Multiplicity Results of Positive Solutions for Nonlinear Three-Point Boundary Value Problems on Time Scales
|
|
- Buddy Copeland
- 5 years ago
- Views:
Transcription
1 Avances in Dynamical Systems an Applications ISSN , Volume 4, Number 2, pp (29) Multiplicity Results of Positive Solutions for Nonlinear Three-Point Bounary Value Problems on Time Scales Bin Zhao Tianshui Normal University Department of Mathematics an Statistics Tianshui, Gansu 74, P. R. China Hong-Rui Sun Lanzhou University School of Mathematics an Statistics Lanzhou, Gansu 73, P. R. China. Abstract Let T be a time scale such that, T T. In this paper, we consier the nonlinear secon-orer three-point bounary value problem u (t) + q(t)f(t, u(t), u (t)) =, t (, T ) T βu() γu () =, αu(η) = u(t ), where β, γ, β + γ >, η (, ρ(t )) T, < α < T/η, an = β(t αη) + γ( α) >. By using a fixe point theorem ue to Avery an Peterson, sufficient conitions are obtaine for the existence of three positive solutions of the above problem. The interesting point is the nonlinear term f which is involve with the first orer erivative explicitly. AMS Subject Classifications: 34B5, 39A. Keywors: Positive solution, fixe point theorem, time scales. Introuction Going back to its founer Stefan Hilger (988), the stuy of ynamic equations on time scales is a fairly new area of mathematics. Motivating the subject is the notion that Receive April 28, 28; Accepte April 3, 29 Communicate by Martin Bohner
2 244 Bin Zhao an Hong-Rui Sun ynamic equations on time scales can buil briges between ifferential equations an ifference equations. Now, the stuy of time scales theory has le to many important applications, for example, in the stuy of insect population moels, neural networks, heat transfer, quantum mechanics, epiemic, crop harvest an stock market [3, 5, 6, 8]. Atici et al present a ynamic optimization problem from economics [3], by constructing a time scale moel, they tol us time scale calculus coul allow exploration of a variety of situations in economics. Very recently, the existence problems of positive solutions for three-point bounary value problems, especially on time scales, have attracte many authors attention an concern [, 2, 5 7, 9 4]. But the nonlinear term f is not involve with the first-orer elta erivative. Many ifficulties occur when the nonlinear term f is involve with the first-orer elta erivative. In [], Sun an Li consiere the existence of positive solution of the problem u (t) + a(t)f(t, u(t)) =, t (, T ) T, βu() γu () =, αu(η) = u(t ) by using fixe point theorem in cones. The authors obtaine the existence of at least one, two an three positive solutions of the above problems. In [7], by a new fixe point theorem, Guo an Ge gave sufficient conitions for the existence of at least one solution to the three-point bounary value problem u (t) + f(t, u, u ) =, < t <, u() =, u() = αu(η). Motivate by those works, in this paper we stuy the existence of multiple positive solutions for the secon-orer nonlinear three-point ynamic equation on time scales u (t) + q(t)f(t, u(t), u (t)) =, t (, T ) T, (.) βu() γu () =, αu(η) = u(t ), (.2) where β, γ, β + γ >, η (, ρ(t )) T, < α < T/η, an = β(t αη) + γ( α) >. We assume a : [, T ] T [, ) is l-continuous, with a (t ) > for at least one t (η, T ) T ; f : [, T ] T [, ) R [, ) is continuous. We establish the existence of at least three positive solutions for bounary value problem (.) an (.2). To our best knowlege, no work has been one for (.) an (.2) on time scales by using the fixe point theorem of Avery an Peterson. 2 Preliminaries In a cone P, let φ an θ be nonnegative continuous convex functionals, ϕ a nonnegative continuous concave functional an ψ a nonnegative continuous functional. Then for
3 Positive Solutions for BVP on Time Scales 245 positive real numbers m, m 2, m 3, m 4, we efine the following convex sets: P (φ, m 4 ) = {u P φ(u) < m 4 }, P (φ, ϕ, m 2, m 4 ) = {u P m 2 ϕ(u), φ(u) m 4 }, P (φ, θ, ϕ, m 2, m 3, m 4 ) = {u P m 2 ϕ(u), θ(u) m 3, φ(u) m 4 }, an a close set R(φ, ψ, m, m 4 ) = {u P m ψ(u), φ(u) m 4 }. Lemma 2. (see [4]). Let P be a cone in a real Banach space E. Let θ an φ be nonnegative continuous convex functionals, ϕ a nonnegative continuous concave functional an ψ a nonnegative continuous functional satisfying ψ(λu) λψ(u) for λ, such that for some positive numbers M an m 4, for all u P (φ, m 4 ). Suppose that ψ(u) ψ(u), u Mφ(u) A : P (φ, m 4 ) P (φ, m 4 ) is completely continuous an there exist positive numbers m, m 2 an m 3 with m < m 2 such that (S) {u P (φ, θ, ϕ, m 2, m 3, m 4 ) ϕ(u) > m 2 } =, an ϕ(au) > m 2 for u P (φ, θ, ϕ, m 2, m 3, m 4 ); (S2) ϕ(au) > m 2 for u P (φ, ϕ, m 2, m 4 ) with θ(au) > m 3 ; (S3) / R(φ, ψ, m, m 4 ) an ψ(au) < m for u P (φ, ψ, m, m 4 ) with ψ(u) = m. Then A has at least three fixe points u, u 2, u 3 P (φ, m 4 ), such that φ(u i ) m 4 for i =, 2, 3; m 2 < ϕ(u ); m < ψ(u 2 ) with ϕ(u 2 ) < m 2 ; ψ(u 3 ) < m. 3 Main Results Now we efine the real Banach space E = C ([, T ] T ) as the set of all -ifferentiable functions with continuous -erivative on [, T ] T with the norm u = max{ u, u }, u E,
4 246 Bin Zhao an Hong-Rui Sun where u := sup{ u(t) : t [, T ] T }, u := sup{ u (t) : t [, T ] T κ}. We efine the cone P E by P = {u E : u(t) is nonnegative an concave on [, T ] T, βu() γu () =, αu(η) = u(t ) }. The following lemma will play an important role in the proof of our main results. Lemma 3.. For u P, there exists a constant M > such that max u(t) M max u (t). t [,T ] T t [,T ] T κ Proof. We first consier the case β >. In this case, by the concavity of u, we have u(t) u() tu () T max u (t). t [,T ] T κ From the bounary conition βu() γu () =, we know u() = γ β u (), thus max t [,T ] T u(t) γ β u () + T Next we consier the case β =. First, by we have ( ) γ max u (t) t [,T ] T κ β + T u(t) = u(t ) t u (s) s, max u (t). t [,T ] T κ max u(t) u(t ) + T max u (t). (3.) t [,T ] T t [,T ] T κ From the bounary conition αu(η) = u(t ) an = γ( α) >, we get ( α)u(t ) = αu(η) αu(t ) α(t η) max u (t). (3.2) t [,T ] T κ In view of (3.) an (3.2), we obtain ( ) α(t η) max u(t) t [,T ] T α + T The proof is complete. max u (t) = t [,T ] T κ ( ) T αη α max u (t). t [,T ] T κ
5 Positive Solutions for BVP on Time Scales 247 For the sake of applying Lemma 2., we let the nonnegative continuous concave functional ϕ, the nonnegative continuous convex functional θ, φ, an the nonnegative continuous functional ψ be efine on the cone P by ϕ(u) = min u(t), φ(u) = max u (t), ψ(u) = θ(u) = max u(t), u P. t [,T ] T κ t [,T ] T For convenience, we introuce the following notations. { β T S = max (T s) q(s) s, q(s) s + αβ η } (η s) q(s) s, L = min{, a}(βη + γ) η (T s) q(s) s, N = βt + γ { α(t η) r = min T αη, αη T, η T }, a = (T s) q(s) s, min{, α}(t η)(βηt + γη), b = at 2 + a [ β(t 2 αη 2 ) + γ( α)σ() ] T +a [ γ(β(t 2 αη 2 ) γ(t αη)σ() ] + r, c = (T + σ(t )) + β(t 2 αη 2 ) + γ( α)σ(). Theorem 3.2. If there exist positive numbers m, m 2 an m 4 with m < m 2 m 4 /c such that (C) f(t, u, v) m 4 /S for (t, u, v) [, T ] T [, Mm 4 ] [ m 4, m 4 ]; (C2) f(t, u, v) > m 2 /L for (t, u, v) [η, T ] T [m 2, bm 2 ] [ m 4, m 4 ]; (C3) f(t, u, v) < m /N for (t, u, v) [, T ] T [, m ] [ m 4, m 4 ], then the problem (.) an (.2) has at least three positive solutions u, u 2, u 3 satisfying max u i (t) m 4 for i =, 2, 3; t [,T ] T κ m 2 < min u (t); m < max t [,T ] T u 2 (t), with min u 2 (t) < m 2 ; max t [,T ] T u 3 (t) < m.
6 248 Bin Zhao an Hong-Rui Sun Proof. Define an operator A : P E by Au (t) = t (t s) q(s)f(s, u(s), u (s)) s + βt + γ α(βt + γ) η (η s) q(s)f(s, u(s), u (s)) s. From [, Lemma 2.2], we have Au P. By a stanar argument, it is easy to see that A : P P is completely continuous. We now show that all the conitions of Lemma 2. are satisfie. Firstly, we show that if (C) is satisfie, then A : P (φ, m 4 ) P (φ, m 4 ). (3.3) For u P (φ, m 4 ), we have φ(u) = max t [,T ] T κ u (t) m 4. With Lemma 3., we have max u(t) Mm 4. Then conition (C) implies that f(t, u(t), u (t)) m 4 /S for t [,T ] T t [, T ] T. On the other han, for u P, we have Au P. Then T u is concave on [, T ] T, so we have γ(au) = max (Au) (t) t [,T ] T κ = max { (Au) (), (Au) (T ) } { β T = max αβ η (η s) q(s)f(s, u(s), u (s)) s, q(s)f(s, u(s), u (s)) s + β αβ η } (η s) q(s)f(s, u(s), u (s)) s max { β, q(s)f(s, u(s), u (s)) s + αβ m { 4 β T S max (T s) q(s) s, = m 4. Therefore, (3.3) hols. η } (η s) q(s)f(s, u(s), u (s)) s q(s) s + αβ η } (η s) q(s) s
7 Positive Solutions for BVP on Time Scales 249 We choose u(t) = am 2 t 2 + a [ β(t 2 αη 2 ) + γ( α)σ() ] m 2 t +a [ γ(t 2 αη 2 ) γ(t αη)σ() ] m 2 for t [, T ] T. It is easy to see that βu() γu () =, αu(η) = u(t ), u(t) an is concave on [, T ] T, so u P. Again ϕ(u) = min u(t) = min{, α}(t η)(βηt + γt + γη γσ())am 2 > m 2, θ(u) = max t [,T ] T u(t) am 2 T 2 + a [ β(t 2 αη 2 ) + γ( α)σ() ] m 2 T +a [ γ(t 2 αη 2 ) γ(t αη)σ() ] m 2 = bm 2, φ(u) [ (T + σ(t )) + β(t 2 αη 2 ) + γ( α)σ() ] m 2 = cm 2 m 4. Hence, u P (φ, θ, ϕ, m 2, bm 2, m 4 ) an {u P (φ, θ, ϕ, m 2, bm 2, m 4 ) ϕ(u) > m 2 }. For u P (φ, θ, ϕ, m 2, bm 2, m 4 ), we have m 2 u(t) bm 2 an u (t) m 4 for t [η, T ] T. Hence by conition (C2), one has that f(t, u(t), u (t)) > m 2 /L for t [η, T ] T. So by the efinition of the functional ϕ, we see that ϕ(au) = min Au(t) = min{au(η), Au(T )} ( βη + γ T = min{, α} βt + γ η ) (η s) q(s)f(s, u(s), u (s)) s ( βη + γ T > min{, α} βt + γ η ) (η s) q(s)f(s, u(s), u (s)) s T η η (βs + γ)q(s)f(s, u(s), u (s)) s ( βη + γ T = min{, α} βη + γ η ) = > min{, a}(βη + γ) min{, a}(βη + γ) η η (T s) q(s) m 2 L s = m 2.
8 25 Bin Zhao an Hong-Rui Sun Therefore, we get ϕ(au) > m 2 for u P (φ, θ, ϕ, m 2, bm 2, m 4 ), an conition (S) in Lemma 2. is satisfie. Next, it follows from [, Lemma 2.4] that for any u P (φ, ϕ, m 2, m 4 ) with θ(au) > bm 2, we have ϕ(au) = min A(u) r max A(u) rθ(au) > m 2. t [,T ] T Thus, conition (S2) of Lemma 2. is satisfie. Finally, we assert that (S3) in Lemma 2. also hols. Clearly, as ψ() = < m, so / R(φ, ψ, m, m 4 ). Assume that u P (φ, ψ, m, m 4 ) with ψ(u) = m. Then by the conition (C3), we obtain that ψ(au) = max (Au)(t) t [,T ] T t = max (t s) q(s)f(s, u(s), u (s)) s + βt + γ α(βt + γ) η (η s) q(s)f(s, u(s), u (s)) s max βt + γ T t [,T ] T < βt + γ (T s) q(s) m N s = m. Therefore, an application of Lemma 2. implies that the BVP (.) an (.2) has at least three positive solutions u, u 2 an u 3 such that max u i (t) m 4 for i =, 2, 3; m 2 < min u (t); t [,T ] T κ m < max t [,T ] T u 2 (t) with The proof is complete. min u 2 (t) < m 2 ; max u 3 (t) < m. t [,T ] T 4 Example [ Let T =, ] { }, n =, 2, 3, β =, γ = 2n 2, α = 2, T =, η = 2. Now we consier the BVP u (t) + f(t, u(t), u (t)) =, t (, ) T, (4.)
9 Positive Solutions for BVP on Time Scales 25 where u() 2 u () =, ( ) 2 u = u(), (4.2) 2 t + 3u3 + sin v, u 2, f(t, u, v) = sin v t , u > 2. It is easy to see by calculating that =, M = 2, r = 4 an S = s L = min{, 2 }( ) N = ( ) 2 s 2 s = 7 6, ( s) s = 2, ( s) s = 2. If we choose m = 4, m 2 =, m 4 = 5, then f(t, u, v) satisfies () f(t, u, v) < = m 4 S for (t, u, v) [, ] T [, 2 5 ] [ 5, 5 ]; (2) f(t, u, v) > 2 = m 2 L for ] (t, u, v) [ 2, T [, 49 3 (3) f(t, u, v) < 3 = m N for (t, u, v) [, ] T [, ] 4 ] [ 5, 5 ]; [ 5, 5 ]. Then all assumptions of Theorem 3.2 hol. Thus by Theorem 3.2, the problem (4.) an (4.2) has at least three positive solutions u, u 2 an u 3 such that max u i (t) 5 for i =, 2, 3, < min u (t); t [,] T κ t [ 2,] T 2 < max t [,] T u 2 (t) with min u 2 (t) < ; t [ 2,] T max t [,] T u 3 (t) < 2.
10 252 Bin Zhao an Hong-Rui Sun References [] Douglas R. Anerson. Solutions to secon-orer three-point problems on time scales. J. Difference Equ. Appl., 8(8): , 22. [2] F. Merivenci Atici an G. Sh. Guseinov. On Green s functions an positive solutions for bounary value problems on time scales. J. Comput. Appl. Math., 4(- 2):75 99, 22. Dynamic equations on time scales. [3] Ferhan M. Atici, Daniel C. Biles, an Alex Lebeinsky. An application of time scales to economics. Math. Comput. Moelling, 43(7-8):78 726, 26. [4] R. I. Avery an A. C. Peterson. Three positive fixe points of nonlinear operators on orere Banach spaces. Comput. Math. Appl., 42(3-5):33 322, 2. Avances in ifference equations, III. [5] Martin Bohner an Allan Peterson. Dynamic equations on time scales. Birkhäuser Boston Inc., Boston, MA, 2. An introuction with applications. [6] Martin Bohner an Allan Peterson. Avances in ynamic equations on time scales. Birkhäuser Boston Inc., Boston, MA, 23. [7] Yanping Guo an Weigao Ge. Positive solutions for three-point bounary value problems with epenence on the first orer erivative. J. Math. Anal. Appl., 29():29 3, 24. [8] Stefan Hilger. Analysis on measure chains a unifie approach to continuous an iscrete calculus. Results Math., 8(-2):8 56, 99. [9] Eric R. Kaufmann. Positive solutions of a three-point bounary-value problem on a time scale. Electron. J. Differential Equations, pages No. 82, pp. (electronic), 23. [] Hua Luo an Qiaozhen Ma. Positive solutions to a generalize secon-orer threepoint bounary-value problem on time scales. Electron. J. Differential Equations, pages no. 7, 4 pp. (electronic), 25. [] Hong-Rui Sun an Wan-Tong Li. Positive solutions for nonlinear three-point bounary value problems on time scales. J. Math. Anal. Appl., 299(2):58 524, 24. [2] Hong-Rui Sun an Wan-Tong Li. Multiple positive solutions for p-laplacian m- point bounary value problems on time scales. Appl. Math. Comput., 82():478 49, 26. [3] İsmail Yaslan. Existence of positive solutions for nonlinear three-point problems on time scales. J. Comput. Appl. Math., 26(2): , 27.
11 Positive Solutions for BVP on Time Scales 253 [4] You-Wei Zhang an Hong-Rui Sun. Three positive solutions for a generalize Sturm-Liouville multipoint BVP with epenence on the first orer erivative. Dynam. Systems Appl., 7(2):33 324, 28.
EXISTENCE OF POSITIVE SOLUTIONS FOR p-laplacian THREE-POINT BOUNDARY-VALUE PROBLEMS ON TIME SCALES
Electronic Journal of Differential Equations, Vol. 28(28, No. 92, pp. 1 14. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp EXISTENCE
More informationMultiple positive solutions for a nonlinear three-point integral boundary-value problem
Int. J. Open Problems Compt. Math., Vol. 8, No. 1, March 215 ISSN 1998-6262; Copyright c ICSRS Publication, 215 www.i-csrs.org Multiple positive solutions for a nonlinear three-point integral boundary-value
More informationPositive Solutions for p-laplacian Functional Dynamic Equations on Time Scales
International Journal of Difference Equations (IJDE. ISSN 973-669 Volume 3 Number 1 (28, pp. 135 151 Research India Publications http://www.ripublication.com/ijde.htm Positive Solutions for p-laplacian
More informationarxiv: v1 [math.ca] 31 Oct 2013
MULTIPLE POSITIVE SOLUTIONS FOR A NONLINEAR THREE-POINT INTEGRAL BOUNDARY-VALUE PROBLEM arxiv:131.8421v1 [math.ca] 31 Oct 213 FAOUZI HADDOUCHI, SLIMANE BENAICHA Abstract. We investigate the existence of
More informationNontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
Advances in Dynamical Systems and Applications ISSN 973-532, Volume 6, Number 2, pp. 24 254 (2 http://campus.mst.edu/adsa Nontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
More informationEven Number of Positive Solutions for 3n th Order Three-Point Boundary Value Problems on Time Scales K. R. Prasad 1 and N.
Electronic Journal of Qualitative Theory of Differential Equations 011, No. 98, 1-16; http://www.math.u-szeged.hu/ejqtde/ Even Number of Positive Solutions for 3n th Order Three-Point Boundary Value Problems
More informationMULTIPLE POSITIVE SOLUTIONS FOR FOURTH-ORDER THREE-POINT p-laplacian BOUNDARY-VALUE PROBLEMS
Electronic Journal of Differential Equations, Vol. 27(27, No. 23, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp MULTIPLE POSITIVE
More informationExistence Of Solution For Third-Order m-point Boundary Value Problem
Applied Mathematics E-Notes, 1(21), 268-274 c ISSN 167-251 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Existence Of Solution For Third-Order m-point Boundary Value Problem Jian-Ping
More informationMULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Fixed Point Theory, 4(23), No. 2, 345-358 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html MULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS
More informationOn some parabolic systems arising from a nuclear reactor model
On some parabolic systems arising from a nuclear reactor moel Kosuke Kita Grauate School of Avance Science an Engineering, Wasea University Introuction NR We stuy the following initial-bounary value problem
More informationResearch Article Multiple Positive Solutions for m-point Boundary Value Problem on Time Scales
Hindawi Publishing Corporation Boundary Value Problems Volume 11, Article ID 59119, 11 pages doi:1.1155/11/59119 Research Article Multiple Positive olutions for m-point Boundary Value Problem on Time cales
More informationNontrivial Solutions for Singular Nonlinear Three-Point Boundary Value Problems 1
Advances in Dynamical Systems and Applications. ISSN 973-532 Volume 2 Number 2 (27), pp. 225 239 Research India Publications http://www.ripublication.com/adsa.htm Nontrivial Solutions for Singular Nonlinear
More informationResearch Article The Existence of Countably Many Positive Solutions for Nonlinear nth-order Three-Point Boundary Value Problems
Hindawi Publishing Corporation Boundary Value Problems Volume 9, Article ID 575, 8 pages doi:.55/9/575 Research Article The Existence of Countably Many Positive Solutions for Nonlinear nth-order Three-Point
More informationOn Two-Point Riemann Liouville Type Nabla Fractional Boundary Value Problems
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 13, Number 2, pp. 141 166 (2018) http://campus.mst.edu/adsa On Two-Point Riemann Liouville Type Nabla Fractional Boundary Value Problems
More informationTRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT BOUNDARY-VALUE PROBLEMS
Electronic Journal of Differential Equations, Vol. 24(24), No. 6, pp. 8. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) TRIPLE POSITIVE
More informationResearch Article New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales
Hindawi Publishing Corporation Boundary Value Problems Volume 20, Article ID 567054, 4 pages doi:0.55/20/567054 Research Article New Fixed Point Theorems of Mixed Monotone Operators and Applications to
More informationEXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO HIGHER-ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 212 (212), No. 234, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationA Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix
More informationPOSITIVE SOLUTIONS FOR NONLOCAL SEMIPOSITONE FIRST ORDER BOUNDARY VALUE PROBLEMS
Dynamic Systems and Applications 5 6 439-45 POSITIVE SOLUTIONS FOR NONLOCAL SEMIPOSITONE FIRST ORDER BOUNDARY VALUE PROBLEMS ERBIL ÇETIN AND FATMA SERAP TOPAL Department of Mathematics, Ege University,
More informationPOSITIVE SOLUTIONS TO SINGULAR HIGHER ORDER BOUNDARY VALUE PROBLEMS ON PURELY DISCRETE TIME SCALES
Communications in Applied Analysis 19 2015, 553 564 POSITIVE SOLUTIONS TO SINGULAR HIGHER ORDER BOUNDARY VALUE PROBLEMS ON PURELY DISCRETE TIME SCALES CURTIS KUNKEL AND ASHLEY MARTIN 1 Department of Mathematics
More informationCOMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED TIME SCALES. Hüseyin Tuna
Indian J. Pure Appl. Math., 47(3): 535-544, September 2016 c Indian National Science Academy DOI: 10.1007/s13226-016-0196-1 COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED
More informationResearch Article On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition
Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 378686, 9 pages doi:10.1155/2011/378686 Research Article On the Existence of Solutions for Dynamic Boundary Value
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationLie symmetry and Mei conservation law of continuum system
Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive
More informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems - Moelling - Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationPositive solutions of BVPs for some second-order four-point difference systems
Positive solutions of BVPs for some second-order four-point difference systems Yitao Yang Tianjin University of Technology Department of Applied Mathematics Hongqi Nanlu Extension, Tianjin China yitaoyangqf@63.com
More informationExistence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions
Electronic Journal of Qualitative Theory of Differential Equations 212, No. 18, 1-9; http://www.math.u-szeged.hu/ejqtde/ Existence and iteration of monotone positive solutions for third-order nonlocal
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationNONLINEAR THREE POINT BOUNDARY VALUE PROBLEM
SARAJEVO JOURNAL OF MATHEMATICS Vol.8 (2) (212), 11 16 NONLINEAR THREE POINT BOUNDARY VALUE PROBLEM A. GUEZANE-LAKOUD AND A. FRIOUI Abstract. In this work, we establish sufficient conditions for the existence
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationOPTIMAL CONTROL OF A PRODUCTION SYSTEM WITH INVENTORY-LEVEL-DEPENDENT DEMAND
Applie Mathematics E-Notes, 5(005), 36-43 c ISSN 1607-510 Available free at mirror sites of http://www.math.nthu.eu.tw/ amen/ OPTIMAL CONTROL OF A PRODUCTION SYSTEM WITH INVENTORY-LEVEL-DEPENDENT DEMAND
More information. ISSN (print), (online) International Journal of Nonlinear Science Vol.6(2008) No.3,pp
. ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.6(8) No.3,pp.195-1 A Bouneness Criterion for Fourth Orer Nonlinear Orinary Differential Equations with Delay
More informationPOSITIVE SOLUTIONS FOR A SECOND-ORDER DIFFERENCE EQUATION WITH SUMMATION BOUNDARY CONDITIONS
Kragujevac Journal of Mathematics Volume 41(2) (2017), Pages 167 178. POSITIVE SOLUTIONS FOR A SECOND-ORDER DIFFERENCE EQUATION WITH SUMMATION BOUNDARY CONDITIONS F. BOUCHELAGHEM 1, A. ARDJOUNI 2, AND
More informationMonotone Iterative Method for a Class of Nonlinear Fractional Differential Equations on Unbounded Domains in Banach Spaces
Filomat 31:5 (217), 1331 1338 DOI 1.2298/FIL175331Z Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Monotone Iterative Method for
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationEXISTENCE OF POSITIVE SOLUTIONS OF A NONLINEAR SECOND-ORDER BOUNDARY-VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 215 (215), No. 236, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF POSITIVE
More informationPositive solutions for singular third-order nonhomogeneous boundary value problems with nonlocal boundary conditions
Positive solutions for singular third-order nonhomogeneous boundary value problems with nonlocal boundary conditions Department of Mathematics Tianjin Polytechnic University No. 6 Chenglin RoadHedong District
More informationAgmon Kolmogorov Inequalities on l 2 (Z d )
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department,
More informationAsymptotic estimates on the time derivative of entropy on a Riemannian manifold
Asymptotic estimates on the time erivative of entropy on a Riemannian manifol Arian P. C. Lim a, Dejun Luo b a Nanyang Technological University, athematics an athematics Eucation, Singapore b Key Lab of
More informationState-Space Model for a Multi-Machine System
State-Space Moel for a Multi-Machine System These notes parallel section.4 in the text. We are ealing with classically moele machines (IEEE Type.), constant impeance loas, an a network reuce to its internal
More informationChaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena
Chaos, Solitons an Fractals (7 64 73 Contents lists available at ScienceDirect Chaos, Solitons an Fractals onlinear Science, an onequilibrium an Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationGLOBAL DYNAMICS OF THE SYSTEM OF TWO EXPONENTIAL DIFFERENCE EQUATIONS
Electronic Journal of Mathematical Analysis an Applications Vol. 7(2) July 209, pp. 256-266 ISSN: 2090-729X(online) http://math-frac.org/journals/ejmaa/ GLOBAL DYNAMICS OF THE SYSTEM OF TWO EXPONENTIAL
More informationINVERSE PROBLEM OF A HYPERBOLIC EQUATION WITH AN INTEGRAL OVERDETERMINATION CONDITION
Electronic Journal of Differential Equations, Vol. 216 (216), No. 138, pp. 1 7. ISSN: 172-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu INVERSE PROBLEM OF A HYPERBOLIC EQUATION WITH AN
More informationSturm-Liouville Theory
LECTURE 5 Sturm-Liouville Theory In the three preceing lectures I emonstrate the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series are just the tip of the iceberg of the theory
More informationMULTIPLE POSITIVE SOLUTIONS FOR DYNAMIC m-point BOUNDARY VALUE PROBLEMS
Dynamic Systems and Applications 17 (2008) 25-42 MULTIPLE POSITIVE SOLUTIONS FOR DYNAMIC m-point BOUNDARY VALUE PROBLEMS ILKAY YASLAN KARACA Department of Mathematics Ege University, 35100 Bornova, Izmir,
More informationGlobal Attractivity of a Higher-Order Nonlinear Difference Equation
International Journal of Difference Equations ISSN 0973-6069, Volume 5, Number 1, pp. 95 101 (010) http://campus.mst.edu/ijde Global Attractivity of a Higher-Order Nonlinear Difference Equation Xiu-Mei
More informationFractional Geometric Calculus: Toward A Unified Mathematical Language for Physics and Engineering
Fractional Geometric Calculus: Towar A Unifie Mathematical Language for Physics an Engineering Xiong Wang Center of Chaos an Complex Network, Department of Electronic Engineering, City University of Hong
More informationBoundary Value Problems For A Differential Equation On A Measure Chain
Boundary Value Problems For A Differential Equation On A Measure Chain Elvan Akin Department of Mathematics and Statistics, University of Nebraska-Lincoln Lincoln, NE 68588-0323 eakin@math.unl.edu Abstract
More informationA global Implicit Function Theorem without initial point and its applications to control of non-affine systems of high dimensions
J. Math. Anal. Appl. 313 (2006) 251 261 www.elsevier.com/locate/jmaa A global Implicit Function Theorem without initial point an its applications to control of non-affine systems of high imensions Weinian
More informationarxiv: v1 [math.ap] 4 Sep 2007
EXISENCE OF POSIIVE SOLUIONS FOR NON LOCAL p-laplacian HERMISOR PROBLEMS ON IME SCALES MOULAY RCHID SIDI AMMI AND DELFIM F. M. ORRES Abstract. We make use of the Guo-Krasnoselskii fixed point theorem on
More informationGLOBAL SOLUTIONS FOR 2D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
Electronic Journal of Differential Equations, Vol. 015 015), No. 99, pp. 1 14. ISSN: 107-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu ftp eje.math.txstate.eu GLOBAL SOLUTIONS FOR D COUPLED
More informationExistence Results for Semipositone Boundary Value Problems at Resonance
Advances in Dynamical Systems and Applications ISSN 973-531, Volume 13, Number 1, pp. 45 57 18) http://campus.mst.edu/adsa Existence Results for Semipositone Boundary Value Problems at Resonance Fulya
More informationSINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES
Communications on Stochastic Analysis Vol. 2, No. 2 (28) 289-36 Serials Publications www.serialspublications.com SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES
More informationCalculus of Variations
16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t
More informationExistence of a Positive Solution for a Right Focal Discrete Boundary Value Problem
Existence of a Positive Solution for a Right Focal Discrete Boundary Value Problem D. R. Anderson 1, R. I. Avery 2, J. Henderson 3, X. Liu 3 and J. W. Lyons 3 1 Department of Mathematics and Computer Science,
More informationPOSITIVE SOLUTIONS OF A NONLINEAR THREE-POINT BOUNDARY-VALUE PROBLEM. Ruyun Ma
Electronic Journal of Differential Equations, Vol. 1998(1998), No. 34, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) POSITIVE SOLUTIONS
More informationOn the Inclined Curves in Galilean 4-Space
Applie Mathematical Sciences Vol. 7 2013 no. 44 2193-2199 HIKARI Lt www.m-hikari.com On the Incline Curves in Galilean 4-Space Dae Won Yoon Department of Mathematics Eucation an RINS Gyeongsang National
More informationPositive Solutions of a Third-Order Three-Point Boundary-Value Problem
Kennesaw State University DigitalCommons@Kennesaw State University Faculty Publications 7-28 Positive Solutions of a Third-Order Three-Point Boundary-Value Problem Bo Yang Kennesaw State University, byang@kennesaw.edu
More informationDissipative numerical methods for the Hunter-Saxton equation
Dissipative numerical methos for the Hunter-Saton equation Yan Xu an Chi-Wang Shu Abstract In this paper, we present further evelopment of the local iscontinuous Galerkin (LDG) metho esigne in [] an a
More informationExponential Tracking Control of Nonlinear Systems with Actuator Nonlinearity
Preprints of the 9th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August -9, Exponential Tracking Control of Nonlinear Systems with Actuator Nonlinearity Zhengqiang
More informationImpulsive Stabilization of certain Delay Differential Equations with Piecewise Constant Argument
International Journal of Difference Equations ISSN0973-6069, Volume3, Number 2, pp.267 276 2008 http://campus.mst.edu/ijde Impulsive Stabilization of certain Delay Differential Equations with Piecewise
More informationarxiv: v1 [math-ph] 5 May 2014
DIFFERENTIAL-ALGEBRAIC SOLUTIONS OF THE HEAT EQUATION VICTOR M. BUCHSTABER, ELENA YU. NETAY arxiv:1405.0926v1 [math-ph] 5 May 2014 Abstract. In this work we introuce the notion of ifferential-algebraic
More informationPeriodic Solutions in Shifts δ ± for a Dynamic Equation on Time Scales
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 9, Number 1, pp. 97 108 (2014) http://campus.mst.edu/adsa Periodic Solutions in Shifts δ ± for a Dynamic Equation on Time Scales Erbil
More informationOscillation criteria for second-order half-linear dynamic equations on time scales
P a g e 46 Vol.10 Issue 5(Ver 1.0)September 2010 Global Journal of Science Frontier Research Oscillation criteria for second-order half-linear dynamic equations on time scales Zhenlai Han a,b, Tongxing
More information2 GUANGYU LI AND FABIO A. MILNER The coefficient a will be assume to be positive, boune, boune away from zero, an inepenent of t; c will be assume con
A MIXED FINITE ELEMENT METHOD FOR A THIRD ORDER PARTIAL DIFFERENTIAL EQUATION G. Li 1 an F. A. Milner 2 A mixe finite element metho is escribe for a thir orer partial ifferential equation. The metho can
More informationDiscrete Operators in Canonical Domains
Discrete Operators in Canonical Domains VLADIMIR VASILYEV Belgoro National Research University Chair of Differential Equations Stuencheskaya 14/1, 308007 Belgoro RUSSIA vlaimir.b.vasilyev@gmail.com Abstract:
More informationPositive Solution of a Nonlinear Four-Point Boundary-Value Problem
Nonlinear Analysis and Differential Equations, Vol. 5, 27, no. 8, 299-38 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/nade.27.78 Positive Solution of a Nonlinear Four-Point Boundary-Value Problem
More informationGlobal Attractivity in a Higher Order Nonlinear Difference Equation
Applied Mathematics E-Notes, (00), 51-58 c ISSN 1607-510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Global Attractivity in a Higher Order Nonlinear Difference Equation Xing-Xue
More informationThree symmetric positive solutions of fourth-order nonlocal boundary value problems
Electronic Journal of Qualitative Theory of Differential Equations 2, No. 96, -; http://www.math.u-szeged.hu/ejqtde/ Three symmetric positive solutions of fourth-order nonlocal boundary value problems
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More informationBOUNDEDNESS AND EXPONENTIAL STABILITY OF SOLUTIONS TO DYNAMIC EQUATIONS ON TIME SCALES
Electronic Journal of Differential Equations, Vol. 20062006, No. 12, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp BOUNDEDNESS
More informationMethod of Lyapunov functionals construction in stability of delay evolution equations
J. Math. Anal. Appl. 334 007) 1130 1145 www.elsevier.com/locate/jmaa Metho of Lyapunov functionals construction in stability of elay evolution equations T. Caraballo a,1, J. Real a,1, L. Shaikhet b, a
More informationAbstract A nonlinear partial differential equation of the following form is considered:
M P E J Mathematical Physics Electronic Journal ISSN 86-6655 Volume 2, 26 Paper 5 Receive: May 3, 25, Revise: Sep, 26, Accepte: Oct 6, 26 Eitor: C.E. Wayne A Nonlinear Heat Equation with Temperature-Depenent
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson
JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises
More informationOn classical orthogonal polynomials and differential operators
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 38 2005) 6379 6383 oi:10.1088/0305-4470/38/28/010 On classical orthogonal polynomials an ifferential
More informationA COMBUSTION MODEL WITH UNBOUNDED THERMAL CONDUCTIVITY AND REACTANT DIFFUSIVITY IN NON-SMOOTH DOMAINS
Electronic Journal of Differential Equations, Vol. 2929, No. 6, pp. 1 14. ISSN: 172-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu ftp eje.math.txstate.eu A COMBUSTION MODEL WITH UNBOUNDED
More informationA Simple Exchange Economy with Complex Dynamics
FH-Kiel Universitf Alie Sciences Prof. Dr. Anreas Thiemer, 00 e-mail: anreas.thiemer@fh-kiel.e A Simle Exchange Economy with Comlex Dynamics (Revision: Aril 00) Summary: Mukherji (999) shows that a stanar
More informationGlobal Solutions to the Coupled Chemotaxis-Fluid Equations
Global Solutions to the Couple Chemotaxis-Flui Equations Renjun Duan Johann Raon Institute for Computational an Applie Mathematics Austrian Acaemy of Sciences Altenbergerstrasse 69, A-44 Linz, Austria
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More informationOn the number of isolated eigenvalues of a pair of particles in a quantum wire
On the number of isolate eigenvalues of a pair of particles in a quantum wire arxiv:1812.11804v1 [math-ph] 31 Dec 2018 Joachim Kerner 1 Department of Mathematics an Computer Science FernUniversität in
More informationA NONLINEAR NEUTRAL PERIODIC DIFFERENTIAL EQUATION
Electronic Journal of Differential Equations, Vol. 2010(2010), No. 88, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu A NONLINEAR NEUTRAL
More informationANALYSIS OF NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS
International Journal of Analysis and Applications ISSN 229-8639 Volume 7, Number (205), 79-95 http://www.etamaths.com ANALYSIS OF NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS JAGAN MOHAN JONNALAGADDA
More informationMartin Luther Universität Halle Wittenberg Institut für Mathematik
Martin Luther Universität alle Wittenberg Institut für Mathematik Weak solutions of abstract evolutionary integro-ifferential equations in ilbert spaces Rico Zacher Report No. 1 28 Eitors: Professors of
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More informationarxiv: v1 [math.dg] 1 Nov 2015
DARBOUX-WEINSTEIN THEOREM FOR LOCALLY CONFORMALLY SYMPLECTIC MANIFOLDS arxiv:1511.00227v1 [math.dg] 1 Nov 2015 ALEXANDRA OTIMAN AND MIRON STANCIU Abstract. A locally conformally symplectic (LCS) form is
More informationMULTIPLE POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES
Fixed Point Theory, 5(24, No. 2, 475-486 http://www.math.ubbcluj.ro/ nodeacj/fptcj.html MULTIPLE POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES
More informationConservation laws a simple application to the telegraph equation
J Comput Electron 2008 7: 47 51 DOI 10.1007/s10825-008-0250-2 Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness
More informationExponential asymptotic property of a parallel repairable system with warm standby under common-cause failure
J. Math. Anal. Appl. 341 (28) 457 466 www.elsevier.com/locate/jmaa Exponential asymptotic property of a parallel repairable system with warm stanby uner common-cause failure Zifei Shen, Xiaoxiao Hu, Weifeng
More informationA nonlinear inverse problem of the Korteweg-de Vries equation
Bull. Math. Sci. https://oi.org/0.007/s3373-08-025- A nonlinear inverse problem of the Korteweg-e Vries equation Shengqi Lu Miaochao Chen 2 Qilin Liu 3 Receive: 0 March 207 / Revise: 30 April 208 / Accepte:
More informationComputing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions
Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5
More informationPositive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations
J. Math. Anal. Appl. 32 26) 578 59 www.elsevier.com/locate/jmaa Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations Youming Zhou,
More informationNontrivial solutions for fractional q-difference boundary value problems
Electronic Journal of Qualitative Theory of Differential Equations 21, No. 7, 1-1; http://www.math.u-szeged.hu/ejqtde/ Nontrivial solutions for fractional q-difference boundary value problems Rui A. C.
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationOptimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations
Optimize Schwarz Methos with the Yin-Yang Gri for Shallow Water Equations Abessama Qaouri Recherche en prévision numérique, Atmospheric Science an Technology Directorate, Environment Canaa, Dorval, Québec,
More informationChapter 2 Lagrangian Modeling
Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie
More informationPartial Differential Equations
Chapter Partial Differential Equations. Introuction Have solve orinary ifferential equations, i.e. ones where there is one inepenent an one epenent variable. Only orinary ifferentiation is therefore involve.
More informationNonlinear Adaptive Ship Course Tracking Control Based on Backstepping and Nussbaum Gain
Nonlinear Aaptive Ship Course Tracking Control Base on Backstepping an Nussbaum Gain Jialu Du, Chen Guo Abstract A nonlinear aaptive controller combining aaptive Backstepping algorithm with Nussbaum gain
More information