First converting the given differential equation to a system of linear equations, then put it in matrix form, after, the diagonalisation of a matrix (Jordan form in general ) is one useful tool for solving differential equations and recurrence sequences. 

Dear Khaled,

The following link leads you to methods for the use of matrix algebra in solving differential equations:

http://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx

Hoping this will be helpful,

Rafik

The following useful link, should answer your questions.

http://www.math.psu.edu/tseng/class/Math251/Notes-LinearSystems.pdf

Also, I suggest you to take a look at the following book:

C.G. Cullen, Matrices and Linear Transformations, Dover Publications, Inc., New York, 1972,  pp.255-271.

First converting the given differential equation to a system of linear equations, then put it in matrix form, after, the diagonalisation of a matrix (Jordan form in general ) is one useful tool for solving differential equations and recurrence sequences. 

Every discretization method turns your differential equation (a problem in "analysis") into a finite set of equations (a problem in "algebra"). This is so for boundary value problems as well as initial value problems. The point of discretization is to construct an algebraic problem which can be solved in finite time. Of course, the algebraic solution will only be an approximation to the the solution of the differential equation.

If the original differential equation is linear, your discrete problem will also be linear, so you have a problem within linear algebra. If the problem is nonlinear, however, you will have to use some iterative technique (usually Newton's method) to find the approximate solution. But even Newton's method will only lead to a sequence of linear algebraic problems to be solved.

In a few cases, matrix algebra can be avoided. This happens in initial value problems u' = f(u) if you use explicit time stepping methods, e.g. Euler's explicit method,

un+1 = un + h*f(un).

You can run this recursion, without using any matrices or linear algebra work. However, as soon as you use an implicit method, e.g. Euler's implicit method,

un+1 = un + h*f(un+1),

you have to do equation solving on each time step, and this requires algebraic work.

In boundary value problems, e.g. u" = f(x), discretization immediately produces a linear system of equations, L*u = f, where the matrix L is large and sparse. So here you have to use linear algebra for the numerical solution.

So basically the only numerical solution technique that doesn't use matrix algebra is explicit time stepping methods in initial value problems. 

I totally agree with Professor Gustaf Söderlind.

A discretization method converts the differential equation into a set of algebraic equations.

Yes numerical methods use matrices to solve a differential equation. However, if your equation is linear then theory of linear algebra, matrices and eigen values come automatically in the picture. The fact that solutions of the equation are linearly independent in that case.

Converting differential quantities to finite difference form will convert the differential equations to matrix equations which can then be solved algebraically.

Dear all Researchers,

    Thank you all for your valuable answers. I'm still waiting for more additional answers and further investigations, specially in the field of nonlinear equations and PDE.

Best regards

For nonlinear equations and PDEs, the framework is the same. Let's say you have a PDE ut = Lu, where L is a linear differential operator in space. Using method of lines, you discretize space to obtain a system of ODEs. With only a slight abuse of notation, let's write it 

ut = Lu,

where the t subscript on the left refers to the time derivative, and L is now a matrix, replacing the former differential operator acting on the space variable(s).

If L is a first order operator in space, you might still get away with an explicit time stepping method (think Explicit Euler), provided that you fulfill the CFL stability condition dt/dx < C, where dt and dx are time and space mesh widths, respectively. So with an explicit scheme, you won't have to use matrix algebra. (This is the only exception.)

If L is a second order operator in space, however, the CFL condition becomes dt/dx2 < C. This is prohibitive (compare stiffness) as you are forced to keep the time-step incredibly small just in order to maintain stability. Therefore, you'll have to use an implicit time stepping method (think Implicit Euler). You then overcome the time step limitation, but at the cost of having to solve a linear system of the form

(I - dt*L) un+1 = un

on every time step. That's when you have to start using linear algebra, and matrix theory. 

The same happens if you have higher order operators in space. The higher order, the more you're in need of implicit methods.

Should L be a nonlinear operator, the same thing happens, only you have to solve your systems using Newton-type methods instead. (Very large systems will have to be addressed using some type of iterative methods, such as GMRES or similar).

And if you don't have any time dependency, it's still the same story. There's no time-stepping then, but you still have an equation system to solve, necessitating the use of linear algebra.

U can do just linear diff equation is a linear operator so by representation according to a suirable basis and jordan canonixal form if xan be represented to a matrix giving solutiobns....

I suggest to use the following link which contains:

Introduction and First Definitions

Modeling via Differential Equations

First Order Differential Equations

Linear Equations

Separable Equations

Qualitative Technique: Slope Fields

Equilibria and the Phase Line

Bifurcations

Bernoulli Equations

Riccati Equations

Homogeneous Equations

Exact and Non-Exact Equations

Integrating Factor technique

Some ApplicationsRadioactive Decay

Newton's Law of Cooling

Orthogonal Trajectories

Population Dynamics

Numerical Technique: Euler's Method

Existence and Uniqueness of Solutions

Picard Iterative Process

Second Order Differential Equations

Nonlinear Equations

Linear Equations

Homogeneous Linear Equations

Linear Independence and the Wronskian

Reduction of Order

Homogeneous Equations with Constant Coefficients

Non-Homogeneous Linear EquationsMethod of Undetermined Coefficients

Method of Variation of Parameters

Euler-Cauchy Equations

Series SolutionsIntroduction

Derivatives and Index Shifting

First Examples

Airy's Equation

The Radius of Convergence of Series Solutions

Hermite's Equation

Higher Order Linear Equations

Introduction and Basic Results

Homogeneous Linear Equations with Constant Coefficients

Non-Homogeneous Linear Equations

Method of Undetermined Coefficients

Method of Variation of Parameters

Laplace Transform

Basic Definitions and Results

Application to Differential Equations

Impulse Functions: Dirac Function

Convolution Product

Table of Laplace Transforms

Systems of Differential Equations

Introduction and Motivation

Second Order Equations and Systems

Euler's Method for Systems

Qualitative Analysis

Linear SystemsIntroduction and First Definitions

Vector Representations of Solutions of Linear Systems

Eigenvalues and Eigenvectors TechniqueReal Eigenvalues

Repeated Eigenvalues

Complex Eigenvalues

Zero Eigenvalues

Qualitative Analysis of Linear Systems

Nonlinear SystemsEquilibrium Point Analysis: Linearization Technique

Fourier Series

Fourier Series: Basic Results

Fourier Sine and Cosine Series

Convergence of Fourier Series

More on Convergence of Fourier Series

Gibbs Phenomenon

Bessel's Inequality and Parseval Formula: The Energy Theorem

Operations on Fourier Series

Application of Fourier Series to Differential Equations

http://www.sosmath.com/diffeq/diffeq.html

Rafik

Finite Element methods.

Here  is  an example  of  the method   that uses  matrix algebra in solving  of  

A LINEAR PARTIAL DIFFERENTIAL EQUATION OF THE HIGHER ORDER IN TWO VARIABLES WITH  INITIAL CONDITION WHOSE COEFFICIENTS ARE  REAL-VALUED SIMPLE STEP FUNCTIONS

   http://arxiv.org/pdf/1505.00896v3.pdf

Perhaps the simplest application of the matrix algebra is this: if A is an

N x N constant matrix and y is an N-component vector, then the (unique) solution of the Cauchy problem dy(t)/dt=A y(t), y(0)=z is given by y(t)=exp(At)z.