[ad_1]

## ABSTRACT

A naive finite difference approximations for singularly

perturbed parabolic reaction-diffusion problems

In this thesis, we treated a Standard Finite Difference Scheme for a singularly

perturbed parabolic reaction-diffusion equation. We proved that the Standard

Finite Difference Scheme is not a robust technique for solving such problems

with singularities. First we discretized the continuous problem in time using the

forward Euler method. We proved that the discrete problem satisfied a stability

property in the l1 ? norm and l2 ? norm which is not uniform with respect to

the perturbation parameter, as the solution is unbounded when the perturbation

parameter goes to zero. Error analysis also showed that the solution of the

SFDS is not uniformly convergent in the discrete l1 ? norm with respect to

the perturbation parameter, (i.e., the convergence is very poor as the parameter

becomes very small). Finally we presented numerical results that confirmed our

theoretical findings.

** **

## TABLE OF CONTENTS

1 Introduction 1

1.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 1

2 Numerical Schemes 3

2.1 Finite difference approximations of (1.1) . . . . . . . . . . . . . . 3

2.2 Some preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Existence and Uniqueness of solution . . . . . . . . . . . . . . . . 9

3 Consistency-Stability 11

3.1 Consistency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Convergence analysis 21

4.1 Convergence of the explicit scheme . . . . . . . . . . . . . . . . . 21

4.2 Convergence of the implicit scheme . . . . . . . . . . . . . . . . . 24

5 Numerical simulations and future works 27

5.1 Numerical examples for (2.9) . . . . . . . . . . . . . . . . . . . . . 28

5.2 Numerical examples for (2.10) . . . . . . . . . . . . . . . . . . . . 33

5.3 Concluding remarks and Future works . . . . . . . . . . . . . . . 38

## CHAPTER ONE

ntroduction

This work falls within the general areas of numerical methods for partial differential

equations (PDE), an area which prominent mathematicians have explored

due to its diverse applications in numerous fields of sciences. This is evident since

most D.Es can not be solved analytically, thus the method gives us useful insights

into the solutions of the D.Es without necessarily solving them analytically.

1.1 Formulation of the problem

Standard Finite Difference Scheme is one of the most frequently used methods

for solving differential equations numerically. To this end, we study a naive finite

difference approximations for singularly perturbed parabolic reaction-diffusion

problems. The governing equation of the problem is given by:

8><

>:

ut ? “uxx + b(x; t)u = f(x; t) (x; t) 2 Q =

(0; T]

u(x; 0) = 0 x 2

= [0; 1]

u(0; t) = u(1; t) = 0 t 2 (0; T];

(1.1)

where b(x; t) > 0 for all (x; t) 2

[0; T], ” is the positive perturbation

parameter and f(x; t) is the external force. The diffusion term is uxx, while the reaction

term is b(x; t)u. The problem (1.1) is generally called singularly perturbed

partial differential equation because of the small parameter ” in front of the second

order derivative term in space uxx. Thus (1.1) is one in which a small positive per?

turbation parameter ” is multiplied to the highest derivative term in the equa?

tion of the problem: Problems of these nature are well known in the literature of

Nnakwe Monday Ogudu 1

partial differential equations as they constitute an element of interest in the area

of population dynamics and chemical reactors, and their numerical analysis is

hard because of the presence of singularity when ” goes to 0: The existence and

uniqueness result of (1.1) is well developed (see [4]). The objective of this thesis

is to show that a naive numerical methods for (1.1) fails when ” goes to 0: To

have an insight into the study, if one takes the stationary problem (as in [5])

8><

>:

” 00 + 0 = 1

2 0 < x < 1 0 < << 1;

(0) = 0;

(1) = 1:

(1.2)

The exact solution of (1.2) is

(x; ) =

1 ? exp?x

2(1 ? exp?1

)

+

x

2

:

Thus the solution as

lim

!0

(x; ) =

1 + x

2

= 0(x)

does not live in C2[0; 1] since 0(x) does not satisfy the boundary condition at

x = 0. So we infer that the solution is badly behaved.

In Chapter 2, we introduced the notion of the classical SFD approximation accompanied

with some basic definitions and results. Then we formulated the classical

SFD schemes for (1.1), an elegant proof of the existence and uniqueness of the

solution of the discrete problem was presented.

In Chapter 3, we investigated the consistency and stability of the schemes of the

continuous problem (1.1). It turned out that the stability was not uniform with

respect to the perturbation parameters “:

In Chapter 4, we studied the convergence of the schemes to our continuous problem

(1.1). It turned out that the convergence was very poor as ” goes to zero.

Basically this is why the classical SFDM failed to approximate (1.1), it had no

control over ” and it found itself in damaging position.

In Chapter 5, computer programs were written and simulated for the several

cases of interest and the numerical investigations corroborated with our theoretical

findings.

2

**GET THE COMPLETE PROJECT»**

Do you need help? Talk to us right now: (+234) 8111770269, 08111770269 (Call/WhatsApp). Email: [email protected]

**IF YOU CAN’T FIND YOUR TOPIC, CLICK HERE TO HIRE A WRITER»**

Disclaimer: This PDF Material Content is Developed by the copyright owner to Serve as a RESEARCH GUIDE for Students to Conduct Academic Research. You are allowed to use the original PDF Research Material Guide you will receive in the following ways: 1. As a source for additional understanding of the project topic. 2. As a source for ideas for you own academic research work (if properly referenced). 3. For PROPER paraphrasing ( see your school definition of plagiarism and acceptable paraphrase). 4. Direct citing ( if referenced properly). Thank you so much for your respect for the authors copyright. Do you need help? Talk to us right now: (+234) 8111770269, 08111770269 (Call/WhatsApp). Email: [email protected]

*Related Current Research Articles*

[ad_2]

###

Purchase Detail

Hello, we’re glad you stopped by, you can download the complete project materials to this project with Abstract, Chapters 1 – 5, References and Appendix (Questionaire, Charts, etc) for ~~N~~4000 ($15) only, To pay with **Paypal**, **Bitcoin** or **Ethereum**; please click here to chat us up via Whatsapp.

You can also call **08111770269** or **+2348059541956** to place an order or use the whatsapp button below to chat us up.

Bank details are stated below.

**Bank:** UBA

**Account No:** 1021412898

**Account Name:** Starnet Innovations Limited

### The Blazingprojects Mobile App

**Tags:**current project topics in mathematics education m.sc maths project topics in differential equations m.sc maths project topics in graph theory pdf m.sc maths project topics in linear algebra m.sc maths project topics in topology pdf m.sc project topics in maths maths project topics in algebra msc maths project topics in algebra msc maths project topics in algebra pdf msc maths project topics in complex analysis msc maths project topics in differential equations msc maths project topics in graph theory msc maths project topics in graph theory pdf msc maths project topics in linear algebra msc maths project topics in number theory msc maths project topics in numerical analysis msc maths project topics in operations research msc maths project topics in operations research pdf msc maths project topics in real analysis msc maths project topics in topology msc maths project topics in topology pdf project ideas on maths project topic in mathematics project topic in mathematics education project topics for applied mathematics project topics for bsc maths students project topics for mathematics students project topics for maths project topics for maths and statistics department project topics for maths for class 8 project topics for msc maths project topics in discrete mathematics project topics in financial mathematics project topics in fuzzy mathematics project topics in industrial mathematics project topics in mathematics and statistics project topics in mathematics department project topics in mathematics education for undergraduate project topics in mathematics education pdf project topics in mathematics for degree students project topics in mathematics for pg project topics in mathematics pdf project topics in maths education project topics in pure mathematics project topics on maths project topics on maths education project topics related mathematics undergraduate project topics in mathematics

## Recent Comments