Second order hyperbolic partial differential equations

The parts of Equation 1.54 are linear second-order hyperbolic partial differential equations called wave equations. The general solutions for the wave equations were given by D Alembert in 1747 [16] as... 

As is well known, many physical phenomena are expressed mathematically as a second-order partial differential equation. Electrical transients associated with a wave-propagation characteristic are mathematically represented by a hyperbolic partial differential equation. 

The theory for classifying linear, second-order, partial-differential equations is well established. Understanding the classification is quite important to understanding solution algorithms and where boundary conditions must be applied. Partial differential equations are generally classified as one of three forms elliptic, parabolic, or hyperbolic. Model equations for each type are usually stated as... 

The three-dimensional, second-order, nonlinear, elliptic partial differential equation may be simplified in the limit of weak electrolyte solutions, where the hyperbolic sine of is well approximated by 4). This yields the linearized Poisson—Boltzmann equation... 

Linear second-order partial differential equations in two independent variables are further classified into three canonical forms elliptic, parabolic, and hyperbolic. The general form of this class of equations is... 

A similar classification for second-order partial differential equations with three independent variables is given by Tychonov and Samarski [21. This classification includes elliptic, parabolic, hyperbolic, and ultrahyperbolic. The majority of partial differential equations in engineering and physics are of second-order with two, three, or four independent... 

Second-order partial differential equations of the hyperbolic type occur principally in physical problems connected with vibration processes. For example the one-dimensional wave equation... 

A brief review of the different classes of second-order partial differential equations is appropriate. Parabolic PDFs are typified by the heat equation duldx = Dd uldy, hyperbolic equations by the wave equation d uldx = c d utdy, and elliptical equations by Laplace s equation d utdx + d uldy = 0. x and y are spatial coordinates, and u is representative of forces, stresses, or similar quantities. The diffusion coefficient D in the heat equation must have units of length, and the "sound speed" in the wave equation is dimensionless. 

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