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 ↓


Separation of Variables


The method of separation of variables is a technique used to solve partial differential equations (PDEs). PDEs can describe a variety of phenomena such as heat flow, wave propagation, fluid dynamics, and more. This method simplifies the process by dividing the PDE into smaller, more manageable ordinary differential equations (ODEs).

Introduction

Partial differential equations involve multiple independent variables. The main goal when using the variable separation technique is to rewrite the PDE so that each independent variable appears separately. This allows the PDE to be decomposed into a set of single-variable ODEs. From there, given certain boundary conditions, the solutions to these ODEs can be combined to form the overall solution to the original PDE.

Why use variable separation?

Separation of variables is particularly useful when dealing with linear PDEs with boundary conditions. It allows mathematicians and scientists to handle complex equations through a systematic approach. This method also works well when the PDE and its conditions exhibit sufficient symmetry. For example, in cases of steady-state temperatures or vibrations at edges, this technique can quickly determine solutions, making it a staple in mathematical physics.

Basic steps

  1. Suppose that the solution to the PDE can be expressed as a product of functions, each of which depends on a single coordinate.
  2. Substitute this form into the original PDE.
  3. Divide the resulting equation into parts, each of which depends only on a single coordinate.
  4. Set each part equal to a constant (called the dissociation constant).
  5. Solve the resulting ODEs.
  6. Use limiting conditions to determine unknown constants.
  7. Construct the complete solution by summing the infinite series of solutions derived from each dissociation constant.

An example with visual representation

Consider the heat equation in one dimension:

∂u/∂t = α² ∂²u/∂x²

Here, ( u(x, t) ) is the temperature at position ( x ) and time ( t ), and ( α ) is a constant related to thermal diffusivity.

For separation of variables assume:

u(x, t) = X(x)T(t)

Substitute this into the heat equation:

X(x) ∂T/∂t = α² T(t) ∂²X/∂x²

Divide both sides by ( α² X(x) T(t) ):

1/T(t) ∂T/∂t = α² / X(x) ∂²X/∂x²

The left side depends only on ( t ) and the right side depends only on ( x ). Set them equal to the dissociation constant, say ( -λ ):

1/T(t) ∂T/∂t = -λ = α² / X(x) ∂²X/∂x²

This results in two ODEs:

  • For ( T(t) ) : ( ∂T/∂t = -λT(t) )
  • For ( X(x) ) : ( α² ∂²X/∂x² = -λX(x) )

Each of these can be solved, subject to marginal conditions, to produce a complete solution.

Example of limiting condition: If the ends of the rod are held rigidly at zero temperature, then these conditions can be expressed as:

u(0, t) = 0 and u(L, t) = 0 for all t

Applying separation of variables, ( X(0)T(t) = 0 ) implies ( X(0) = 0 ), and so on ( X(L) = 0 ). Solve:

X(x) = C sin(nπx/L)

Here, ( n ) is an integer (1, 2, 3, ...) and ( C ) is a constant. This equation satisfies the limiting conditions.

More in-depth example

Example: wave equation

Consider the wave equation:

∂²u/∂t² = c² ∂²u/∂x²

Assume a solution ( u(x, t) = X(x)T(t) ). Substituting, we get:

X(x) ∂²T/∂t² = c² T(t) ∂²X/∂x²

Divide by ( c² X(x) T(t) ):

1/T(t) ∂²T/∂t² = 1/c² X(x) ∂²X/∂x² = -λ

The isolated ODEs are as follows:

  • ( ∂²T/∂t² = -λT(t) )
  • ( c² ∂²X/∂x² = -λX(x) )

By handling these ODEs differently and applying boundary conditions, solutions to the original wave equation can be found. This method shows the power of separation of variables to manage complex equations with many variables.

Example: Laplace equation

Consider the Laplace equation, widely used in electromagnetism and fluid dynamics:

∇²u = 0

In two dimensions, it is:

∂²u/∂x² + ∂²u/∂y² = 0

Let ( u(x, y) = X(x)Y(y) ) and substitute:

Y(y) ∂²X/∂x² + X(x) ∂²Y/∂y² = 0

Discrete Variables:

1/X(x) ∂²X/∂x² = -1/Y(y) ∂²Y/∂y² = λ

The resulting ODEs:

  • ( ∂²X/∂x² = λX(x) )
  • ( ∂²Y/∂y² = -λY(y) )

The solution, given specific boundary conditions for the problem, allows the construction of a complete solution of the Laplace equation for the region of interest.

Closing thoughts

Separation of variables is a powerful and integral method for solving PDEs that is commonly seen in a variety of fields such as mathematics, physics, and engineering. Although this technique has its limitations, particularly requiring simple boundaries and linearity, its effectiveness in decomposing complex problems makes it an essential tool. This method is an example of how breaking down problems into simpler, component parts can lead to effective solutions, just as other disciplines have modular problem-solving strategies.

By understanding and using separation of variables, one can obtain practical solutions to problems that model important physical processes. The main thing is to recognize the patterns and symmetries inherent in such problems that allow effective use of this method.


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Separation of Variables

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