Eigenvalues and eigenfunctions of a non-local boundary value problem of Sturm Liouville differential equation
|
|
- Ήρα Πολίτης
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Global Journal of Pure and Applied Mathematics. ISSN Volume 12, Number 5 (2016, pp Research India Publications Eigenvalues and eigenfunctions of a non-local boundary value problem of Sturm Liouville differential equation A. M. A El-Sayed, Zaki. F. A. EL-Raheem Faculty of Science, Alexandria University, Alexandria, Egypt. amasayed@alexu.edu.eg, N. A. O. Buhalima Faculty of Science, Omar AlMukhtar University, Albeda, Libya. Abstract In this paper we study the existence and properties of eigenvalues and eigenfunctions of a non-local boundary value problem of the Sturm-Liouville equation. AMS subject classification: Keywords: Sturm Liouville equation, eigenvalues, eigenfunctions, nonlocal boundary condition. 1. Introduction Problems with non-local boundary conditions arise in various fields of mathematical, physical [2] [7] and biological [8], [9] sciences. The solvability of boundary value problems with non-local conditions for Sturm-Liouville equation has received great attention from many authors and become a very hot research topic. A. A. Samarskii and A. V. Bitsadze were originators of such problems. They formulated and investigated the non-local boundary problem for an ellibtic equation [1]. Recently, the authors studied in [11] the existence of eigenvalues and eigenfunctions of the boundary value problem of the Sturm- Liouville differential equation y + q(xy = 2 y, 0 x π (1.1
2 3886 A. M. A El-Sayed, et al. with each one of the two non-local conditions and y(0 = 0, y(ξ = 0, ξ (0,π], (1.2 y( = 0, y(π = 0, [0,π (1.3 Consider the non-local boundary value problem of the Sturm-Liouville equation (1.1 with the nonlocal boundary condition y( = 0 and y(ξ = 0, and [0, π, ξ (0,π]. (1.4 Here we study the existence and some general properties of the eigenvalues and eigenfunctions of the non-local boundary value problems (1.1 and (1.4. Comparison with the local boundary value problem problem of equation (1.1 with the local boundary value problem y(0 = 0, y(π = 0 will be given. 2. General properties Here we prove some results concerning the eigenvalues and eigenfunctions of the nonlocal problem (1.1 (1.4. Lemma 2.1. The eigenvalues of the non-local boundary value problem (1.1 and (1.4 are real. Proof. Let y 0 (x be the eigenfunction that corresponds to the eigenvalue 0 of the problem (1.1 and (1.4, then and y 0 + q(x y 0 = 2 0 y 0 (0 x π, (2.5 y 0 ( = y 0 (ξ = 0 (2.6 Multiplying both sides of (2.5 by y 0 and then integrating form to ξ with respect to x, we have y 0 y 0 ξ + y 0 2 dx + q(x y 0 2 dx = 2 0 y 0 2 dx. using the boundary condition (2.6, we have 2 0 = [q(x y y 0 2 ]dx y. 0 2 dx
3 Eigenvalues and eigenfunctions From which it follows the reality of 2 0. Lemma 2.2. The eigenfunctions that corresponds to two different eigenvalues of the non-local boundary value problem (1.1 and (1.4 are orthogonal. Proof. Let 1 = 2 be two different eigenvalues of the non-local boundary value problem (1.1 and (1.4. Let y 1 (x, y 2 (x be the corresponding eigenfunctions, then and y 1 + q(x y 1 = 2 1 y 1 (0 x π, (2.7 y 1 ( = y 1 (ξ = 0 (2.8 y 2 + q(x y 2 = 2 2 y 2 (0 x π, (2.9 y 2 ( = y 2 (ξ = 0 (2.10 Multiplying both sides of (2.7 by ȳ 2 and integrating with respect to x, we obtain y 1 ȳ 2 dx + q(x y 1 ȳ 2 dx = 2 1 y 1 ȳ 2 dx. (2.11 By taking the complex conjugate of (2.9 and multiply it by y 1 and integrate the resulting expression with respect to x,wehave y 1 ȳ 2 dx + q(x y 1 ȳ 2 dx = 2 1 y 1 ȳ 2 dx. (2.12 Subtracting (2.11 from (2.12 and using the boundary conditions of (2.8 and (2.10 we obtain ( y 1 ȳ 2 dx = 0, 2 1 = 2 2. which completes the proof. 3. The asymptotic formulas for the solution Here we study the asymptotic formulas for the solutions of problem (1.1 and (1.4. Lemma 1.1 deals with the nature the eigenvalues. Let be φ(x, the solution of equation (1.1 and (1.4 satisfying the initial conditions φ(, = 0, φ (, = 1 (3.13 and by ϑ(x, the solution of the same equation, satisfying the initial conditions ϑ(, = 1, ϑ (, = 0 (3.14
4 3888 A. M. A El-Sayed, et al. We notes that φ(x, and ϑ(x, are linearly independent if and only if ω( = 0. The characteristic equation will be ω( = φ(x, ϑ (x, φ (x, ϑ(x,. ω( = φ(ξ, (3.15 Lemma 3.1. The solution φ(x, of problem (1.1 and (1.4 satisfy the integral equations sin (x x sin (x τ φ(x, = + q(τφ(τ,dτ. (3.16 Proof. First we obtain formula (3.16 Indeed,with solution of the form q(x = 0. (1.1 becomes becomes y = 2 y by means of variation of parameter method, we have φ(x, = C 1 (x, cos x + C 2 (x, sin x (3.17 and the direct calculation of C 1 (x, s and C 2 (x, s, wehave sin C 1 (x, = sin τ q(τφ(τ,dτ, (3.18 C 2 (x, = cos + cos τ q(τφ(τ,dτ. substituting from (3.18 into (3.17 equation (3.16 follows. Second we show that the integral representation (3.16 satisfies the problem (1.1 and (3.13. Let ϕ(x, be the solution of (1.1, so that We multiply both sides by q(x φ(x, = φ (x, + 2 φ(x,. sin (x τ and integrating with respect to τ from to x we obtain sin (x τ q(τ φ(τ, dτ = sin (x τ φ (τ, dτ ( sin (x τ φ(τ, dτ.
5 Eigenvalues and eigenfunctions Integrating by parts twice and using the condition (3.13, we have sin (x τ φ (τ, dτ (3.20 sin (x x = φ(x, sin (x τ φ(τ, dτ. By substituting from (3.20 into (3.19 we get the required formula (3.16. Lemma 3.2. Let = σ + it. Then there exists 0 > 0, such that > 0 the following inequalities for the solutions φ(x,of boundary value problem (1.1 and (1.4 hold true ( sin (x e Im (x φ(x, = + O 2, (3.21 Proof. We show first that t (x φ(x, = O. where the inequality is uniformly with respect to x. Form the integral equation (3.16 we have φ(x, e t (x + e t (x e t (x q(τ φ(τ, dτ. (3.22 By using the notation φ(x, e t (x = F(x,, equation (3.22 takes the form F(x, π Let µ = max F(x,, so that from (3.23 it follows that 0 x π For > 0 = π 0 µ q(τ F(τ, dτ. (3.23 π 0 q(τ dτ. q(τ dτ it follows from the last inequality that F(x, constant / and this implies that t (x φ(x, = O, (3.24 By the aid of (3.23 we find that sin (x τ t (x q(τ φ(τ, dτ = O 2. (3.25
6 3890 A. M. A El-Sayed, et al. From (3.16 and (3.23 it follows that, ϕ(x, has the asymptotic formula (3.21. Theorem 3.3. Let = σ + it and suppose that q(x has a second order piecewise differentiable derivatives on [0,π]. Then the solution φ(x, of non-local boundary value (1.1 and (1.4 have the following asymptotic formula where sin (x φ(x, = α 1(x 2 cos (x α ( 2(x e Im (x 3 sin (x + O 4 α 1 (x = 1 q(t dt, 2 α 2 (x = 1 ( 4 2 q(t dt (3.26 [q(x + q(0]. ( Proof. By substituting from (3.21 into the integral equation (3.16, we have sin (x cos (x φ(x, = 2 2 q(t dt cos (x + 2t q(t dt Im (x + O 3. (3.28 Integrating the last integration of (3.28 by parts and noticing that there exists q (x such that q L 1 [0,π] cos (x + 2t q(t dt = 1 sin (x [q(x + q(] sin (x + 2t q (t dt Im (x = O 3. (3.29 substituting from (3.29 into (3.28, we get sin (x φ(x, = α ( 1(x e Im (x 2 cos (x + O 3. (3.30 where α 1 (x is defined by (3.27. In order to make φ(x, more precise we repeat this procedure again by substituting from the last result (3.30 into the same integral equation
7 Eigenvalues and eigenfunctions (3.16, we have sin (x sin (x tsin (t φ(x, = + 2 q(t dt sin (x tcos (t q(tα 1 (t dt + sin (x t 3 Im (x q(to 3 dt. (3.31 Now we estimate each term in (3.31. Integrating by parts twice the first term of (3.31, and noticing that q L 1 [0,π],wehave sin (x tsin (t 2 = α 1(x 2 Similarly, we have cos (x + q(t dt q(x + q( 4 3 Im (x sin (x + O 4. (3.32 sin (x tcos (t 3 q(t α 1 (t dt = 1 ( sin (x e Im (x α 1 (t q(t dt O 4 (3.33 Substituting from (3.32 and (3.33 into (3.31 we get the required formula (3.26. Now inserting the values of the functions ϕ(x, from the estimate (3.26 into the second of the boundary conditions in (1.4, we obtain the following equation for the determination of the eigenvalues: Equation (3.21 is the characteristic equation which gives roots of 0 n = nπ, n = 0, ±1, ±2,... (ξ Then the ω( has the same root of the function sin ξ (By Rouche s theorem n = 0 n + ε n, n = 0, 1, 2,... (3.34 Theorem 3.4. Let q L 1 (0,π, then we have the following asymptotic formulas for n of non-local boundary value (1.1 and (1.4 n = nπ (ξ + α ( 1 1 nπ + O n 2. (3.35 where α 1 (x defined in (3.27.
8 3892 A. M. A El-Sayed, et al. Proof. ω(x, = sin (ξ α 1 2 cos (ξ α 2 sin (ξ +O 3 Im (ξ 4 (3.36 It follows from (3.36 that sin (ξ α 1 cos (ξ α ( 2 e Im (ξ sin (ξ + O 2 3 = 0 (3.37 From equation (3.37, we have [ 1 α ] 2 2 sin (ξ α 1 cos (ξ = 0 (3.38 Dividing (3.38 by cos (ξ we obtain since imaginary = O [ 1 α ] 2 2 tan (ξ = α 1 ( 1, then n tan n (ξ = α ( O n n 2 From (3.34, (3.39 after elementary calculation, we obtain ε n = α ( 1 1 nπ + O n 2 (3.39 (3.40 From (3.34 and (3.40, we have n = nπ (ξ + α ( 1 1 nπ + O n 2. Corollary 3.5. If = 0 and ξ = π, then the eigenvalues of (3.35, we obtain n = n + α ( 1 1 nπ + O n 2. Which meets with the result obtained in [10].
9 Eigenvalues and eigenfunctions References [1] A. V. Bitsadze and A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems. Dokl. Akad. Nauk SSSR, 185: , [2] A. M.A. El-Sayed, E. M. Hamdallah, Kh.W. Elkadeky, Solution of a class of internal nonlocal Cauchy problems for the differential equation x (t = f(t,x(t,x (t, Fixed Piont Theorem Vol. 15(2, (2014. [3] A. M. A. EL-Sayed, M. S. EL-Azab, A. Elsaid and S. M. Helal, Eigenvalue problem for Elliptic partial differential equations with nonlocal boudary conditions, Journal of Fractional Calculus and Applications, Vol. 5(3S No. 14, pp (2014. [4] A. M. A. EL-Sayed and F. M. Gaafar, Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions, Advances in Difference Equations, / (2011 [5] W. A. Day. Extensions of a property of the heat equation to linear thermoelasticity and order theories. Quart. Appl. Math., 40: , [6] N. Gordeziani. On some non-local problems of the theory of elasticity. Buletin of TICMI, 4:43 46, [7] N. I. Ionkin. The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition. Differ. Equ., 13(2: , (in Russian [8] A. M. Nakhushev. Equations of Mathematical Biology. Vysshaya Shkola, Moscow, (in Russian [9] K. Schuegerl. Bioreaction Engineering. Reactions Involving Microorganisms and Cells, volume 1. John Wiley and Sons, [10] G. Freiling, V. Yurko, Inverse Sturm-Liouville Problems and their Applications, Nova Science, New York, [11] A. M. A. EL-Sayed, Z. F. A. EL-Raheem AND N. A. O. Buhalima, Eigenvalues and eigenfunctions of non-local boundary value problems of the Sturm-Liouville equation, Electronic Journal of Mathematical Analysis and Applications, Vol. 5(1 Jan. 2017, pp
10
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραThe Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points
Applied Mathematical Sciences, Vol. 3, 009, no., 6-66 The Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points A. Neamaty and E. A. Sazgar Department of Mathematics,
Διαβάστε περισσότεραCoefficient Inequalities for a New Subclass of K-uniformly Convex Functions
International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραHomomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραD Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραThe k-α-exponential Function
Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραQuadratic Expressions
Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots
Διαβάστε περισσότεραDifferential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότεραCHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραAppendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραA summation formula ramified with hypergeometric function and involving recurrence relation
South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραPROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1)
GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 995, 535-545 PROPERTIES OF CERTAIN INTEGRAL OPERATORS SHIGEYOSHI OWA Abstract. Two integral operators P α and Q α for analytic functions in the open unit disk
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραA REGULARIZED TRACE FORMULA FOR A DISCONTINUOUS STURM-LIOUVILLE OPERATOR WITH DELAYED ARGUMENT
Electronic Journal of Differential Equations, Vol. 217 217, No. 14, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu A REGULARIZED TRACE FORMULA FOR A DISCONTINUOUS
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότερα1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραES440/ES911: CFD. Chapter 5. Solution of Linear Equation Systems
ES440/ES911: CFD Chapter 5. Solution of Linear Equation Systems Dr Yongmann M. Chung http://www.eng.warwick.ac.uk/staff/ymc/es440.html Y.M.Chung@warwick.ac.uk School of Engineering & Centre for Scientific
Διαβάστε περισσότεραOn Numerical Radius of Some Matrices
International Journal of Mathematical Analysis Vol., 08, no., 9-8 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/ijma.08.75 On Numerical Radius of Some Matrices Shyamasree Ghosh Dastidar Department
Διαβάστε περισσότεραA General Note on δ-quasi Monotone and Increasing Sequence
International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in
Διαβάστε περισσότεραDifferentiation exercise show differential equation
Differentiation exercise show differential equation 1. If y x sin 2x, prove that x d2 y 2 2 + 2y x + 4xy 0 y x sin 2x sin 2x + 2x cos 2x 2 2cos 2x + (2 cos 2x 4x sin 2x) x d2 y 2 2 + 2y x + 4xy (2x cos
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραLecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραMA 342N Assignment 1 Due 24 February 2016
M 342N ssignment Due 24 February 206 Id: 342N-s206-.m4,v. 206/02/5 2:25:36 john Exp john. Suppose that q, in addition to satisfying the assumptions from lecture, is an even function. Prove that η(λ = 0,
Διαβάστε περισσότεραThe semiclassical Garding inequality
The semiclassical Garding inequality We give a proof of the semiclassical Garding inequality (Theorem 4.1 using as the only black box the Calderon-Vaillancourt Theorem. 1 Anti-Wick quantization For (q,
Διαβάστε περισσότεραA Discontinuous Sturm-Liouville Operator With Indefinite Weight
www.ccsenet.org/jmr Journal of Mathematics Research Vol., No. 3; August A Discontinuous Sturm-Liouville Operator With Indefinite Weight Qiuxia Yang (Corresponding author) Mathematics Science college, Inner
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραDiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Διαβάστε περισσότεραSOME PROPERTIES OF FUZZY REAL NUMBERS
Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραOn the k-bessel Functions
International Mathematical Forum, Vol. 7, 01, no. 38, 1851-1857 On the k-bessel Functions Ruben Alejandro Cerutti Faculty of Exact Sciences National University of Nordeste. Avda. Libertad 5540 (3400) Corrientes,
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραParametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραMemoirs on Differential Equations and Mathematical Physics
Memoirs on Differential Equations and Mathematical Physics Volume 31, 2004, 83 97 T. Tadumadze and L. Alkhazishvili FORMULAS OF VARIATION OF SOLUTION FOR NON-LINEAR CONTROLLED DELAY DIFFERENTIAL EQUATIONS
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραThe Pohozaev identity for the fractional Laplacian
The Pohozaev identity for the fractional Laplacian Xavier Ros-Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya (joint work with Joaquim Serra) Xavier Ros-Oton (UPC) The Pohozaev
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραTrigonometric Formula Sheet
Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ
Διαβάστε περισσότεραGreen s Function for the Light Scattering Equations
Bulletin of TICMI Vol. 2, No. 1, 216, 25 31 b Green s Function for the Light Scattering Equations Dazmir Shulaia a,b and Giorgi Makatsaria c a Georgian Technical University 77 M. Kostava St., 175, Tbilisi,
Διαβάστε περισσότεραPalestine Journal of Mathematics Vol. 2(1) (2013), Palestine Polytechnic University-PPU 2013
Palestine Journal of Matheatics Vol. ( (03, 86 99 Palestine Polytechnic University-PPU 03 On Subclasses of Multivalent Functions Defined by a Multiplier Operator Involving the Koatu Integral Operator Ajad
Διαβάστε περισσότεραThe k-bessel Function of the First Kind
International Mathematical Forum, Vol. 7, 01, no. 38, 1859-186 The k-bessel Function of the First Kin Luis Guillermo Romero, Gustavo Abel Dorrego an Ruben Alejanro Cerutti Faculty of Exact Sciences National
Διαβάστε περισσότεραCommutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραn=2 In the present paper, we introduce and investigate the following two more generalized
MATEMATIQKI VESNIK 59 (007), 65 73 UDK 517.54 originalni nauqni rad research paper SOME SUBCLASSES OF CLOSE-TO-CONVEX AND QUASI-CONVEX FUNCTIONS Zhi-Gang Wang Abstract. In the present paper, the author
Διαβάστε περισσότεραTrigonometry 1.TRIGONOMETRIC RATIOS
Trigonometry.TRIGONOMETRIC RATIOS. If a ray OP makes an angle with the positive direction of X-axis then y x i) Sin ii) cos r r iii) tan x y (x 0) iv) cot y x (y 0) y P v) sec x r (x 0) vi) cosec y r (y
Διαβάστε περισσότεραEvaluation of some non-elementary integrals of sine, cosine and exponential integrals type
Noname manuscript No. will be inserted by the editor Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type Victor Nijimbere Received: date / Accepted: date Abstract
Διαβάστε περισσότεραOn a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότεραA Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering
Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix
Διαβάστε περισσότεραF19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Διαβάστε περισσότερα