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First Darboux problem for nonlinear hyperbolic equations of second order
Mathematical Notes, 2008zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Dzhokhadze, O. M. +1 more
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A solvability result for a nonlinear weakly hyperbolic equation of second order
Nonlinear Differential Equations and Applications NoDEA, 1995The author considers the Cauchy problem \[ u_{tt} - u^{2k} \sum^n_{i,j = 1} a_{ij} (t,x,u) u_{x_i x_j} = f(t,x,u,u_t), \quad u (0,x) = \Phi (x),\;u_t(0,x) = \Psi (x), \] where \(\Phi\), \(\Psi \in C_0^\infty (\mathbb{R}^n)\), \(k \in \mathbb{N}\), \(a_{ij} = a_{ji}\), \(f\) are \(C^\infty\)-functions, \(f(t,x,0,0) = 0\), and \[ \sum^n_{i,j = 1} a_{ij} \
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Long time effects of nonlinearity in second order hyperbolic equations
Communications on Pure and Applied Mathematics, 1986This is an expository paper based on a talk given by the author on October 21, 1985. The author considers initial value problems for the quasilinear equation \[ u_{tt}-2b_ i(u')u_{tx_ i}- a_{ik}(u')u_{x_ ix_ i}=0,\quad u'=(u_ t,u_{x_ i},...,u_{x_ n}) \] and describes some of the methods used to discuss existence of solutions for large t.
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Singularities of solutions for nonlinear hyperbolic equations of second order
2000We consider the Cauchy problem for nonlinear hyperbolic partial differential equations of second order. Then the Cauchy problem does not generally admit a classical solution in the large, that is to say, singularities generally appear in finite time. The typical example of singularity is “shock wave”.
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International Journal of Bifurcation and Chaos, 2015
If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a ...
Liangliang Li +3 more
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If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a ...
Liangliang Li +3 more
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Korean Journal of Computational & Applied Mathematics, 1998
The authors consider the initial-boundary value problem for a nonlinear hyperbolic equation of second order in flux formulation, \[ \text{grad }p+ \underline b(\underline u)= \underline O\quad\text{in }\Omega\times (0,T], \] \[ p_{tt}+ \text{div }\underline u= f\quad\text{in }\Omega\times (0,T], \] where \(p\), \(f\) are scalars and \(\underline u\), \(
Jiang, Ziwen, Chen, Huanzhen
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The authors consider the initial-boundary value problem for a nonlinear hyperbolic equation of second order in flux formulation, \[ \text{grad }p+ \underline b(\underline u)= \underline O\quad\text{in }\Omega\times (0,T], \] \[ p_{tt}+ \text{div }\underline u= f\quad\text{in }\Omega\times (0,T], \] where \(p\), \(f\) are scalars and \(\underline u\), \(
Jiang, Ziwen, Chen, Huanzhen
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Reduction of Nonlinear Wave Equations to a Second-Order Quasi-linear Hyperbolic System
2017As stated before, this book is concerned with the Cauchy problem of nonlinear wave equations with small initial data.
Tatsien Li, Yi Zhou
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Nonlinear PDE Model for Image Restoration Using Second-Order Hyperbolic Equations
Numerical Functional Analysis and Optimization, 2015In this article we consider a novel nonlinear PDE-based image denoising technique. The proposed restoration model uses second-order hyperbolic diffusion equations. It represents an improved nonlinear version of a linear hyperbolic PDE model developed recently by the author, providing more effective noise removal results while preserving the edges and ...
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Numerical Methods for Partial Differential Equations, 2020
AbstractWe have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one‐ and two‐dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one‐ and two‐dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods.
Muhammad Asif +3 more
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AbstractWe have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one‐ and two‐dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one‐ and two‐dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods.
Muhammad Asif +3 more
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Second-order finite-volume schemes for a non-linear hyperbolic equation: error estimate
Mathematical Methods in the Applied Sciences, 2000Second-order finite volume schemes for multidimensional nonlinear hyperbolic equations one derived and studied. The main result is an error estimate for the approximation to the entropy solution of the equation. A discrete entropy inequality is introduced and proved under natural assumptions on the problem. An error estimate of order \(h^{n/k}\) (\(h\)
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