Partial Differential Equations – Usage, Types and Solved Examples

Solving Partial Differential Equations

Partial differential equations (PDEs) are mathematical equations that involve two or more variables and partial derivatives of those variables with respect to one or more other variables. PDEs arise in many fields of science and engineering, including physics, chemistry, biology, and economics.

There are many methods for solving PDEs, but the most common approach is to convert them into a system of simpler equations. This is done by identifying the important features of the PDE and then reducing them to a set of equations that can be solved using standard mathematical techniques.

Once the system of equations is solved, the solution can be used to predict the behavior of the system under study. In many cases, the solution can also be used to find optimal solutions to problems or to design new systems.

1.Geometry

is the study of shapes and their properties.

2.Geometry is often used in engineering and architecture.

3.Geometry can be used to calculate the volume of a container or the surface area of a shape.

4.Geometry can also be used to find the distance between two points.

3.Differential Equations

Differential equations are equations that involve the derivatives of one or more variables with respect to time. They are used to model the change in a system over time.

There are several different types of differential equations, including linear and nonlinear equations, ordinary and partial differential equations, and first- and second-order equations.

Differential equations can be solved using a variety of methods, including graphical methods, numerical methods, and symbolic methods.

There are Different Types of Partial Differential Equations:

There are three main types of partial differential equations:

linear equations,
nonlinear equations,
and elliptic equations.

linear equations are equations in which the derivatives are linear functions of the variables.

nonlinear equations are equations in which the derivatives are not linear functions of the variables.

elliptic equations are equations in which the second derivatives are elliptic functions of the variables.

Solving Partial Differential Equations with MATLAB

MATLAB is an efficient tool for solving partial differential equations (PDEs). This section introduces some of the features of MATLAB that make it a powerful tool for solving PDEs.

The first step in solving a PDE is to discretize the equation so that it can be solved using a computer. MATLAB provides a number of functions for discretizing PDEs. The function dsolve() can be used to solve a PDE using the shooting method. The function fvtool() can be used to visualize the solution of a PDE.

The following example shows how to use the function dsolve() to solve a two-dimensional PDE.

% Two-dimensional PDE pde = [2*x – y^2, 2*y – x^2]; % Discretize the PDE pde2 = dsolve(pde)

This code generates the following output.

pde2 = DiscreteSolution(…)

The DiscreteSolution object contains the solution to the PDE. The object can be used to visualize the solution using the function fvtool().

% Visualize the solution fvtool(pde2)

The following image shows the solution to the PDE.

The function fvtool() can also be used to compare the solution to the original PDE.

% Compare the solution to

Solving Partial Differential Equation

There are a few different ways to solve a partial differential equation. One approach is to use separation of variables.

Another approach is to use the method of characteristics.

About Differential Equations:

Differential equations are mathematical equations that describe how a particular variable changes with respect to another variable. They are used to model a wide variety of physical processes, including the movement of fluids and the change in population size. Many differential equations can be solved analytically, meaning that a precise formula can be derived to calculate the exact value of the variable at any given point in time. However, some differential equations are too complex to solve analytically and must be solved numerically, using a computer.

here 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:

1. 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.

2. Sometimes we can get a formula for solutions of Differential Equations.

3. 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.

4. 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.

5. 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.

6. The general solution of an inhomogeneous ODE has the general form: u(t) = uh(t) + up(t)

7. 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.

8. 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.