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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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1986
In this chapter we shall study the standard examples of elliptic equations, which bear the name of the French mathematician Laplace. Our setting will be, in general, a region in En. We shall, however, restrict ourselves to E1, E2, or E3 in various instances for computational purposes, since the techniques to be used are easily extended to higher ...
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In this chapter we shall study the standard examples of elliptic equations, which bear the name of the French mathematician Laplace. Our setting will be, in general, a region in En. We shall, however, restrict ourselves to E1, E2, or E3 in various instances for computational purposes, since the techniques to be used are easily extended to higher ...
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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
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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
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2011
In this chapter we will consider the solution of Laplace’s Equation $${\nabla }^{2}\Phi = 0$$ (9.111) using separation of variables. We can claim that this chapter is required by the preceding Chap. 8, which indicated the importance of the homogeneous solution to Poisson’s Equation. However, as with Chap. 8 it is not integral to a study of the
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In this chapter we will consider the solution of Laplace’s Equation $${\nabla }^{2}\Phi = 0$$ (9.111) using separation of variables. We can claim that this chapter is required by the preceding Chap. 8, which indicated the importance of the homogeneous solution to Poisson’s Equation. However, as with Chap. 8 it is not integral to a study of the
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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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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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