UMERICAL METHODS FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS

UMERICAL METHODS FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS

DECLARATION

I hereby declare that the work described in this thesis, entitled “NUMERICAL METHODS FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS” which is being submitted by me in partial fulfillment for the award of the degree of Doctor of Philosophy (Ph.D.) in the Department of Mathematics to the Jawaharlal Nehru Technological University Hyderabad, Kukatpally, Hyderabad – 500 085, Telangana, is the result of investigations carried out by me under the supervision of Dr. Madhulatha K, Associate Professor of Mathematics, Kamala Institute of Technology & Science, Huzurabad, Karimnagar and co-supervision of Dr. B. Ravindra Reddy, Associate Professor of Mathematics & Additional Controller of Examinations, JNTUH, Kukatpally, Hyderabad.

The work is original and has not submitted for any Degree/Diploma of this or any other University.

Place: Warangal                      Signature of the Candidate

Date:                               (V VIDYASAGAR)

CERTIFICATE

This is to certify that the thesis entitled “NUMERICAL METHODS FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS” which is being submitted by Mr. V VIDYASAGAR in partial fulfillment for the award of the degree of Doctor of Philosophy (Ph.D.) in Mathematics to the Jawaharlal Nehru Technological University Hyderabad, Kukatpally, Hyderabad – 500 085, Telangana, is a record of bonafide work carried out by him under our guidance and supervision.

The results embodied in this thesis have not been submitted to any other University or Institute for the award of any degree or diploma.

Signature of the Co-Supervisor                Signature of the Supervisor     

(Dr. B Ravindra Reddy)                     (Dr. Madhulatha K)

KAMALA INSTITUTE OF TECHNOLOGY & SCIENCE

(Sponsored by Vodithala Education Society, Approved by AICTE, Affiliated to JNTU, Hyderabad and NAAC B++)

Singapuram, Huzurabad, Karimnagar-505468, Telangana, India

CERTIFICATE

This is to certify that the thesis entitled “NUMERICAL METHODS FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS” which is being submitted by Mr. V VIDYASAGAR in partial fulfillment for the award of the degree of Doctor of Philosophy (Ph.D.) in Mathematics to the Jawaharlal Nehru Technological University Hyderabad, Kukatpally, Hyderabad – 500 085, Telangana, is a record of bonafide work carried out by him at our institution.

Signature of Head/Director of

Organization/Institution

                        ACKNOWLEDGEMENTS

The successful completion of this thesis would not have been possible without the support of all my well-wishers.

First and foremost, I would like to express my sincere gratitude to my Research Supervisor Dr. Madhulatha K, Associate Professor of Mathematics, Kamala Institute of Technology &  Science, Huzurabad, Karimnagar, my co-supervisor Dr. B. Ravindra Reddy, Associate Professor of Mathematics & Additional Controller of Examinations, JNTUH, Kukatpally, Hyderabad, for their continuous guidance from the very early stage of this research as well as giving me extraordinary experiences throughout the work. Above all and the most needed, they provided me with constant encouragement and support in various ways. Without them, this thesis would not have been completed.

I express my heartfelt gratitude to Prof. M. A. Srinivas, Head, Department of Mathematics, JNTUHCE, Hyderabad, Prof. M. N. Raja Shekar, Chairman, BOS, Department of Mathematics, JNTUH, Kukatpally, Hyderabad, for their academic support.

I am profoundly grateful to the authorities of my institute, in particular to Sri Capt. V. Lakshmikantha Rao, Chairman,  Sri V. Satish Kumar, Secretary, Dr. K. Shanker, Principal, and                  Prof. V. Rajeshwar Rao, Head, Department of Humanities & Sciences, for permitting to complete my research.

I express my gratitude to Dr. K. Phaneendra, Assistant Professor, Department of Mathematics, Osmania University, for his support in my research work.

I express my thanks to my friends Dr. K. Yugendhar, Mr. N. Raghu, Mr. D. Prashanth Kumar, Mr. B. Yakhoob, Mr. N. Goutham Kumar, Mr. K. Praveen Kumar Rao, Mrs. N. Sasikala, and Mrs. M.Sridevi for providing me a good atmosphere to carry out research work.

I am also thankful to all my colleagues at Kamala Institute of Technology & Science, Huzurabad, Karimnagar.

I express my gratitude to my beloved parents, Shri. V.Nagesh and Smt. V.Saraswathi, for their love and blessings.

I express my thanks to my younger brothers, Mr. V.Sridhar and Dr.V.Naresh, my sister-in-laws Mrs.V.Sreeja and Mrs.V. Navya Sri for their love, affection, moral support, and encouragement.

My Special thanks to my dear wife, Mrs. V.Vishnu Priya, for her love, patience. Finally, my beloved children V.Sai Sankeerth and V.Sanjay Ram, for their love and affection.

I wish to acknowledge everyone who had a direct or indirect contribution to do my thesis.

V VIDYASAGAR

ABSTRACT

In general, any differential equation in which the highest order derivative is multiplied by a small positive parameter is called the Singular Perturbation Problem. A differential equation in which the highest order derivative is multiplied by a small positive parameter and has at least one shift term (delay or advance) is called singularly perturbed differential-difference equation (SPDDE).  Here the negative shift is used for the delay, and a positive shift is used for advance. When we apply the existing standard numerical methods to this SPDDE, we get oscillatory/unsatisfactorily results when step size h is greater than the value of the perturbation parameter . As a result of this, finding solutions for SPDDE has become the most exciting and challenging task.  Thus it is of considerable scientific interest to the researchers to develop simple and efficient computational methods for singularly perturbed differential-difference equations.

We have proposed and illustrated some simple and efficient numerical techniques for finding the solution of singularly perturbed differential-difference equations. It consists of six chapters. Chapter-I includes definition and survey of singularly perturbed differential-difference equations. Chapters-II deals with a non-symmetric special second-order fitted method to solve singularly perturbed differential-difference equations with layer behaviour at one end. Chapter-III deals with a special finite difference method to solve singularly perturbed differential-difference equations with layer behaviour at one end. Chapter-IV consists of the numerical integration method to solve singularly perturbed differential-difference equations with layer behaviour at one end. Chapter-V deals with an exponentially fitted spline method to solve singularly perturbed differential-difference equations with layer behaviour at one end and both ends. Chapter-VI consists of the fourth-order numerical method to solve singularly perturbed differential-difference equations with dual-layer behaviour.

In a nutshell, the methods presented in this for solving singularly perturbed differential-difference equations are observed to be simple, accurate, and efficient than the conventional methods. We have implemented all these proposed methods on several linear models with left layer, right layer, and dual layers for different values of the step size h, perturbation parameter, delay parameter, and the advanced parameter .  Computational results are presented and compared with exact/approximate solutions. It is observed that these methods approximate the exact solution very well.  Further, it is found that the accuracy predicted can always be achieved with a little computational effort. Above all, these methods are conceptually simple, easy to use, and are readily adaptable for computer implementation with a modest amount of problem preparation.

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