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Laplace's Equation and Network Flows
Operations Research, 1967This paper shows that the problem of Laplace's equation can be formulated as a minimum cost network flow problem with quadratic cost. Then an algorithm for solving quadratic cost network flow problems can be modified to solve the standard Laplace's equation.
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2013
The Laplace equation Δu = 0 occurs frequently in applied sciences, in particular in the study of the steady state phenomena. Its solutions are called harmonic functions. For instance, the equilibrium position of a perfectly elastic membrane is a harmonic function as it is the velocity potential of a homogeneous fluid. Also, the steady state temperature
Sandro Salsa +3 more
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The Laplace equation Δu = 0 occurs frequently in applied sciences, in particular in the study of the steady state phenomena. Its solutions are called harmonic functions. For instance, the equilibrium position of a perfectly elastic membrane is a harmonic function as it is the velocity potential of a homogeneous fluid. Also, the steady state temperature
Sandro Salsa +3 more
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1977
Let Ω be a domain in ℝ n and u a C 2(Ω) function. The Laplacian of u, denoted ⊿u, is defined by $$\Delta u = \sum\limits_{i = 1}^n {{D_u}u = div} Du.$$ (2.1)
David Gilbarg, Neil S. Trudinger
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Let Ω be a domain in ℝ n and u a C 2(Ω) function. The Laplacian of u, denoted ⊿u, is defined by $$\Delta u = \sum\limits_{i = 1}^n {{D_u}u = div} Du.$$ (2.1)
David Gilbarg, Neil S. Trudinger
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2009
In Chapter 4 we have discussed the PDEs that control the heat flow in two and three dimensional spaces given by $$ \begin{gathered} u_t= \bar k(u_{xx}+ u_{yy} ), \hfill \\ u_t= \bar k(u_{xx}+ u_{yy}+ u_{zz} ), \hfill \\ \end{gathered} $$ (7.1) respectively, where \( \bar k \) is the thermal diffusivity.
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In Chapter 4 we have discussed the PDEs that control the heat flow in two and three dimensional spaces given by $$ \begin{gathered} u_t= \bar k(u_{xx}+ u_{yy} ), \hfill \\ u_t= \bar k(u_{xx}+ u_{yy}+ u_{zz} ), \hfill \\ \end{gathered} $$ (7.1) respectively, where \( \bar k \) is the thermal diffusivity.
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2014
Potential theory has its origins in gravitational theory and electromagnetic theory. The common element of these two is the inverse square law governing the interaction of two bodies. The concept of potential function arose as a result off the work done in moving a unit charge from one point of space to another in the presence of another charged body ...
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Potential theory has its origins in gravitational theory and electromagnetic theory. The common element of these two is the inverse square law governing the interaction of two bodies. The concept of potential function arose as a result off the work done in moving a unit charge from one point of space to another in the presence of another charged body ...
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2017
The Laplace equation is the archetypal elliptic equation. It appears in many applications when studying the steady state of physical systems that are otherwise governed by hyperbolic or parabolic operators. Correspondingly, elliptic equations require the specification of boundary data only, and the Cauchy (initial-value) problem does not arise.
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The Laplace equation is the archetypal elliptic equation. It appears in many applications when studying the steady state of physical systems that are otherwise governed by hyperbolic or parabolic operators. Correspondingly, elliptic equations require the specification of boundary data only, and the Cauchy (initial-value) problem does not arise.
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2000
In Chapter 11 we discussed the technique of the separation of variables for the most important PDEs encountered in introductory physics and engineering courses.
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In Chapter 11 we discussed the technique of the separation of variables for the most important PDEs encountered in introductory physics and engineering courses.
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1991
The subject of this book is the study of steady and unsteady porous media flow. In order to study a physical problem, one can describe it by a mathematical model. In the present case, this process leads to the Laplace equation (see previous chapter) which is one of the fundamental equations of engineering analysis.
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The subject of this book is the study of steady and unsteady porous media flow. In order to study a physical problem, one can describe it by a mathematical model. In the present case, this process leads to the Laplace equation (see previous chapter) which is one of the fundamental equations of engineering analysis.
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