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Partial Differential Equations

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Partial Differential Equation (PDE) is an equation 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.

In this article, we will learn the definition of Partial Differential Equations, the representation of PDE, the order of PDE, the Types of Partial Differential Equations, how to solve PDE, and many more details.

Partial Differential Equations Definition

Partial Differential Equation also called PDE is a differential equation containing partial derivatives of the dependent variable with one independent variable. For any function f(x1, x2,…,xn) its partial differential equation(PDE) is,

U(x1, x2,…,xn, ∂f/∂x1, ∂f/∂x2,…,∂f/∂xn) = 0

Suppose we have a linear function of u then its PDE is,

∂u/∂x (x,y) = 0

Order of Partial Differential Equations

Order of the highest derivative term which occurs in a given partial differential equation is called the order of the said equation.

Let’s say ∂z/∂x + ∂y/∂x = x + zy is a partial differential equation. As the order of the highest derivative is 1, hence, this is a first-order partial differential equation.

Examples of Partial Differential Equations

Various examples of partial differential equation are,

  • 3ux + 5uy – uxy + 7 = 0
  • 2uxy + 3uy – 8ux + 11 = 0

Degree of Partial Differential Equations

Degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation ∂z/∂x + ∂y/∂x = x + zy has 1 as the highest derivative of the first degree.

General Form of Partial Differential Equations

The general form of Partial Differential Equation is,

f(x_1,....x_n;u,\frac{\partial u}{\partial x},...,x_1,....x_n;u,\frac{\partial u}{\partial x_n};x_1,....x_n;u,\frac{\partial^2 u}{\partial x_1\partial x_1},.....,\frac{\partial^2 u}{\partial x_1\partial x_n})=0

Representing Partial Differential Equation

Partial Differntial Equations are represented using subcript and ∂ or ∇ symbol. suppose we have a function f then Partial Differential Equations are given as:

  • fx = ∂f/∂x
  • fxx = ∂2f/∂x2
  • fxy = ∂2f/∂x∂y = ∂/∂y(∂f/∂x)

We use, ∂ and ∇ symbols to represent the Partial Differential Eqautions.

Types of Partial Differential Equations

Various types of Differential Equations are,

  • First-Order Partial Differential Equations
  • Second-Order Partial Differential Equations
  • Quasi-Linear Partial Differential Equations
  • Homogeneous Partial Differential Equations

Now let’s learn about the same in detail.

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 Auxx + 2Buxy + Cuyy + 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 PDEs
  • Hyperbolic PDEs
  • Elliptic PDEs

Now lets learn about the same,

Parabolic PDE

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 economic, and financial topics such as derivative investments, particle diffusion, heat induction, etc.

Hyperbolic PDE

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 PDE

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,

uxx = ut

  • Light and sound waves and the concept surrounding their propagation can also be explained easily by way of a partial differential equation given as,

uxx – uyy = 0

  • PDEs are also used in the areas of accounting and economics. For example, the Black-Scholes equation is used to construct financial models.

Solve 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.

∂z/∂x = yf'(x) + g(y)           —(1)

∂z/∂y = f(x) + xg'(y)           —(2)

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

2z/∂x∂y = f'(x) + g'(y) 

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

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

From Step II, we have,

x∂z/∂x + y∂z/∂y = z + xy(∂2z/∂x∂y)

Thus, partial differential equations are solved using the steps added above.

Read More,

Partial Differential Equations Solved Examples

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

Solution:

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

∂u/∂x = 2x.f'(x2 – y2)…(1)

 ∂u/∂y = -2y.f'(x2 – y2)…(2)

Dividing (1) by (2), we get

 (∂u/∂x)/(∂u/∂y)= -x/y

Thus, differential equation is given as: y.∂u/∂x+ x.∂u/∂y = 0

Example 2: Prove that u(x,t) = sin(at)cos(x) is a solution to ????2u/????t2 = a2(????2u/????x2, given that a is constant.

Solution:

Differentiate both LHS and RHS

∂u/∂t = acos(at)cos(x)

2u/∂t2 = -a2sin(at)cos(x)

Since,

  • ux = – sin (at) sin (x)
  • uxx = – sin (at)cos(x)

????2u/????t2 = a2(????2u/????x2)

Thus, u(x,t) = sin(at) cos(x) is a solution of ????2u/????t2 = a2(????2u/????x2)

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 is, (x – a)2 + (y – b)2 + z2 = r2

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

2z{∂z/∂x} = -2(x – a)

2z{∂z/∂y} = -2(y – a)

(x – a) = -z{∂z/∂x}

(y – a) = -z{∂z/∂y}

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

z^2 = \frac{r^{2}}{(\frac{\partial z}{\partial x})^{2} + (\frac{\partial z}{\partial y})^{2} + 1}

Example 4: Prove that ????2p/????t2 = b2????2p/????x2 if p(x, t) = sin(bt)cosx.

Solution:

????p/????t = b cos(bt) cos(x)

⇒ ????2p/????t2 = -b2 sin(bt) cos(x)

Now,

????p/????x = -sin(bt) sin(x)

⇒ ????2p/????x2 = -sin(bt) cos(x)

b2????2p/????x2 = -b2sin(bt) cos(x)

Hence proved.

Example 5: Reduce uxx + 5uxy + 6uyy = 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

Then,

  • μx = 3
  • ηx = 2
  • μy = −1
  • ηy = −1

u = u(μ(x, y), η(x, y))

ux = uμμx + uηηx = 3uμ + 2uη

uy = uμμy + uηηy = −uμ − uη

uxx = (3uμ + 2uη)x = 3(uμμμx + uμηηx) + 2(uημμx + uηηηx)

uxx = 9uμμ + 12uμη + 4uηη…(1)

uxy = (3uμ + 2uη)y = 3(uμμμy + uμηηy) + 2(uημμy + uηηηy

uxy = −3uμμ − 5uμη − 2uηη…(2)

uyy = −(uμ + uη)y = −(uμμμy + uμηηy + uημμy + uηηηy

uyy = uμμ + 2uμη + uηη…(3)

Thus, canonical form is given as, uμη = 0

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

Practice Questions on Partial Differential Equations

Q1. Solve PDE ux + 2uy – 4 = 0

Q2. Solve PDE uxy + 3uy – 6ux = 0

FAQs on Partial Differential Equations

1. What are Partial Differential Equations?

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

2. Are Partial Differential Equations Linear?

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

3. What are Applications of Partial Differential Equations?

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

4. What are Ordinary Differential Equations?

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.

5. How to Solve Partial Differential Equation?

We use various methods, such as variable substitution and change of variables, to find the general, specific, or singular solution of a partial differential equation.



Last Updated : 28 Dec, 2023
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