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Partial differential equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
where ∇=∂/∂x,∂/∂y $ \nabla = \left\langle {\partial /\partial x,\partial /\partial y} \right\rangle $ is the gradient operator, and uxx and uyy are shortcuts for partial derivatives, so uxx = ∂ 2u/ ∂ x2 and uyy = ∂ 2u/ ∂ y2. Laplace’s equation also occurs in other branches of mathematical physics, in‐ cluding electrostatics, hydrodynamics, elasticity, and many others. A smooth function ux,y $ u\left( {x, y} \right) $ that satisfies Laplace’s equation is called a harmonic function.
Numerical Methods for Elliptic PDEs
Published in Victor G. Ganzha, Evgenii V. Vorozhtsov, Numerical Solutions for Partial Differential Equations, 2017
Victor G. Ganzha, Evgenii V. Vorozhtsov
In the presence of the heat sources we obtain the equation () Δu=−f(x,y) where f = F/k, F is the density of the heat sources, and k is the thermal conductivity coefficient; f(x,y), is usually a given function of its arguments. The inhomogeneous Laplace’s equation is often called the Poisson equation.
Conformal Mapping
Published in Sivaji Chakravorti, Electric Field Analysis, 2017
Because both the real and imaginary parts of F(z), namely, ϕ(x,y) and ψ(x,y), are harmonic functions, they satisfy Laplace’s equation and hence either one of these two could be used to find potential. Thus, the complex analytic function F(z) is known as complex potential. Laplace’s equation is one of the most important partial differential equations in engineering and physics. The theory of solutions of Laplace’s equation is known as potential theory. The concept of complex potential relates potential theory closely to complex analysis.
An efficient computational approach for the solution of time-space fractional diffusion equation
Published in International Journal of Modelling and Simulation, 2023
Sudarshan Santra, Jugal Mohapatra
The Poisson’s equation is an inhomogeneous form of the Laplace’s equation, which has a vast application in the formation of numerous models in the fields of Mechanics and Physics. It also describes models in fluid flows, in the theory of gravitation, etc. For further studies, one may refer [1–3] and references therein. The Poisson’s equation was used in [4] to represent a nonequilibrium state of an ideal quantum gas confined in the cavity under a moving piston. The nonplanar very-large-scale integration was analyzed in [5] by using the solutions of the Poisson’s equations. The integral transform method, described in [6], can be used to solve the Poisson’s equations analytically and also, Arzani and Afshar in [7] investigated the numerical solution of the Poisson’s equations by the discrete least square meshless method. Shojaei et al. [8] developed the geometrical transform method and the graph product rules to find out an efficient numerical solution of the Laplace’s and the Poisson’s equations. The numerical solution of the Laplace’s and the Poisson’s equation was also discussed in [9]. The biquadratic finite volume element method was considered in [10] for solving two-dimensional Poisson’s equations.