What is Laplace's equation: Definition and 116 Discussions

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as





{\displaystyle \nabla ^{2}\!f=0\qquad {\mbox{or}}\qquad \Delta f=0,}




{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}
is the Laplace operator,

{\displaystyle \nabla \cdot }
is the divergence operator (also symbolized "div"),

{\displaystyle \nabla }
is the gradient operator (also symbolized "grad"), and


{\displaystyle f(x,y,z)}
is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
If the right-hand side is specified as a given function,


{\displaystyle h(x,y,z)}
, we have


{\displaystyle \Delta f=h.}
This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.
The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.

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  1. chwala

    A The boundary conditions in reference to Laplace's equation

    We have inhomogenous dirichlet boundary conditions (well understood)....the laplace equation is a steady state equation and we can clearly see that in 2D..it will be defined by 4 boundary conditions and NO initial condition...having said that; kindly have a look at the continuation below... I...
  2. Harikesh_33

    I Question regarding Laplace's Equation for regions with charges

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  3. Ahmed1029

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  4. Ahmed1029

    I Laplace's equation in presence of a dipole (perfect or physical)

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  5. Stefan H

    A Solving Laplace's equation in polar coordinates for specific boundary conditions

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  6. yucheng

    I Unraveling Laplace's Equation: Exploring Valid Domains and Boundary Conditions

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  7. Rlwe

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  8. HansBu

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  9. L

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  10. M

    I PDEs: Laplace's Equation over a Parallelogram

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  11. N

    Laplace's equation vs Principle of Least Action?

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  12. Faizan Samad

    Electrostatics Problem using Laplace's Equation

    Homework Statement Prove that in a region free of electric charge, the value of the potential at a point is equal to its average over any sphere centered at that point. Homework Equations V(r) = 1/(4piR^2) Integral(V * da) The Attempt at a Solution I defined a point outside the region where a...
  13. Another

    I Solve Laplace's Equation with Laplace Transform

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  14. Another

    I Exploring Solutions to Laplace's Equation in 3D

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  15. B

    I Can the Schrodinger equation satisfy Laplace's equation?

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  16. B

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  17. C

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  18. D

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  19. R

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  20. R

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  21. DrPapper

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  22. J

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  23. H Smith 94

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  24. A

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    Homework Statement Consider solutions to the One Dimensional Laplace's Equation in Cartesian Coordinates Let the range of x be from x1 to x2 (x1 > x2) and the boundary conditions are V[x1] = V1 and V[x2] = V2 Find the equation for V[x] Homework Equations V[x] = 1/2 (V(x+a)+V(x-a)) V[x] = mx...
  25. SquidgyGuff

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  26. S

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  27. H

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  28. B

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  29. I

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  30. V

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  31. V

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  32. P

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  33. I

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  34. N

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  35. T

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  36. T

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  37. P

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  38. M

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  39. T

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  40. C

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  41. A

    Brownian motion solves Laplace's Equation?

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  42. S

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  43. S

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  44. C

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  45. J

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  46. H

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  47. L

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  48. J

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  49. T

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  50. J

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