme for rectangular and non-rectangular computational domains. The proposed method solutions are converging quickly in comparison with the different existing numerical methods in the recent literature.

CORRECTNESS OF A MIXED PROBLEM FOR DEGENERATE THREE-DIMENSIONAL HYPERBOLIC-PARABOLIC EQUATIONS

In mathematical modeling of electromagnetic fields in space, the nature of electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate three-dimensional hyperbolic equations. If the medium has a high conductivity, then we come to degeneratethree-dimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the mediums conductivity changes) is reduced to degenerate three-dimensional hyperbolic-parabolic equations. The mixed problem for multidimensional hyperbolic equations is well studied and has been previously considered in the works of various authors. In the articles of Professor S.A. Aldashev, the unique solvability of the mixed problem for degenerate multidimensional hyperbolic equations is proved. It is known that mixed problems for multidimensional hyperbolic-parabolic equations have not been studied much. The paper finds a new class of degenerate three-dimensional hyperbolic-parabolic equations for which the mixed problem has a unique solution and gives an explicit representation of its classical solution.

Computational Identification of the Time Dependence of the Right-Hand Side of a Hyperbolic Equation

Many applied problems lead to the necessity of solving inverse problems for partial differential equations. In particular, much attention is paid to the problem of identifying coefficients of equations using some additional information. The problem of determining the time dependence of the right-hand side of a multidimensional hyperbolic equation using information about the solution at an interior point of the computational domain is considered. An approximate solution is obtained using a standard finite element spatial approximation and implicit schemes for time approximations. The computational algorithm is based on a special decomposition of the solution of the inverse problem when the transition to a new time level is ensured by solving standard elliptic problems. Numerical results for a model two-dimensional problem are given, which demonstrate the potentialities of the computational algorithms proposed to approximately solve inverse problems.

The Bitsadze Problem for a Multidimensional Hyperbolic Equation

We prove the unique solvability of the Bitsadze problem for multidimensional hyperbolic equation.

Бикомпактные схемы для многомерных уравнений гиперболического типа на декартовых сетках с адаптацией к решению

Бикомпактные схемы для многомерных уравнений гиперболического типа на декартовых сетках с адаптацией к решению

КОРРЕКТНОСТЬ СМЕШАННОЙ ЗАДАЧИ ДЛЯ ОДНОГОКЛАССА ВЫРОЖДАЮЩИХСЯ МНОГОМЕРНЫХГИПЕРБОЛИЧЕСКИХ УРАВНЕНИЙ

Oscillations of elastic membranes in 3D are modelled as degenerate multi-dimensional hyperbolic equations. For applied work, it is important to obtain explicit representations of solution of the studied boundary-value problems. This paper shows the unique solvability and obtains the explicit form of the classical solution of the mixed problem for degenerate multi-dimensional hyperbolic equations.

Bicompact schemes for multidimensional hyperbolic equations on Cartesian meshes with solution-based AMR

Bicompact schemes for multidimensional hyperbolic equations on Cartesian meshes with solution-based AMR

On Determination of the Coefficient of the Lower Term of the Multidimensional Hyperbolic Equation of the Second Order

On Determination of the Coefficient of the Lower Term of the Multidimensional Hyperbolic Equation of the Second Order

Recovery of a Rapidly Oscillating Absolute Term in the Multidimensional Hyperbolic Equation

The paper is devoted to the development of the theory of inverse problems for evolution equations with summands rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints (overdetermination conditions) are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.