What Is a Partial Differential Equation Definition Types Formulas and Solved Examples
Solving Partial Differential Equations
Today we’ll be discussing Partial Differential Equations. A topic like Differential Equations is full of surprises and fun but at the same time is considered quite difficult. So, to fully understand the concept let’s break it down to smaller pieces and discuss them in detail. Do you know what an equation is? An equation is a statement in which the values of the mathematical expressions are equal.
For eg. : 2x-3=10 is an equation.
Well, equations are used in 3 fields of mathematics and they are:
(i) geometry
(ii) algebra
(iii) differential
1.Geometry
Equations are used in geometry to describe geometric shapes. Equations are considered to have infinite solutions. So in geometry, the purpose of equations is not to get solutions but to study the properties of the shapes. Analytic Geometry deals mostly in Cartesian equations and Parametric Equations.
For eg. to explain a circle there is a general equation: (x – h)2 + (y – k)2 = r2
2.Algebra
In algebra, mostly two types of equations are studied from the family of equations. The most common one is polynomial equations and this also has a special case in it called linear equations. Polynomial equations are generally in the form P(x)=0 and linear equations are expressed ax+b=0 form where a and b represents the parameter. Algebra also uses Diophantine Equations where solutions and coefficients are integers.
3.Differential
Differential equations are the equations which have one or more functions and their derivatives. These are used for processing model that includes the rates of change of the variable and are used in subjects like physics, chemistry, economics, and biology. There are two types of differential equations:
A) Ordinary Differential Equations
B) Partial Differential Equations
A) Ordinary Differential Equations
Ordinary Differential Equations or ODE are equations which have a function of an independent variable and their derivatives. A variable is used to represent the unknown function which depends on x. In the equation, X is the independent variable. There are many other ways to express ODE.
B) Partial Differential Equations
The definition of Partial Differential Equations (PDE) is a differential equation that has many unknown functions along with their partial derivatives. It is used to represent many types of phenomenons like sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation, and quantum mechanics.
There are Different Types of Partial Differential Equations:
(i) Equations of First Order/ Linear Partial Differential Equations
(ii) Linear Equations of Second Order Partial Differential Equations
(iii) Equations of Mixed Type
Furthermore, the classification of Partial Differential Equations of Second Order can be done into parabolic, hyperbolic, and elliptic equations.
uxx [+] uyy = 0 (2-D Laplace equation)
uxx [=] ut (1-D heat equation)
uxx [−] uyy = 0 (1-D wave equation)
The following is the Partial Differential Equations formula:
Solving Partial Differential Equations
We will do this by taking a Partial Differential Equations example.
Example 1.
(y + u) ∂u ∂x + y ∂u∂y = x − y in y > 0, −∞ < x < ∞,
with u =(1 + x) on y = 1.
Solving Partial Differential Equation
We first look for the general solution of the PDE before applying the initial conditions. Combining the characteristic and compatibility equations,
dxds = y + u, (2.11)
dyds = y, (2.12)
duds = x − y (2.13)
we seek two independent first integrals.
Equations (2.11) and (2.13) give
d(x + u)ds= x + u,
and equation (2.12)
1y dyds = 1.
Now, consider dds (x + uy) = 1y dds(x + u) − x + uy2 dyds , = x + uy − x + uy = 0.
So,(x + u)y = c1 is constant.
This defines a family of solutions of the PDE; so, we can choose φ(x, y, u) = x + uy
Example 2. A partial differential equation requires
a)exactly one independent variable
b) two or more independent variables
c) more than one dependent variable
d) an equal number of dependent and independent variables
Solution:
The correct answer is (B).
If a differential equation has only one independent variable then it is called an ordinary differential equation. A partial differential equation has two or more unconstrained variables.
Fun Facts About Differential Equations:
A Differential Equation can have an infinite number of solutions as a function also has an infinite number of antiderivatives. The reason for both is the same.
Sometimes we can get a formula for solutions of Differential Equations.
Since we can find a formula of Differential Equations, it allows us to do many things with the solutions like devise graphs of solutions and calculate the exact value of a solution at any point.
For multiple essential Differential Equations, it is impossible to get a formula for a solution, for some functions, they do not have a formula for an anti-derivative.
Even though we don’t have a formula for a solution, we can still Get an approx graph of solutions or Calculate approximate values of solutions at various points.
The general solution of an inhomogeneous ODE has the general form: u(t) = uh(t) + up(t)
A linear ODE of order n has precisely n linearly independent solutions. There are many ways to choose these n solutions, but we are certain that there cannot be more than n of them.
The ‘=’ sign was invented by Robert Recorde in the year 1557.He thought to show for things that are equal, the best way is by drawing 2 parallel straight lines of equal lengths.
FAQs on Partial Differential Equations Explained with Concepts and Applications
1. What is a partial differential equation (PDE)?
A partial differential equation (PDE) is an equation that involves partial derivatives of a function of two or more independent variables. In other words, it relates a multivariable function (such as u(x, y) or u(x, t)) to its partial derivatives.
- Example: ∂u/∂t = k ∂²u/∂x² (the heat equation).
- PDEs model physical phenomena like heat flow, wave motion, and fluid dynamics.
- They differ from ordinary differential equations (ODEs), which involve derivatives with respect to only one variable.
2. What is the general form of a second-order partial differential equation?
The general form of a second-order PDE in two variables is A u_xx + B u_xy + C u_yy + D u_x + E u_y + F u = G, where the coefficients may depend on x and y. Here:
- u_xx, u_xy, u_yy are second-order partial derivatives.
- A, B, C determine the type (elliptic, parabolic, hyperbolic).
- G may be zero (homogeneous PDE) or non-zero (non-homogeneous PDE).
3. How are partial differential equations classified?
Second-order linear PDEs are classified as elliptic, parabolic, or hyperbolic based on the discriminant B² − 4AC. For the equation A u_xx + B u_xy + C u_yy = 0:
- If B² − 4AC < 0, it is elliptic (e.g., Laplace’s equation).
- If B² − 4AC = 0, it is parabolic (e.g., heat equation).
- If B² − 4AC > 0, it is hyperbolic (e.g., wave equation).
4. What is the difference between a PDE and an ODE?
The main difference is that a PDE involves partial derivatives with respect to multiple variables, while an ODE involves derivatives with respect to only one variable.
- Example of ODE: dy/dx = 3x².
- Example of PDE: ∂u/∂t = k ∂²u/∂x².
- PDEs describe systems depending on space and time, whereas ODEs typically describe single-variable change.
5. What is the heat equation in partial differential equations?
The heat equation is a parabolic PDE given by ∂u/∂t = k ∂²u/∂x², where k is the thermal diffusivity constant. It models how heat diffuses over time in a medium.
- u(x, t) represents temperature.
- k > 0 controls the rate of heat conduction.
- It is typically solved using separation of variables and Fourier series.
6. What is Laplace’s equation and why is it important?
Laplace’s equation is the elliptic PDE ∇²u = 0, or in two variables, u_xx + u_yy = 0. It describes steady-state phenomena such as electrostatics and steady heat flow.
- It has no time dependence.
- Solutions are called harmonic functions.
- It is fundamental in potential theory and boundary value problems.
7. How do you solve a PDE using separation of variables?
The separation of variables method assumes the solution can be written as a product of single-variable functions, such as u(x, t) = X(x)T(t). The steps are:
- Substitute u(x, t) = X(x)T(t) into the PDE.
- Divide to separate variables so each side depends on only one variable.
- Set both sides equal to a constant (separation constant).
- Solve the resulting ODEs for X and T.
8. What is the wave equation in partial differential equations?
The wave equation is a hyperbolic PDE given by ∂²u/∂t² = c² ∂²u/∂x², where c is the wave speed. It models vibrations and wave propagation in strings, sound, and light.
- u(x, t) represents displacement.
- c is the constant wave velocity.
- Its general solution can often be written using d’Alembert’s formula.
9. What are boundary conditions in partial differential equations?
Boundary conditions are additional constraints that specify the solution of a PDE on the boundary of the domain. Common types include:
- Dirichlet condition: specifies the value of the function (u = f).
- Neumann condition: specifies the derivative (∂u/∂n = g).
- Mixed condition: combination of value and derivative.
10. Can you give a simple example of solving a first-order PDE?
A simple first-order PDE is ∂u/∂x = 0, whose general solution is u(x, y) = f(y), where f is any function of y. This is because:
- If the partial derivative with respect to x is zero, the function does not depend on x.
- So u remains constant in the x-direction.
- The solution can vary only with the other variable (y).
