NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS WITH DISTRIBUTIONAL DERIVATIVES LI YONGFENG (B.Sc., Jilin University) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF COMPUTATIONAL SCIENCE NATIONAL UNIVERSITY OF SINGAPORE 2002 Acknowledgments I would like to thank my current supervisor, Associate Professor Wei Guowei, who guided me into the field of numerical computation, gave me the opportunity to work on such an interesting problem and shared a lot of good ideas and experience with me, and thank him for his patience and much time on reviewing my thesis as well I would also like to thank Professor Yi Yingfei, who was ever my supervisor until he left NUS, for his strong support and sustained help that I will never forget It is Professor Yi who suggested by his acute academic insight that I touch this field to strengthen and enrich my background of applied mathematics My special thanks go to my wife, Sun Fangfang, for her patient and valuable help since I touched this field, and for her loving care as well My sincere thanks go to all of my department-mates for their friendship and to the secretaries of our department, Lindah, Lucy and Hwee Sim for their kindly assistance ii Acknowledgments iii My thanks are also due to the National University of Singapore for the financial support by offering me the Research Scholarship during the past two years Finally, I present my thanks to my parents for their love, encouragement and support Li Yongfeng October 2002 Table of Contents Acknowledgments ii Summary vi Introduction 1 Differential Equations with Delta Distribution 1.1 Introduction 1.2 Modeling Euler-Bernoulli Beam with Discontinuities Discrete Singular Convolution 12 2.1 Singular Convolution and Regularization 12 2.2 Discrete Singular Convolution And Sampling Theory 15 2.2.1 Discrete Singular Convolution 15 2.2.2 Sampling Theory 18 Choosing DSC kernel 23 2.3 iv Table of Contents Approximation to Delta Distribution v 26 3.1 Some Sequences of Delta type 27 3.2 Convergence Rate of Sequence of Delta Type 31 Computational Results and Conclusions 39 4.1 Estimation of Regularization Error 39 4.2 Numerical Results 43 4.2.1 Example 43 4.2.2 Example 44 4.3 Conclusion 45 4.4 Further Research 46 A Tables and Figures 48 Bibliography 54 Summary Discrete singular convolution(DSC) method is a new and robust numerical method of solving many kinds of high order partial differential equations Using the singular convolution theory as the starting point, the main idea of the DSC method is to approximate the delta distribution by classical functions On the other hand, the DSC method is closely related to the sampling theory For example, one of DSC kernels is the regularized version of the Shannon sampling kernel In this thesis the DSC method is employed to solve a class of differential equations with the delta distribution and its distributional derivatives Here the governing equation of the Euler-Bernoulli beam with jump discontinuities is considered as an example Since such an differential equation holds in the distributional sense, some regularization is necessary first of all Chapter of this thesis contains the derivation of the total governing equation of Euler-Bernoulli beam with jump discontinuities by using the singular function method The exact solution can be obtained for some simple examples by the Laplace transform and its inverse transform In Chapter 2, the DSC method is introduced It will be studied from the different vi Summary points of view, of distribution theory or precisely singular convolution theory and of sampling theory, respectively The choice of a DSC kernel is discussed there Chapter is the most important part of this thesis Since the regularization of the distributional differential equation and DSC method are all related to the classical approximation to the delta distribution, how to approximate the delta distribution will be extremely important Throughout this chapter, the construction of classical delta sequences and their convergence rates are studied in details Finally in Chapter 4, two cases of Euler-Bernoulli beam are used as examples for the numerical computation by using the DSC method with the RSK kernel The estimation of regularization error is given Taking the exact solution as the standard, the numerical results are compared by using different delta sequences vii Introduction In practical applications, sometimes one has to analyze beam with jump discontinuities in slope, deflection, or flexural stiffness and in some instances the beams are under discontinuous loading conditions Subsequently the governing equation of a beam cannot be written in the classical sense because of the discontinuity In order to study this problem analytically, the traditional method is to partition the beam into beam segments on each of which the solution is continuous, and then solve the problems by applying continuity conditions at the interface of the segments One drawback of this method is that many differential equations must be solved and thus, many continuity conditions must be applied if many discontinuities are involved This makes the method cumbersome This problem can be simplified by using singular function method which has rigorous mathematical foundation, i.e., the theory of distribution or generalized function, see [35] The main idea is to write a single expression for the whole beam (or beam moment) in terms of Macaulay bracket (the same as Heaviside function) and then establish the governing equation for it In this case, only one single differential equation need be solved Singular function method was utilized widely Introduction in the beam or plate bending analysis, see [32] and the references therein In a recent work, Yavari, etc used this method to analyze Euler-Bernoulli beams and Timoshenko beams with jump discontinuities and obtained the exact solutions by the Laplace transform, see [30], [31], [32], [33] By using singular function method, the resulting governing equation will involve the Heaviside function H(x), Dirac delta function (or delta distribution) δ(x) and its n-th order distributional derivative δ (n) (x) in the forcing term and some auxiliary conditions at the interfaces The use of distributions depends on the number of discontinuities Thus, if there are so many discontinuities that the corresponding governing equation becomes very complicated, this method will be very tedious even if the exact solution may be obtained Moreover, not all such governing equations can be solved to obtain exact solution by the Laplace transform, for example, as l = µ in the governing equation (1.8), or see [32] Thus the numerical method is indispensable in this case However, the Dirac δ function is not a classical function but a generalized function or distribution, which results in that the governing equation holds exactly in the distributional sense but not in classical sense Thus some of numerical schemes such as finite difference method are not applicable to the distributions of delta type, since the latter cannot be discretized directly due to its strong singularities Nevertheless finite element method(FEM) still works on this case because FEM involves the integration which balances the singularity of the delta distribution One example is following d4 u L 0≤x≤L = P δ(x − ) + P δ(x − L), dx du d2 u d3 u u(0) = (0) = 0, (L) = (L) = dx dx2 dx3 (1) Physically, system (1) describes a model for a cantilevered beam (clamper at s = 0) L deformed by two point forces at x = and x = L Equation (1) can be written as Introduction the necessary condition to be satisfied by the solution of the variational problem J[ϕ] = extremum with L J[ϕ] = d2 ϕ dx2 L dx − 2P ϕ( ) − 2P ϕ(L), (2) (3) where the admissible functions ϕ should satisfy the essential boundary conditions of the problem: ϕ(0) = ϕ (0) = Then some approximation of ϕ can be chosen to obtain the approximate solution to equation (1) Please refer to [12] for more details Other than finite element method, in this paper, we will discuss an alternative numerical method to handle the equations with distributions For the purpose of solving this problem numerically, the governing equation under study must be approximately regularized In other word, a classical equation which can be solved numerically should be found to approximate the original one in distributional sense Since the problem exists in the distribution of delta type, the essential part of this regularization process is how to approximate the delta distribution by using classical functions After regularization, the second problem is which numerical scheme should be chosen to solve this regularized equation numerically Since the governing equation of the beam bending problem is a high order differential equation, a good high order numerical method is required to obtain the good numerical result Recently the discrete singular convolution (DSC) method has emerged as a potential approach to the computer realization of singular convolutions, see [18], [19], [20] Coincidentally, DSC method has the distribution theory as its underlying mathematical framework and deals with the approximation of the delta distribution as well Consider a singular convolution Φ(x) = (L ∗ ϕ)(x) = R L(x − t)ϕ(t)dt (4) 4.4 Further Research 46 delta sequences are constructed to approximate the Dirac delta function Theoretically, except Poisson kernel which has convergence rate of one order less than that of others, all of these approximations have the same convergence rates in H α with α < However, the numerical results shows that all of these approximations are commonly good only in smooth case, in which Mollifier is the best one compared with others; and SINC kernel and Gaussian are good at continuous case, while RSK kernel has excellent performance particularly at the discontinuous case 4.4 Further Research From all the tables of data and figures, it is obviously that the difference between the numerical solution and exact solution is much larger around the singular point x0 than that at other smooth points The main reason is that the change of high frequency of classical approximation of the delta distribution around singularity results in much loss of information by using uniform grids In order to collect more information around x0 , more grid points around it are needed To this end, an appropriate transformation of coordinate is worth considering For example, let x(y) be a monotone smooth function satisfying x(y0 ) = 0, x(0) = x0 , x(yN ) = L (4.32) and < x (y) < (4.33) in a neighborhood of 0, where y0 < < yN Under such a transformation, more grid points x(yi ) accumulate around x0 and the corresponding computational grid points yi remains uniform grid Here the requirement x (0) = is to guarantee that the change of variable for δ(x − x0 ) makes sense, refer to [11] 4.4 Further Research Nevertheless, such a transformation can make the original equation more complicated Thus whether such a kind of transformations are applicable in this numerical method or not is the main point of our further research On the other hand, the example under study in this thesis is relatively simple because it is linear and the delta distribution is in the forcing term It makes it easy to obtain the estimation of regularization error The following work may be focused on the Schrăodinger equation with delta well potential It will be an interesting problem 47 Appendix A Tables and Figures In this section, the numerical results are listed and showed in the following tables and figures, respectively In each set of table and figures, the L2 errors with relative L2 errors (denoted by r − L2 ) and L∞ errors with relative L∞ errors (denoted by r − L∞ ) are listed in the tables on the left The numerical solutions and exact solution are plotted in the top figure and the differences between them are given at the bottom For β = 0.3 and β = 0.6 in Example 2, the numerical solutions produced by the Shannon kernel, Mollifier and Poisson kernel are far away from the exact solution, thus only the numerical solutions by the Gaussian and RSK kernel are plotted in Figures A.6 and A.7 There the letters in each table represent S −− Shannon kernel G −− Gaussian M −− Mollifier P Poisson kernel −− R −− RSK kernel 48 49 S G R M P S G R M P N L∞ r − L∞ L2 r − L2 101 6.22(-1) 1.36(-2) 6.65(-1) 9.92(-3) 201 3.21(-1) 7.02(-3) 3.34(-1) 4.98(-3) 401 1.63(-1) 3.55(-3) 1.67(-1) 2.50(-3) 801 7.89(-2) 1.72(-3) 8.25(-2) 1.23(-3) 100 1.81(-1) 3.97(-2) 8.72(-1) 1.30(-2) 200 9.73(-1) 2.13(-2) 3.26(-1) 4.86(-3) 400 5.05(-1) 1.10(-2) 1.18(-1) 1.77(-3) 800 2.59(-1) 5.63(-3) 4.25(-2) 6.34(-4) 100 11.283 2.48(-1) 15.276 2.28(-1) 200 16.897 3.69(-1) 22.665 3.38(-1) 400 22.324 4.86(-1) 29.823 4.45(-1) 800 26.541 5.78(-1) 35.391 5.28(-1) 101 42.214 9.27(-1) 60.647 9.05(-1) 201 42.267 9.23(-1) 60.356 9.01(-1) 401 42.297 9.22(-1) 60.214 8.99(-1) 801 42.312 9.21(-1) 60.144 8.98(-1) 101 31.319 6.87(-1) 41.939 6.26(-1) 201 30.460 6.65(-1) 40.093 5.98(-1) 401 27.567 6.01(-1) 34.974 5.22(-1) 801 11.965 2.60(-1) 10.009 1.49(-1) N L∞ r − L∞ L2 r − L2 101 2.37(-2) 8.99(-3) 2.30(-2) 5.47(-3) 201 1.23(-2) 4.64(-3) 1.16(-2) 2.75(-3) 401 6.26(-3) 2.36(-3) 5.79(-3) 1.38(-3) 801 3.17(-3) 1.19(-3) 2.90(-3) 6.91(-4) −1.5 100 7.73(-2) 2.93(-2) 3.67(-2) 8.75(-3) −2 200 4.22(-2) 1.60(-2) 1.40(-2) 3.34(-3) 400 2.21(-2) 8.34(-3) 5.16(-3) 1.23(-3) 800 1.13(-2) 4.27(-3) 1.86(-3) 4.43(-4) 100 7.95(-2) 3.01(-2) 7.02(-2) 1.67(-2) 200 1.43(-1) 5.41(-2) 1.22(-1) 2.91(-2) 400 2.28(-1) 8.61(-2) 1.92(-1) 4.57(-2) 800 3.21(-1) 1.21(-1) 2.68(-1) 6.39(-2) 101 1.17(0) 4.44(-1) 1.41(0) 3.35(-1) 201 1.15(0) 4.35(-1) 1.36(0) 3.24(-1) 401 1.14(0) 4.30(-1) 1.34(0) 3.18(-1) 0.6 −10 −20 Shannon Gaussian RSK kernel Mollifier Possion Exact solution −30 −40 −50 6 Example 1, β=0.3 50 40 30 20 10 0 Table A.1: and Figure A.1 −0.5 −1 Shannon Gaussian RSK kernel Mollifier Possion Exact solution −2.5 −3 6 Example 1, β=0.6 1.2 0.8 801 1.13(0) 4.28(-1) 1.33(0) 3.16(-1) 0.4 101 4.96(-1) 1.88(-1) 4.22(-1) 1.00(-1) 0.2 201 4.85(-1) 1.84(-1) 3.99(-1) 9.49(-2) 401 4.44(-1) 1.67(-1) 3.52(-1) 8.39(-2) 801 3.31(-1) 1.25(-1) 2.89(-1) 6.88(-2) 0 Table A.2: and Figure A.2 50 S G R M P S G R M P N L∞ r − L∞ L2 r − L2 101 5.16(-5) 1.90(-4) 7.83(-5) 1.39(-4) 201 1.29(-5) 4.75(-5) 1.96(-5) 3.49(-5) 401 3.22(-6) 1.18(-5) 4.88(-6) 8.69(-6) 801 7.38(-7) 2.72(-6) 1.05(-6) 1.87(-6) 100 5.86(-5) 2.16(-4) 9.43(-5) 1.68(-4) 200 1.45(-5) 5.33(-5) 2.34(-5) 4.17(-5) 400 3.61(-6) 1.33(-5) 5.85(-6) 1.04(-5) −0.1 −0.15 −0.2 −0.25 800 9.26(-7) 3.41(-6) 1.50(-6) 2.67(-6) 100 5.80(-5) 2.13(-4) 9.32(-5) 1.66(-4) 200 1.42(-5) 5.24(-5) 2.27(-5) 4.05(-5) 400 3.50(-6) 1.29(-5) 5.55(-6) 9.88(-6) 800 8.87(-7) 3.27(-6) 1.39(-6) 2.48(-6) 101 5.02(-5) 1.85(-4) 8.49(-5) 1.51(-4) 201 1.25(-5) 4.61(-5) 2.10(-5) 3.74(-5) 401 3.12(-6) 1.15(-5) 5.21(-6) 9.27(-6) 801 7.15(-7) 2.63(-6) 1.07(-6) 1.90(-6) 101 4.98(-5) 1.83(-4) 7.54(-5) 1.34(-4) 201 1.24(-5) 4.58(-5) 1.88(-5) 3.35(-5) 401 3.10(-6) 1.14(-5) 4.67(-6) 8.32(-6) 801 7.09(-7) 2.61(-6) 9.85(-7) 1.75(-6) N L∞ r − L∞ L2 r − L2 101 1.47(-4) 2.12(-3) 1.01(-4) 6.69(-4) −0.01 201 7.89(-5) 1.13(-3) 4.90(-5) 3.24(-4) −0.02 401 4.12(-5) 5.92(-4) 2.44(-5) 1.62(-4) −0.03 1.22(-5) 8.09(-5) −0.04 1.99(-4) 1.31(-3) −0.05 200 2.52(-4) 3.62(-3) 8.35(-5) 5.53(-4) −0.06 400 1.39(-4) 2.00(-3) 3.25(-5) 2.15(-4) −0.07 800 7.35(-5) 1.06(-3) 1.21(-5) 8.00(-5) 100 9.86(-5) 1.42(-3) 8.44(-5) 5.59(-4) 200 2.08(-4) 2.99(-3) 1.36(-4) 8.97(-4) 400 3.80(-4) 5.46(-3) 2.35(-4) 1.55(-3) 800 6.16(-4) 8.85(-3) 3.73(-4) 2.47(-3) 101 3.32(-3) 4.78(-2) 2.63(-3) 1.74(-2) 2.5 201 3.16(-3) 4.54(-2) 2.41(-3) 1.59(-2) 401 3.10(-3) 4.45(-2) 2.31(-3) 1.53(-2) 1.5 2.27(-3) 1.50(-2) 9.03(-4) 5.97(-3) 201 1.41(-3) 2.03(-2) 8.87(-4) 5.87(-3) 401 1.40(-3) 2.02(-2) 8.63(-4) 5.71(-3) 801 1.32(-3) 1.90(-2) 8.16(-4) 5.40(-3) 6 x 10 3.02(-4) 4.41(-2) Table A.3: and Figure A.3 6.05(-3) 1.97(-2) 2.10(-5) 3.07(-3) 4.21(-4) 1.37(-3) Example 1, β=1.0 801 801 −5 100 101 Shannon Gaussian RSK kernel Mollifier Possion Exact solution −0.05 Shannon Gaussian RSK kernel Mollifier Possion Exact solution 6 Example 1, β=1.4 −3 3.5 x 10 0.5 0 Table A.4: and Figure A.4 51 S G R M P S G R M P N L∞ r − L∞ L2 r − L2 101 2.66(-5) 8.31(-4) 2.70(-5) 3.82(-4) −0.005 201 1.43(-5) 4.47(-4) 9.74(-6) 1.38(-4) −0.01 401 7.54(-6) 2.36(-4) 4.30(-6) 6.09(-5) −0.015 801 2.08(-6) 1.21(-4) 2.08(-6) 2.94(-5) 100 7.18(-5) 2.25(-3) 3.96(-5) 5.61(-4) 200 4.46(-5) 1.40(-3) 1.56(-5) 2.22(-4) 400 2.52(-5) 7.88(-4) 6.01(-6) 8.52(-5) 800 1.34(-5) 4.20(-4) 2.23(-6) 3.76(-5) 100 1.86(-5) 5.83(-4) 2.35(-5) 3.34(-4) 200 2.59(-5) 8.10(-4) 1.64(-5) 2.33(-4) 400 4.95(-5) 1.55(-3) 2.79(-5) 3.96(-4) 800 8.30(-5) 2.60(-3) 4.57(-5) 6.49(-4) 100 5.22(-4) 1.63(-2) 3.83(-4) 5.43(-3) 200 4.87(-4) 1.52(-2) 3.39(-4) 4.81(-3) 400 4.74(-4) 1.48(-2) 3.23(-4) 4.58(-3) 800 4.69(-4) 1.47(-2) 3.15(-4) 4.47(-3) 100 2.06(-4) 6.45(-3) 1.26(-4) 1.79(-3) 200 2.15(-4) 6.72(-3) 1.23(-4) 1.75(-3) 400 2.15(-4) 6.73(-3) 1.20(-4) 1.71(-3) 800 2.07(-4) 6.46(-3) 1.15(-4) 1.63(-3) N L∞ r − L∞ L2 r − L2 101 1.86(0) 2.47(-1) 7.48(0) 6.96(-2) 201 9.58(0) 1.26(0) 17.1(0) 1.59(0) 401 19.0(0) 2.50(0) 34.1(0) 3.16(0) 801 38.0(0) 4.98(0) 68.2(0) 6.31(0) 101 2.53(0) 3.35(-1) 1.86(0) 1.73(-1) 201 2.08(0) 2.73(-1) 1.05(0) 9.78(-2) 401 1.86(0) 2.44(-1) 6.24(-1) 5.78(-2) 801 1.99(0) 2.60(-1) 3.90(-1) 3.60(-2) 101 2.02(0) 2.68(-1) 5.43(-1) 5.05(-2) 201 2.07(0) 2.73(-1) 3.84(-1) 3.56(-2) 401 2.09(0) 2.75(-1) 2.72(-1) 2.52(-2) 801 2.10(0) 2.76(-1) 1.92(-1) 1.78(-2) 2.5 101 2.15(0) 2.85(-1) 3.00(0) 2.79(-1) 201 2.13(0) 2.81(-1) 3.00(0) 2.78(-1) 401 2.12(0) 2.79(-1) 3.00(0) 2.78(-1) 801 2.12(0) 2.78(-1) 3.00(0) 2.77(-1) 101 2.82(0) 3.74(-1) 3.53(0) 3.29(-1) 0.5 201 2.55(0) 3.35(-1) 3.39(0) 3.14(-1) 401 2.37(0) 3.11(-1) 3.33(0) 3.08(-1) 801 2.26(0) 2.97(-1) 3.30(0) 3.06(-1) Shannon Gaussian RSK kernel Mollifier Possion Exact solution −0.02 −0.025 −0.03 −0.035 6 Example 1, β=1.7 −4 x 10 0 Table A.5: and Figure A.5 −2 −4 −6 −8 Gaussian RSK kernel Exact solution 6 Example 2, β=0.3 1.5 0 Table A.6: and Figure A.6 52 S G R M P S G R M P N L∞ r − L∞ L2 r − L2 101 6.82(-1) 2.31(-1) 2.24(-1) 5.03(-2) 201 2.27(0) 7.61(-1) 4.03(0) 9.02(-1) 401 4.52(0) 1.51(0) 8.05(0) 1.80(0) 801 9.02(0) 3.00(0) 16.09 3.59(0) 101 6.94(-1) 2.35(-1) 4.03(-1) 9.04(-2) −2 201 6.40(-1) 2.14(-1) 2.55(-1) 5.70(-2) −2.5 401 6.83(-1) 2.28(-1) 1.70(-1) 3.79(-2) −3 801 7.04(-1) 2.34(-1) 1.16(-1) 2.60(-2) 101 7.10(-1) 2.40(-1) 1.85(-1) 4.14(-2) 201 7.18(-1) 2.40(-1) 1.31(-1) 2.94(-2) 401 7.22(-1) 2.41(-1) 9.31(-2) 2.08(-2) 801 7.24(-1) 2.41(-1) 6.60(-2) 1.47(-2) 101 7.16(-1) 2.42(-1) 8.42(-1) 1.89(-1) 201 7.20(-1) 2.41(-1) 8.43(-1) 1.89(-1) 401 7.23(-1) 2.41(-1) 8.43(-1) 1.88(-1) 801 7.24(-1) 2.41(-1) 8.43(-1) 1.88(-1) 101 8.31(-1) 2.81(-1) 9.31(-1) 2.09(-1) 201 7.90(-1) 2.65(-1) 9.17(-1) 2.05(-1) 401 7.64(-1) 2.55(-1) 9.10(-1) 2.03(-1) 801 7.48(-1) 2.49(-1) 9.08(-1) 2.03(-1) N L∞ r − L∞ L2 r − L2 100 7.87(-3) 1.53(-2) 9.65(-3) 1.11(-2) 200 3.93(-3) 7.65(-3) 4.82(-3) 5.53(-3) 400 1.96(-3) 3.83(-3) 2.41(-3) 2.76(-3) −0.2 −0.3 800 9.79(-4) 1.91(-3) 1.20(-3) 1.38(-3) 100 7.97(-3) 1.55(-2) 9.18(-3) 1.05(-2) 200 3.97(-3) 7.72(-3) 4.75(-3) 5.45(-3) 400 1.97(-3) 3.83(-3) 2.40(-3) 2.75(-3) 800 9.80(-4) 1.91(-3) 2.40(-3) 1.38(-3) 100 7.88(-2) 1.53(-2) 9.72(-3) 1.11(-2) 200 3.93(-3) 7.65(-3) 4.83(-3) 5.54(-3) 400 1.96(-3) 3.82(-3) 2.41(-3) 2.76(-3) 800 9.79(-4) 1.96(-3) 1.20(-3) 1.38(-3) 100 8.30(-3) 1.62(-2) 9.97(-3) 1.14(-2) 200 4.15(-3) 8.09(-3) 4.97(-3) 5.69(-3) 400 2.08(-3) 4.04(-3) 2.48(-3) 2.84(-3) 800 1.04(-3) 2.02(-3) 1.24(-3) 1.42(-3) 100 7.89(-3) 1.53(-2) 9.65(-3) 1.11(-2) 200 3.94(-3) 7.67(-3) 4.81(-3) 5.51(-3) 400 1.97(-3) 3.84(-3) 2.40(-3) 2.75(-3) 800 9.86(-4) 1.92(-3) 1.20(-3) 1.37(-3) −0.5 −1 −1.5 Gaussian RSK kernel Exact solution 6 Example 2, β=0.6 0.8 0.6 0.4 0.2 0 Table A.7: and Figure A.7 −0.1 Shannon Gaussian RSK kernel Mollifier Possion Exact solution −0.4 −0.5 6 Example 2, β=1.0 −3 x 10 0 Table A.8: and Figure A.8 53 S G R M P S G R M P N L∞ r − L∞ L2 r − L2 101 2.33(-3) 5.66(-2) 1.30(-3) 1.82(-2) 201 1.82(-3) 4.43(-2) 9.57(-4) 1.34(-2) 401 1.56(-3) 3.79(-2) 8.38(-4) 1.17(-2) 801 1.43(-3) 3.47(-2) 8.02(-4) 1.12(-2) 101 2.23(-3) 5.42(-2) 1.15(-3) 1.61(-2) 201 1.81(-3) 4.39(-2) 9.01(-4) 1.26(-2) 401 1.56(-3) 3.78(-2) 8.27(-4) 1.15(-2) −0.05 −0.15 801 1.43(-3) 3.47(-2) 8.00(-4) 1.12(-2) 101 2.36(-3) 5.74(-2) 1.35(-3) 1.88(-2) 201 1.83(-3) 4.45(-2) 9.66(-4) 1.35(-2) 401 1.56(-3) 3.79(-2) 8.40(-4) 1.17(-2) 801 1.43(-3) 3.47(-2) 8.02(-4) 1.12(-2) 101 1.37(-3) 3.33(-2) 8.50(-4) 1.19(-2) 201 1.28(-3) 3.11(-2) 7.97(-4) 1.11(-2) 401 1.26(-3) 3.05(-2) 7.87(-4) 1.10(-2) 801 1.25(-3) 3.04(-2) 7.84(-4) 1.10(-2) 101 1.38(-3) 3.36(-2) 1.05(-3) 1.46(-2) 201 1.27(-3) 3.09(-2) 8.89(-4) 1.24(-2) 401 1.24(-3) 3.02(-2) 8.32(-4) 1.16(-2) 801 1.24(-3) 3.02(-2) 8.09(-4) 1.13(-2) N L∞ r − L∞ L2 r − L2 101 2.93(-2) 4.56(-1) 2.19(-2) 1.97(-1) 201 1.70(-2) 2.65(-1) 1.68(-2) 1.52(-1) 401 1.53(-2) 2.38(-1) 1.87(-2) 1.68(-1) 801 2.24(-2) 3.49(-1) 2.67(-2) 2.40(-1) 101 2.21(-2) 3.43(-1) 1.89(-2) 1.69(-1) 201 1.34(-2) 2.09(-1) 1.04(-2) 9.35(-2) 401 7.54(-3) 1.17(-1) 5.51(-3) 4.95(-2) 801 6.81(-3) 1.06(-1) 2.90(-3) 2.61(-2) 101 3.09(-2) 4.81(-1) 1.84(-2) 1.65(-1) 201 1.76(-2) 2.74(-1) 9.94(-3) 8.94(-2) 401 9.78(-3) 1.52(-1) 5.20(-3) 4.67(-2) 801 6.84(-3) 1.07(-1) 2.68(-3) 2.41(-2) 0.03 101 6.98(-3) 1.09(-1) 4.63(-3) 4.16(-2) 0.025 201 6.88(-3) 1.07(-1) 4.56(-3) 4.10(-2) 0.02 401 6.85(-3) 1.07(-1) 4.55(-3) 4.09(-2) 0.015 0.01 801 6.84(-3) 1.07(-1) 4.55(-3) 4.09(-2) 101 6.59(-3) 1.03(-1) 4.88(-3) 4.38(-2) 201 6.69(-3) 1.04(-1) 4.71(-3) 4.24(-2) 401 6.77(-3) 1.05(-1) 4.64(-3) 4.18(-2) 801 6.81(-3) 1.06(-1) 4.61(-3) 4.15(-2) Shannon Gaussian RSK kernel Mollifier Possion Exact solution −0.1 6 Example 2, β=1.4 0.04 0.03 0.02 0.01 0 Table A.9: and Figure A.9 −0.02 −0.04 Shannon Gaussian RSK kernel Mollifier Possion Exact solution −0.06 −0.08 6 Example 2, β=1.7 0.035 0.005 0 Table A.10: and Figure A.10 Bibliography [1] G Bao, G.W Wei and A.H Zhou, Analysis of regularized Whittaker- Kotel’nikov-Shannon sampling expansion, preprint [2] John J Benedetto, Paulo J.S.G Ferreira, Modern sampling theory, Birkhăauser, Boston(2000) [3] Schomburg Bernd, On the approximation of the delta distribution in Sobolev spaces of negative order, Appl Anal 36 no 1-2(1990) 89–93 [4] B.F Feng and G.W Wei, A comparison of the spectral and the discrete singular convolution schemes for the KdV-type equations, J Comput Appl Math., 145 (2002) 183-188 [5] Frank Stenger, Numerical Methods Based On Sinc and Analytic Functions, Springer-Verlag 1993 [6] Marek A Kowalski, Krszysztof A Sikorski and Frank Stenger, Selected Topics in Approximation and Computation, Oxford University 1995 54 Bibliography [7] S Guan, C.-H Lai and G.W Wei, Bessel-Fourier analysis of patterns in a circular domain, Physica D 151 (2001) 83 [8] S Guan, C.-H Lai and G.W Wei, A wavelet method for the characterization of 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Timoshenko beams with jump discontinuities: applications of distribution theory, Int J Solids and Structures, 38 8389-8406(2001) [34] S.Y Yang, Y.C Zhou, and G W Wei, Comparison of the discrete singular convolution algorithm and the Fourier pseudospectral method for solving partial differential equations, Comput Phys Commun 143, 113 (2002) [35] A.H Zemanian, Distribution Theory and Transform Analysis, McGraw-Hill, New York(1965) [36] Y.B Zhao, G.W Wei and Y Xiang, Discrete singular convolution for the prediction of high frequency vibration of plates, Int J Solids and Structures, 39(2002) 65-88 [37] Y.B Zhao, G.W Wei and Y Xiang, Plate vibration under irregular internal supports, Int J Solids and Structures, 39(2002) 1361-1383 [38] Y.C Zhou, G.W Wei, High-resolution conjugate filters for the simulation of flows, J Comput Phys in press 57 Name: Li Yongfeng Degree: Master of Science Department: Computational Science Thesis Title: Numerical Methods for Differential Equations with Distributional Derivatives Abstract A recently developed numerical method, the discrete singular convolution (DSC) method, for solving high order differential equations with distributional derivatives is presented in the different framework of distribution theory and sampling theory, respectively In order to use this method to solve a class of differential equations with the delta distribution and its distributional derivatives, the classical approximations to the delta distribution and their convergence rates in Sobolev space H α of negative order α are studied in details As an example, the model of EulerBernoulli beam with jump discontinuities is used to test the efficiency of some delta sequences and the DSC method Keywords: Discrete singularity convolution(DSC), sampling theory, regularized Shannon kernel(RSK), delta distribution, sequence of delta type, convergence rate NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS WITH DISTRIBUTIONAL DERIVATIVES LI YONGFENG NATIONAL UNIVERSITY OF SINGAPORE 2002 NUMERICAL METHOD FOR DIFFERENTIAL EQUATIONS WITH DISTRIBUTIONAL DERIVATIVES LI YONGFENG 2002 ... this thesis, we will mainly consider the numerical method for differential equations with the delta distribution and its distributional derivatives Such equations have very strong physical backgrounds... why we will consider numerical methods for solving this kind of differential equations However, the exact solution in hand can be used as benchmark to evaluate numerical methods In the following... This makes any differential equation with the delta distribution and its distributional derivatives (e.g equation (1.8)) hold only in the distributional sense As a result, such a differential equation