**APPLICATION OF PARTIAL DIFFERENTIAL EQUATION IN ENGINEERING**

**ABSTRACT**

*The focus is on the wave equation since it has well known properties and it is representative of many types of PDE system. This distinction usually makes PDEs much harder to solve than ODEs but here again there will be simple solution for linear problems. It is well known that PDEs are applicable in areas such as Wave equation, Heat conduction, Laplace equation, Electrostatics, Electrodynamics, Fluid flow, Machines and in various areas of science and engineering.*

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**CHAPTER ONE**

**GENERAL INTRODUCTION**

**Background of Study**

In mathematics a Partial Differential Equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives (A special Case are ordinary differ entail equations. PDEs are used to formulate problem involving function of several variable and are either solved by hand or used to create a relevant computer model.(Evans L.C) [8]

PDEs are equation that involves rate of change with respect to continues variable. The position of a rigid body is specified by six number, but the configuration of a fluid is given by the continuous distribution of several parameter, such as the temperature, pressure and so forth.(Jost.J.)[18]

Lewy, Hans [16] also suggested that the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than Ordinary Differential Equation (ODEs) but here again there will be simple solution for linear problems.

Real –time distributed simulation has been used extensively in many application areas such as the defense industry automotive and aerospace applications. However, there are currently few fundamental approaches to guide the development of distributed real-time simulation.

A mechanical system consists of complex interconnection of heterogeneous mechanical model that involve different types of equations with different method of solution. Simulation of these problems generally requires the solution of large number of Partial Differential Equation (PDEs) and Ordinary Differential Equation (ODEs) with algebraic constraints.

This thesis investigates innovative methods for real- time distributed simulation of PDEs including realistic visualization of distributed simulation results. The Performance & Scalability of the method is also studied.

**Objective of the Study**

The objective of study of application of PDEs in Engineering is as follow;

- To show areas where PDEs is applicable in science and engineering areas like Sound heat, Wave Equation, Heat Conduction Equation, Laplace’s Equation.
- To describe a wide variety of phenomena such as electrostatics, electrodynamics, fluid flow, elasticity or quantum, mechanics.

**Definition of Terms**

**Differential Equation**

Any equation involving differentials or derivatives is called a differential equation. In order words, a differential equation is a relationship between an independent variable x, a dependent variable y, and one or more derivatives of y with respect to x.

e.g x^{2} = y sin x =

**Order of Differential Equation**

The order of a differential equation is defined as the largest positive integer n for which an nth derivative occurs in the equation.

x – y^{2} = 0 is an equation of the 1^{st} order

xy – y^{2} sin = 0 is an equation of the 2^{nd} order

– y + e^{4x} = 0 is an equation of the 3^{rd} order

The order of a differential equation is divided into two, namely First order and second order differential equation.

**Degree of Differential Equation**

Is the degree of the highest derivative that appears. E.g.

y – 2y^{2} = Ax^{3} is of degree 1

(y^{1})^{3} + 2y^{4} = 3x^{5} is of degree 3

**First Order Differential Equation**

These are equations that contain only the First derivatives y^{1 }and may contain y and any given functions of x. For this reason, we can write them as:

F(x,y,y^{1}) = 0

Or often in the form

Y^{1}** = **F(x,y)

**Second Order Differential Equation**

Many practical problems in engineering give rise to second order differential equations of the form.

a + b + cy = F(x)

Where a, b and c are constant coefficients and F(x) is a given function of x.

Considering a case where f(x) = 0, so that the equation becomes:

a+ b__dy__ + cy = 0………………………………………… (*)

Let y = u and also let y = v (u & v are functions of x) be two solutions of the equation.

a + b + cu = 0…………………………………………. (1)

a + b + cv = 0…………………………………………. (2)

Adding equation (1) & (2) we get

a[] + b[+ ] + c (u+v) = 0…………………… (3)

now, = + ………………………………………(4)

= + ………………………………………….(5)

Therefore the equation (4) & (5) can be written as

a + b + c (u + v) = 0…………………………(6)

Which is our original equation with y replaced by (u + v).

If a = 0 in our original equation (*), we get the first order equation of the same family.

b + cy = 0

i.e + ky = 0

where k =

Solving this by the same method of separating variables, we have:

= -ky :.= y

= ln y = -kx + c

:- y = e^{-kx+c} = e^{-kx} e^{c} = Ae^{-kx }(where e^{c} is a constant)

i.e y = Ae^{-kx}

If we write the symbol m for –k, the solution is y=Ae^{mx}. In the same way, y=Ae^{mx} will be a solution of the second-order equation.

a + b + cy = 0, if it satisfies this equation.

Now, if y = Ae^{mx}

= Ame^{mx}

= Am^{2}e^{mx}

And substituting these expressions for the differential coefficients in the left-hand side of the equation, we obtain:

aAm^{2}e^{mx} +bAme^{mx }+ cAe^{mx} = 0

Dividing both side by Ae^{mx} we obtain

am^{2} + bm + c = 0.

**Degree of Differential Equation**

The degree of a differential equation is the degree of the highest derivative that appears. E.g.

y – 2y^{2} = Ax^{3} is of degree 1

(y^{1})^{3} + 2y^{4} = 3x^{5} is of degree 3

**Types of Differential Equation**

Differential equations are of two types for the purpose of this work, namely: Ordinary Differential Equations and Partial Differential Equations.

**Ordinary Differential Equations (ODEs)**

An ordinary differential equation is an equation that contains one or several derivatives of an unknown function, which we usually call y(x) (or sometimes y(t) if the independent variable is time t). The equation may also contain y itself, known functions of x (or t), and constants. For example:

(i) yˡ = cos x

(ii) yˡˡ + ay = e^{-2x}

(iii) yˡyˡˡˡ – __3__ yˡ^{2} =0

They are also referred to as equations whose unknowns are functions of a single variable and are usually classified according to their **order.**

**Partial Differential Equations (PDEs)**

These are differential equations in which the unknown function depends on more than one variable. Which can also be describe as an equation relating an unknown function (the dependent variable) of two or more variables with one or more of its partial derivatives with respect to these variables.

A(x,y) U_{xx} + B(x,y) U_{xy} + C(x,y) U_{yy} + D(x,y) U_{x} + E(x,y) U_{y} + F(x,y) U = G(x,y).

For the purpose of this work, a detailed explanation will be thrown on Partial Differential Equation in chapter three.

**APPLICATION OF PARTIAL DIFFERENTIAL EQUATION IN ENGINEERING**

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