Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Differential Equations → Partial Differential Equations ↓


Laplace Equation


The Laplace equation is one of the most important equations in the field of mathematics and physics. It is a type of partial differential equation used to describe various physical phenomena such as heat conduction, fluid flow, and electric and gravitational potentials. Named after French mathematician Pierre-Simon Laplace, this equation is a second-order linear elliptic partial differential equation.

What is Laplace equation?

Mathematically, the Laplace equation can be expressed as:

∇²φ = 0

where ∇² (nabla squared) is the Laplace operator, also known as the Laplacian. The symbol φ usually represents the scalar field potential. The equation implies that the divergence of the gradient of a field is zero in the region where it is valid.

In two dimensions, the Laplace equation has the form:

∂²φ/∂x² + ∂²φ/∂y² = 0

Its expansion in three dimensions is as follows:

∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z² = 0

In simple terms, the Laplace equation states that the sum of the second derivatives of the potential function with respect to each spatial variable is equal to zero. This implies that the potential function is a harmonic function.

Importance and applications

Laplace's equation is very important in many fields of science and engineering. Its applications are as follows:

  • Electrostatics: In electrostatics, the Laplace equation governs the electric potential in a space free of charge.
  • Fluid dynamics: In fluid dynamics, it describes potential flow, which is an idealized flow where viscous forces are ignored.
  • Heat conduction: This is the steady state heat distribution model, where the temperature does not change with time.
  • Gravitational field: It describes the gravitational potential in massless regions.
  • Quantum Mechanics: This equation is used to solve the Schrödinger equation under certain circumstances.

Visual representation and examples

I electric potential Equipotential lines

The above diagram shows a simple depiction of the electric potential and equipotential lines in a region, which characterizes the solution of the Laplace equation.

Text example: heat distribution

Consider a thin rectangular metal plate with temperatures defined at its edges. Our goal is to find the temperature distribution across the plate. The temperature at any point on the plate is represented by u(x, y).

Assumptions:

  • The plate reaches thermal equilibrium, which means the heat flow is constant, and the temperature does not change with time.
  • This plate is perfect and uniform, ensuring uniform thermal properties.

The Laplace equation for this problem is:

∂²u/∂x² + ∂²u/∂y² = 0

Suppose the boundary conditions are such that the temperature is 100 degrees at the left edge of the plate and 0 degrees at the right edge, with the top and bottom edges insulated. We can solve the Laplace equation under these boundary conditions to find u(x, y).

This type of problem is usually solved using methods such as separation of variables or numerical techniques. The uniform micro-scale properties of the plate result in homogeneity and predictability in the solution.

Properties of Laplace Equation

The solutions of the Laplace equation, known as harmonic functions, have some important properties:

  • Mean Value Property: In a region where a harmonic function is defined, the value at any point within that region is the average of the values in any surrounding region completely contained within that region.
  • Maximum and Minimum Principle: In a bounded domain, the maximum and minimum values of a harmonic function occur on the boundary. Therefore, a non-constant harmonic function cannot have a local maximum or minimum inside its domain.
  • Uniqueness Theorem: Given the domain and boundary conditions, the solution to the Laplace equation is unique. If two solutions exist, they must be identical over the entire domain.

Solving the Laplace equation

Solving the Laplace equation usually involves appropriate mathematical techniques for dealing with partial differential equations in the complex variable domain. Here are some basic approaches:

Separation of variables

Separation of variables is a general method for solving partial differential equations, including the Laplace equation. It assumes that the solution can be written as a product of functions, each of which depends on a single coordinate:

φ(x, y) = X(x)Y(y)

By substituting this form into the Laplace equation and separating the variables, we can obtain ordinary differential equations for each function. Solving these ODEs and applying marginal conditions yields the solution of the original PDE.

Numerical methods

In cases where analytical solutions are difficult or impossible to obtain, numerical methods such as finite difference methods or finite element methods can be used. These methods partition the domain and estimate the partial derivatives in the equation to obtain a solution.

Consider a circular metal disk whose edge is maintained at a constant temperature. The temperature distribution inside the disk is governed by the Laplace equation. In practice, one can approximate this disk as a grid of point temperatures and use numerical techniques to solve for each point.

Conclusion

The Laplace equation is a foundational element in the study of potential theory and many areas of science and engineering. Understanding the properties and solutions of the Laplace equation facilitates the study of a variety of physical and theoretical phenomena, enabling the prediction and manipulation of complex systems in both natural and designed environments.

Throughout history and into the modern era, Laplace's equation has remained a pillar of mathematical analysis, influencing subjects from electromagnetism to thermodynamics and beyond.


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