NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS .
NUMERICAL SOLUTIONS OF PARTIALDIFFERENTIAL EQUATIONS USING B-SPLINEBySaima ArshedA THESISSUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYINMATHEMATICSSupervised ByProf. Dr. Shahid S. SiddiqiUNIVERSITY OF THE PUNJABQUAID-E-AZAM CAMPUS, LAHORESEPTEMBER, 2014
CERTIFICATEI certify that the research work presented in this thesis isthe original work of Miss. Saima Arshed D/O MuhammadArshed Karim and is carried out under my supervision. Iendorse its evaluation for the award of Ph.D. degree throughthe official procedure of University of the Punjab.Prof. Dr. Shahid S. Siddiqi(Supervisor)ii
DECLARATIONI, Miss. Saima Arshed D/O Muhammad ArshedKarim, hereby declare that the matter printed in this thesisis my original work. This thesis does not contain any materialthat has been submitted for the award of any other degree inany university and to the best of my knowledge, neither doesthis thesis contain any material published or written previouslyby any other person, except due reference is made in the text ofthis thesis.Saima Arshediii
DedicationsTo my loving Parents, Husbandand Daughteriv
Table of ContentsTable of ContentsvAbstractxiiiAcknowledgementsxiv1 Introduction1.1 Differential Equations . . . . . . . . . . . . . . . . . . . . .1.1.1 Types of Differential Equations . . . . . . . . . . . .1.2 Introduction to Partial Differential Equation . . . . . . . . .1.2.1 Partial Integro-Differential Equation (PIDE) . . . . .1.2.2 Fractional Partial Differential Equation (FPDE) . . .1.3 Applications of PDEs . . . . . . . . . . . . . . . . . . . . . .1.3.1 Partial Differential Equation in Shock Wave . . . . .1.3.2 Partial Differential Equations in Fluid Mechanics . .1.3.3 Partial Differential Equations in Solar Dynamics . . .1.3.4 Partial Differential Equations in Quantum mechanics1.4 Classification of Linear Second-Order PDEs . . . . . . . . .1.4.1 Parabolic PDEs . . . . . . . . . . . . . . . . . . . . .1.4.2 Hyperbolic PDEs . . . . . . . . . . . . . . . . . . . .1.4.3 Elliptic PDEs . . . . . . . . . . . . . . . . . . . . . .1.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . .1.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . .1.7 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . .1.7.1 Numerical Techniques for Solving PDEs . . . . . . .1.7.2 Discretization . . . . . . . . . . . . . . . . . . . . . .1.7.3 Consistency, Convergence and Stability . . . . . . . .1.7.4 Lax’s Equivalence Theorem . . . . . . . . . . . . . .1.7.5 Fractional Calculus . . . . . . . . . . . . . . . . . . .1.7.6 B-spline . . . . . . . . . . . . . . . . . . . . . . . . .1.7.7 B-spline Basis Functions . . . . . . . . . . . . . . . .v1123456677889101111121313151617171819
1.81.7.8 Collocation Method . . . . . . . . . . . . . . . . .1.7.9 Collocation Method with B-spline Basis Functions1.7.10 Cubic B-spline Collocation Method . . . . . . . .1.7.11 Quintic B-spline Collocation Method . . . . . . .Literature Survey . . . . . . . . . . . . . . . . . . . . . .2 Quintic B-Spline for the NumericalBoussinesq Equation2.1 Introduction . . . . . . . . . . . . .2.2 Temporal discretization . . . . . . .2.3 Quintic B-spline collocation method2.4 The Initial Vector . . . . . . . . . .2.5 Stability Analysis . . . . . . . . . .2.6 Numerical Results . . . . . . . . . .2.7 Conclusion . . . . . . . . . . . . . .2122222426Solution of the Good.3 Solution of Fourth-Order Partial Differential Equation3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .3.2 Discretization in time . . . . . . . . . . . . . . . . . . . .3.3 Quintic B-spline Collocation Method . . . . . . . . . . .3.4 The Initial Vector . . . . . . . . . . . . . . . . . . . . . .3.5 Stability Analysis . . . . . . . . . . . . . . . . . . . . . .3.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . .3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .3131333537394243.44444547555758614 Numerical Solution of Partial Integro-Differential Equation4.1 Parabolic Integro-Differential Equations . . . . . . . . . . .4.1.1 Temporal Discretization . . . . . . . . . . . . . . . .4.1.2 Cubic B-spline Collocation Method . . . . . . . . . .4.1.3 Numerical Results . . . . . . . . . . . . . . . . . . .4.2 Convection-Diffusion Integro-Differential Equation . . . . . .4.2.1 Temporal Discretization . . . . . . . . . . . . . . . .4.2.2 Discretization in Space . . . . . . . . . . . . . . . . .4.2.3 Numerical Results . . . . . . . . . . . . . . . . . . .4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .636465687380828487965 Numerical Solution of Time-Fractional Fourth-order PartialDifferential Equations995.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . 1025.3 Discretization in space . . . . . . . . . . . . . . . . . . . . . 112vi
5.45.55.6Time-fractional semilinear fourth-order partial differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 116Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296 Numerical Solution of Time-Fractional Convection-DiffusionEquation1316.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.2 The cubic B-spline . . . . . . . . . . . . . . . . . . . . . . . 1346.3 Discretization in time . . . . . . . . . . . . . . . . . . . . . . 1356.4 Stability and convergence analysis . . . . . . . . . . . . . . . 1396.5 Discretization in space . . . . . . . . . . . . . . . . . . . . . 1446.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 1476.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 B-Spline Solution of Time-Fractional Integro Partialential Equation With a Weakly Singular Kernel7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .7.2 Discretization in time . . . . . . . . . . . . . . . . . .7.3 Discretization in space . . . . . . . . . . . . . . . . .7.4 Numerical Results . . . . . . . . . . . . . . . . . . . .7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . .Differ152. . . . 153. . . . 154. . . . 164. . . . 167. . . . 172References174Appendix182vii
List of Figures4.1The results at M 500 for Example 4.1. . . . . . . . . . . . .754.2The exact and numerical solutions at M 10 . . . . . . . . .754.3The results at M 500 for Example 4.2. . . . . . . . . . . . .784.4The exact and numerical solutions at M 10 . . . . . . . . .784.5The results at M 500 for Example 4.3. . . . . . . . . . . . .804.6The exact and numerical solutions at M 10. . . . . . . . . .814.7The results at K 500 for Example 4.4. . . . . . . . . . . . .904.8The exact and numerical solutions at K 10 . . . . . . . . .904.9The results at K 500 for Example 4.5. . . . . . . . . . . . .924.10 The exact and numerical solutions at K 10 . . . . . . . . .934.11 The results at K 500 for Example 4.6. . . . . . . . . . . . .954.12 The exact and numerical solutions at K 10 . . . . . . . . .954.13 The results at K 500 for Example 4.7. . . . . . . . . . . . .974.14 The exact and numerical solutions at K 10 . . . . . . . . .975.1The results at M 40, K 1000 and t 0.00001 for Example5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.2The exact and numerical solution at K 1000. Dotted line:numerical solution, Solid line: exact solution . . . . . . . . . 1185.3Errors as a function of the time t for α 0.75 . . . . . . . 1195.4The results at M 40, K 500 and t 0.00001 for Example5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.5The exact and numerical solution at K 1000. Dotted line:numerical solution, Solid line: exact solution . . . . . . . . . 121viii
ix5.6Errors as a function of the time t for α 0.50 . . . . . . . 1225.7The results at M 80, K 1000 and t 0.00001 for Example5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.8The exact and numerical solution at K 1000. Dotted line:numerical solution, Solid line: exact solution . . . . . . . . . 1255.9Errors as a function of the time t for α 0.90 . . . . . . . 125
List of Tables1.1Coefficients of cubic B-spline and its derivatives at knots xi .1.2Coefficients of quintic B-spline and its derivatives at knots xi . 242.1Numerical results for Example 2.1 . . . . . . . . . . . . . . .3.1Coefficients of quintic B-spline and its derivatives at knots xj . 483.2Absolute errors for h 0.05 at points x 0.1, x 0.2, x 0.3, x 0.4, x 0.5 for example 3.1. . . . . . . . . . . . . . .3.36060Absolute errors at midpoints, x 0.5, for h 0.05 for example 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.659Absolute errors at midpoints, x 0.5, for h 0.05 andr 2.0 for example 3.1 . . . . . . . . . . . . . . . . . . . . .3.542Absolute errors at midpoints, x 0.5, for h 0.05 andr 0.5 for example 3.1 . . . . . . . . . . . . . . . . . . . . .3.42361Comparison of proposed method with H. Caglar and N. Caglar[7] in maximum absolute errors for example 3.1 . . . . . . .613.7Maximum absolute errors with h 0.05 for example 3.2 . . .624.1The errors keK k and keK k2 when M 60 and t 0.0001for example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . .4.2The errors keK k and keK k2 when M 60 and t 0.001for example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . .4.37474The errors keK k and keK k2 when K 10 and t 0.0001for example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . .x75
xi4.4The errors keK k and keK k2 when M 60 and t 0.0001for example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . .4.5The errors keK k and keK k2 when M 60 and t 0.001for example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . .4.67777The errors keK k and keK k2 when K 10 and t 0.0001for example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . .774.7Maximum norm errors keK k for M 40 for example 4.3 .794.8L2 norm errors keK k2 for M 40 for example 4.3 . . . . . .804.9keK k and keK k2 for t 0.0001 for example 4.4 . . . . . .894.10 keK k and keK k2 for t 0.00001 for example 4.4 . . . . .894.11 keK k and keK k2 for t 0.0001 for example 4.5 . . . . . .924.12 keK k and keK k2 for t 0.00001 for example 4.5 . . . . .924.13 keK k and keK k2 for t 0.0001 for example 4.6 . . . . . .944.14 keK k and keK k2 for t 0.00001 for example 4.6 . . . . .944.15 L2 norm errors keK k2 for M 100 for example 4.7 . . . . .984.16 Maximum norm errors keK k for M 100 for example 4.7 .985.1The errors keK k and keK k2 for different K taken t 0.00001.1175.2The errors keK k and keK k2 of different time steps withM 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.3The errors keK k and keK k2 for different M taken t 0.00001.1215.4The errors keK k and keK k2 of different time steps withM 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.5The errors keK k and keK k2 for different M taken t 0.00001.1245.6The errors keK k and keK k2 of different time steps withM 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.7The errors keK k and keK k2 for different M taken t 0.00001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.8The errors keK k and keK k2 of different time steps withM 80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.9The errors keK k and keK k2 for different M taken t 0.00001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
xii5.10 The errors keK k and keK k2 of different time steps withM 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.1Coefficients of cubic B-spline and its derivatives at knots xi . 1356.2The errors keK k and keK k2 for different M taken t 0.00001.1496.3The errors keK k and keK k2 of different time steps withM 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.4The errors keK k and keK k2 for different M taken t 0.00001.1506.5The errors keK k and keK k2 of different time steps withM 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.1The errors L and L2 of different time steps with M 100 . 1697.2The errors L and L2 of different time steps with M 60 . 1707.3Results for u with h 1/50 and T 1.0 . . . . . . . . . . . 1717.4The errors L when t 0.000017.5The errors L when t 0.000001. . . . . . . . . . . . . . 172. . . . . . . . . . . . . 172
AbstractThe main objective of the thesis is to develop the numerical solution ofpartial differential equations, partial integro-differential equations with aweakly singular kernel, time-fractional partial differential equations andtime-fractional integro partial differential equations.The numerical solutions of these PDEs have been obtained using cubicand quintic B-spline collocation method. The time derivatives have beenapproximated by finite difference formulas and the time-fractional derivativehas been described in the Caputo sense. The stability and convergenceproperties related to the time discretization have been discussed and proven,theoretically.The main advantage of B-spline collocation method is that many partialdifferential equations which are not simple to solve, can be solved easilyby this method. The collocation method with B-spline as basis functionsrepresents an economical alternative, since it only requires the evaluationof the unknown parameters at the grid points.It has been observed from the numerical results that the presented Bspline collocation method exhibits high level of efficiency and accuracy.Moreover, the numerical results approximate the exact solutions very efficiently.xiii
AcknowledgementsIn the name of ALLAH Almighty the most Benevolent, the most Merciful, the Creator of the universe and the Master of Life and Death, whoinculcated His countless blessings upon me to fulfill the requirements ofthis dissertation. I offer my extremely humblest, sincerest Darood-O-Salamto our beloved Prophet Hazrat Muhammad (peace be upon him) who isforever a symbol of complete guidance in every walk of life for humanity.I would like to express my sincere gratitude and supreme warm thanksto my supervisor Prof. Dr. Shahid Saeed Siddiqi for the continuous supportof my Ph.D study and research, for his patience, motivation and immenseknowledge. His guidance helped me in all the time of research and writingof this thesis. I could not have imagined having a better advisor, overseerand mentor for my Ph.D study. I warmly thank Dr. Ghazala Akram, forher valuable advice and friendly help. Her extensive discussions aroundmy work and interesting explorations in difficult concepts have been veryhelpful for this study.In the first place I would like to record my gratitude to Prof. Dr.Muhammad Sharif, the Chairman, Department of mathematics, Universityof the Punjab, Lahore, Pakistan, for his advices, invaluable and invigorating encouragement and support in various ways. I further categoricallyacknowledge to all my honorable teachers without whom I would not beable to touch this stage of academic zenith.My special thanks to my research mate Miss Shamaila Rani, and mycolleagues Dr. Uzma Ahmad, Ms. Saira Hameed and Ms. MaasoomahSadaf for their constantly varying encouragement and cooperation.It is not possible for me to name all those who have contributed, directlyor indirectly, towards the completion of my work. I am grateful to all mywell-wishers for their sincere support. I express my apology to all those notmentioned personally one by one. Words wane in expressing my venerationfor my loving, grateful, and delicate parents and all my family membersxiv
xvI owe my heartiest gratitude for their assistance and never ending prayersfor my success. I would never have been able to stand today without theircontinuous support and generous help.LahoreSeptember, 2014Saima Arshed
Chapter 1Introduction1.1Differential EquationsMathematics is the mother of all sciences and is widely used in physics,chemistry, biology, statistics, engineering, economics and astronomy etc. Infact mathematics is tool to study nature and make predictions. But ofcourse to achieve this goal, the physical situations must first be writtenin the language of mathematics. The process of describing physical phenomena using suitable mathematical relations is termed as mathematicalmodeling.The word differential and equations indicate that any equation involving derivatives. The rate of change of any quantity with respect to otherquantity (quantities) is expressed by a differential equation. Differentialequations play an important role in mathematical modeling of physical phenomena occurring in science, economics, engineering and medicine etc.Solving various problems in science and engineering requires differential1
Ch 1: Introduction2equations. Many physical, chemical, mathematical models, biological phenomena, economics, financial forecasting, image processing and other fieldsare based on differential equation.1.1.1Types of Differential EquationsThe differential equations are, generally, classified in two types Ordinary differential equation (ODE)It is an equation in which the function depends only on one independent variabledy y 2 x2 xy.dx Partial differential equation (PDE)A partial differential equation is an equation in which the dependentvariable (unknown function) must depends on more than one independent variables.Examples of the PDEs are given by w 2w k 2 t x w k t w k tµµ 2w 2w x2 y 2(1.1.1)¶, 2w 2w 2w x2 y 2 z 2(1.1.2)¶.(1.1.3)
Ch 1: Introduction3It may be noted that the heat flow in one dimensional space, two dimensional space, and three dimensional space are described by equations (1.1.1), (1.1.2) and (1.1.3), respectively.1.2Introduction to Partial Differential EquationEuler’s work in the 18th century marked the beginning of development onthe PDEs. Then d’Alembert, Lagrange, Laplace and Riemann made spectacular progress in the field. Some important PDEs of the 19th century areLaplace equation, Poisson equation, heat equation, Maxwell’s equations,Navier-Stoke equations, KdV equation and many other. The mathematicalmodeling of many scientific problems involving rates of change w.r.t. two ormore independent variables, like time, length or angle, may lead to a PDEor to a system of PDEs. Several phenomena that occur in mathematicalphysics and different fields of engineering can be modeled by partial differential equations. In physics for example, the heat flow problems and the wavepropagation phenomena are well formulated by partial differential equations [4], [16], [25] and [28]. In addition, most physical phenomena of fluiddynamics, quantum mechanics, electricity, plasma physics, propagation ofshallow water waves and many other models are controlled within its domain of validity by partial differential equations [60]. In general, PDEs are,
Ch 1: Introduction4sometimes, more difficult to solve analytically than the ODEs can be. PDEsmay be solved analytically by methods such as Bäcklund transformation,methods of characteristics, Green’s function, integral transform, Lax pairand separation of variables etc. It may be noted that the analytical methods, providing exact solutions, are more laborious ones. Analytical methodsbecome much harder to solve complex problems. Numerical methods havebecome very popular among the researchers in the last few decades. Thereis a variety of numerical techniques for solving PDEs, such as the finitedifference method [55], finite element method [57], finite volume method[29], meshfree method [31] and the spectral method [19]. The finite elementmethod and finite volume method are efficiently used in different branchesof engineering
The main objective of the thesis is to develop the numerical solution of partial difierential equations, partial integro-difierential equations with a weakly singular kernel, time-fractional partial difierential equations and time-fractional integro partial difierential equations. The numerical solutions of these PDEs have been obtained .
Introduction to Advanced Numerical Differential Equation Solving in Mathematica Overview The Mathematica function NDSolve is a general numerical differential equation solver. It can handle a wide range of ordinary differential equations (ODEs) as well as some partial differential equations (PDEs). In a system of ordinary differential equations there can be any number of
Numerical solution of partial differential equations in science and engineering. "A Wiley-Interscience publication." Includes index. 1. Science—Mathematics. 2. Engineering. mathematics. 3. Differential equations, Partial— Numerical solutions. I. Pinder, George Francis, 1942- II. Title. Q172.L36 515.3'53 81-16491 ISBN 0-471-09866-3 AACR2
Chapter 12 Partial Differential Equations 1. 12.1 Basic Concepts of PDEs 2. Partial Differential Equation A partial differential equation (PDE) is an equation involving one or more partial derivatives of an (unknown) function, call it u, that depends on two or
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Math 5510/Math 4510 - Partial Differential Equations Ahmed Kaffel, . Text: Richard Haberman: Applied Partial Differential Equations . Introduction to Partial Differential Equations Author: Joseph M. Mahaffy, "426830A jmahaffy@sdsu.edu"526930B Created Date:
2.2.2. Survey of numerical algorithms From an algorithmic standpoint, a successful numerical simulation of electrodeposition requires numerical techniques to track moving interfaces, as well as schemes to solve partial differential equations on regions bounded by moving boundaries. Many different approaches have been developed to tackle these .
Chapter 10.02 Parabolic Partial Differential Equations . After reading this chapter, you should be able to: 1. Use numerical methods to solve parabolic partial differential eqplicit, uations by ex implicit, and Crank-Nicolson methods. The general second order linear PDE with two independent variables and one dependent variable is given by . 0 .
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