Partial Differential Equations

Partial differential equations are made up of a function with variables and their derivatives. Such equations aid in the relationship of a function with several variables to their partial derivatives. They are extremely important in analyzing natural phenomena such as sound, temperature, flow properties, and waves. They are used to express issues that include an unknown function with numerous dependent and independent variables, as well as the second derivative of this function with respect to the independent variables.

• Order of Partial Differential Equations: The order of the highest derivative term which occurs in a given partial differential equation is called the order of the said equation. Say,  is a partial differential equation. As the order of the highest derivative is 1, hence, this is a first-order partial differential equation.
• Degree of Partial Differential Equations: The degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation  has 1 as the highest derivative is of the first degree.

General Form of Partial Differential Equations

Types of Partial Differential Equations

• First-Order Partial Differential Equations: First-order partial differential equations are those in which the highest partial derivatives of the unknown function are of the first order. They can be both linear and non-linear. The derivatives of these variables are neither squared nor multiplied.
• Second-Order Partial Differential Equations: Second-order partial differential equations have the highest partial derivatives of the order. These equations can be linear, semi-linear, or non-linear. Linear second-order partial differential equations are much more complicated than non-linear and semi-linear second-order PDEs.
• Quasi-Linear Partial Differential Equations: The highest rank of partial derivatives arises solely as linear terms in quasilinear partial differential equations. First-order quasi-linear partial differential equations are commonly utilized in physics and engineering to solve a variety of problems.
• Homogeneous Partial Differential Equations: The nature of the variables in terms determines whether a partial differential equation is homogeneous or non-homogeneous. A non-homogeneous PDE is a partial differential equation that contains all terms including the dependent variable and its partial derivatives.

Classification of Partial Differential Equations

Say there is a linear second-order partial differential equation of second degree given as Au_xx + 2Bu_xy + Cu_yy + constant = 0. Its discriminant is B² – AC. On the basis of different values of such discriminant, the partial differential equations can be classified as follows:

• Parabolic: Such partial equations whose discriminant is zero, i.e., B² – AC = 0, are called parabolic partial differential equations. These types of PDEs are used to express mathematical, scientific as well as economical, and financial topics such as derivative investments, particle diffusion, heat induction, etc.
• Hyperbolic: Such partial equations whose discriminant exceeds zero, i.e., B² – AC > 0, are called hyperbolic partial differential equations. These types of PDEs are used to express wave progressions and other such concepts and fundamentals which pertain to waves.
• Elliptic: Such partial equations whose discriminant is less than zero, i.e., B² – AC < 0, are called elliptic partial differential equations. The most common example of an elliptic PDE is the Laplace equation.

Applications of Partial Differential Equations

PDEs are applied in a lot of fields like mathematics, engineering, physics, finance, etc. Some of their applications are as follows:

• The concept of heat waves and their propagation can be conveniently expressed by way of a partial differential equation, given as u_xx = u_t.
• Light and sound waves and the concept surrounding their propagation can also be explained easily by way of a partial differential equation given as u_xx – u_yy = 0.
• PDEs are also used in the areas of accounting and economics. For example, the Black-Scholes equation is used to construct financial models.

Solving Partial Differential Equations

Various methods, such as variable substitution and change of variables, can be used to identify the general, specific, or singular solution of a partial differential equation. Say we have an equation: z = yf(x) + xg(y). The partial differential equation from the equation can be made as follows:

Steps for Solving Partial Differential Equations

Step I: Differentiate both LHS and RHS w.r.t.x.

 = yf'(x) + g(y)           —(1)

 = f(x) + xg'(y)           —(2)

Step II: Differentiate eq. (1) w.r.t.y and eq. (2) w.r.t.x.

 = f'(x) + g'(y) 

Step III: Multiply the first equation by x and the second equation by y then add the resultant.

x + y = xg(y) + yf(x) + xy(f'(x) + g'(y))

= z + xy(f'(x) + g'(y))

From Step II, we have:

x + y = z + xy()

Solved Examples on Partial Differential Equations

Example 1: Given the function c = f(x² – y²), find its partial differential equation.

Solution:

Differentiate both LHS and RHS w.r.t.x.

 = 2x . f'(x² – y²)                 ..(1)

 = -2y . f'(x² – y²)               ..(2)

Dividing (1) by (2), we get

 = -x/y

The differential equation is given as: y + x = 0.

Example 2: Prove that u(x,t) = sin(at)cos(x) is a solution to , given that a is constant.

Solution:

Differentiate both LHS and RHS w.r.t.t.

 = acos(at)cos(x)

= -a²sin(at)cos(x)

Since, u_x = – sin (at) sin (x) and u_xx = – sin (at)cos(x), so 

Thus, u(x,t) = sin(at)cos(x) is a solution to

Example 3: Form the partial differential equation for all such spheres having a center in the x-y plane and fixed radii.

Solution:

General equation of such spheres = (x – a)² + (y – b)² + z² = r²

Differentiate LHS and RHS w.r.t.x and w.r.t.y

2z  = -2(x – a)

2z  = -2(y – a)

(x – a) = -z 

(y – a) = -z \frac{\partial z}{\partial y}

Substituting these values in the general form of equation, the partial differential equation is:

Example 4: Prove that if p(x, t) = sin(bt)cosx.

Solution:

Hence proved.

Example 5: Reduce u_xx + 5u_xy + 6u_yy = 0. to its canonical form and solve it.

Solution:

Since, b² − 4ac = 1 > 0 for the given equation, it is hyperbolic.

Let μ(x, y)=3x − y, η(x, y)=2x − y

μ_x = 3, η_x = 2

μ_y = −1, η_y = −1

u = u(μ(x, y), η(x, y))

u_x = u_μμ_x + u_ηη_x = 3u_μ + 2u_η

u_y = u_μμ_y + u_ηη_y = −u_μ − u_η

u_xx = (3u_μ + 2u_η)x = 3(u_μμμ_x + u_μηη_x) + 2(u_ημμ_x + u_ηηη_x)

=9u_μμ + 12u_μη + 4u_ηη                 …..(1)

u_xy = (3u_μ + 2u_η)y = 3(u_μμμ_y + u_μηη_y) + 2(u_ημμ_y + u_ηηη_y) 

= −3u_μμ − 5u_μη − 2u_{ηη            .}…(2)

u_yy = −(u_μ + u_η)y = −(u_μμμ_y + u_μηη_y + u_ημμ_y + u_ηηη_y) 

= u_μμ + 2u_μη + u_{ηη            .}…(3)

Thus, the canonical form is given as: u_μη = 0.

The general solution is: u(x, y) = F(3x − y) + G(2x − y).

FAQs on Partial Differential Equations

Question 1: What are Partial Differential Equations?

Solution:

Partial differential equations are differential equations that have an unknown function, numerous dependent and independent variables, and their partial derivatives.

Question 2: Are Partial Differential Equations Linear?

Solution:

It is not necessary for all partial differential equations to be linear. Semi and non- linear partial differential equations also exist.

Question 3: What are the Applications of Partial Differential Equations?

Solution:

In engineering and science, partial differential equations are commonly used to simulate natural processes such as heat transfer, wave propagation, diffusion, and electrostatics.

Question 4: What are Ordinary Differential Equations?

Solution:

Ordinary differential equations (ODE) are equations with only one variable’s differentials. Partial derivatives exist for numerous independent variables in partial differential equations. ODEs are a type of PDE.

Related Resources

• Solving Differential Equation
• Exact Differential Equation