euler apos s method for initial value problems

... y, tend ); This code is used in chunks 20, 35, 40, and 68 25 We check if an integration is successful by calling Solver successful ( ): check if success 25 ≡ if (¬Solver successful ( )) cout ... contains descriptions of test cases Various listings are given in Chapter 6 Chapter Introduction Chapter Installation In this Chapter, we list the utilities and packages necessary for installing ... which derivatives of the solution y(t) exist to some order; that is, y(t) is sufficiently smooth As a consequence, the code list of f should not contain functions such as branches, abs, or In practice,...

Ngày tải lên: 12/01/2014, 22:07

... 1 s, t≤T r s f s, ys T m max G1 t, s r s f s, ys M t∈ 1,T s m Φy , M us us 3.15 Boundary Value Problems 13 Lemma 3.4 Suppose that (H1 ) holds Then Φ : K → K is completely continuous We assume that ... respectively Thus, 1,T 3.14 It follows from the definition of K that T Φy t G1 t, s r s f s, ys t∈ 1,T t∈ 1,T ≥m us s T r s f s, ys us s ≥ ≥ which implies that Φ K ⊂ K m T Ms max G1 t, s 1 s, t≤T ... well-posedness of BVP 1.9 , since the function f depends on the term ut i.e., past values of u As usual, a sequence {u −τ , , u T } is said to be a positive solution of BVP 1.9 if it satisfies...

Ngày tải lên: 21/06/2014, 11:20

... it suffices to restrict our consideration to orthogonal projections onto KerAn , as was done in [3] However, in the mentioned paper, a singularvalue decomposition (SVD) of An is employed for constructing ... DAEs and linear index IDEs is the fact that the pencil {A(t),B(t)} in the continuous case is always of index for all t ∈ J, while for n 1, {An ,Bn } is not necessarily of index Now, we describe ... (3.3) is uniquely solvable n It can be shown that the explicit Euler method applied to nonlinear transferable DAEs [4] leads to nonlinear index IDEs This and other problems related to connections...

Ngày tải lên: 23/06/2014, 00:20

... Since t /s > 1, it is obvious that as1 m s i=1 since asi m s i=1 t /s asm m s i=1 + ··· + as1 m s i=1 t /s < + ··· + asm m s i=1 = 1, ∈ (0, 1) It follows that ( at1 m s t /s i=1 ) + ··· + atm m s ... assumption that at least one of the sets Ω, Ω1 , , Ωm is bounded What remains to be shown is the uniqueness of this solution We consider two cases In the first case, we assume that m ¯ Observe i=1 (Ω ... also assume that at least one of the sets Ω, Ω1 , , Ωm is bounded which guarantees the existence of an optimal solution; see [16] These are our standing assumptions for the remainder of this section...

Ngày tải lên: 14/10/2015, 08:03

... Mathematics Student, vol 63, no 1–4, pp 123–145, 1994 18 T Suzuki, “Strong convergence of Krasnoselskii and Mann s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” ... obtaining the result that limn → ∞ xn − However, many authors have used Suzuki s lemma 18 for obtaining the result that xn in the process of studying the similar algorithms For example, see limn → ... Applications, vol 143, no 1, pp 37–58, 2009 S. -S Chang, Y J Cho, and J K Kim, “Approximation methods of solutions for equilibrium problem in Hilbert spaces,” Dynamic Systems and Applications, vol...

Ngày tải lên: 21/06/2014, 11:20

... equations of second order is proved by studying a full discretization The article is organized as follows In Section 2, we specify some hypotheses, precise sense of the weak solution, then we state ... effects Several problems of thermoelasticity and viscoelasticity can also be reduced to this type of problems A variety of problems arising in mechanics, elasticity theory, molecular dynamics, and ... uniqueness of solution as well as some regularity results are obtained via Rothe–Galerkin method Keywords: Rothe s method; a priori estimate; integrodifferential equation; Galerkin method; weak solution...

Ngày tải lên: 21/06/2014, 17:20

... Arts RMUTR Research Fund and King Mongkut s Diamond scholarship for fostering special academic skills by KMUTT The second author was supported by the Thailand Research Fund and the Commission ... converges strongly to some z ∈ F S ∩ V I C, B Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, 21–24 and the references ... Applications Remark 1.1 It is obvious that any β-inverse-strongly monotone mapping B is monotone and 1/β-Lipschitz continuous An operator A is strongly positive on H if there exists a constant γ...

Ngày tải lên: 21/06/2014, 20:20

... Applications 14 S Takahashi and W Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol ... Φ is nonempty and proved a strong convergence theorem In 2003, Iusem and Sosa 10 presented some iterative algorithms for solving equilibrium problems in finite-dimensional spaces They have also ... Lipschitz continuous A mapping S : C → C is called nonexpansive if Sx − Sy ≤ x − y 1.6 for all x, y ∈ C We denote by F S the set of fixed points of S, that is, F S {x ∈ C : x S x } If C ⊂ H is...

Ngày tải lên: 21/06/2014, 20:20

... Applications inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases; see, for example, 1–5 Some methods have been proposed to solve the equilibrium problems, see, ... following: i Sr is single-valued; ii Sr is nonexpansive if K is Lipschitz continuous with constant ν > such that σ ≥ λν and K x1 − K x2 , η u1 , u2 where ui iii Fix Sr Sr xi for i Ω; iv Ω is closed and ... nonexpansive mappings, the quasi-nonexpansive mappings and the strictly pseudocontractive mappings as special cases Let us also recall that T is called demiclosed if for any sequence {xn } ⊂ H and x...

Ngày tải lên: 21/06/2014, 20:20

... difference equations Here this condition connects the history x0 with the single value Δx(0) This is suggested by the well posedness of the BVP (1.6), since the function f depends on the terms xt and x(t) ... it is easy to see that BVP (1.6) has two positive solutions S J Yang et al 11 Proof of Theorem 2.6 By (3.1)–(3.4) and Lemma 1.1, it is easy to see that BVP (1.6) has three positive solutions An ... m-Laplacian boundary value problems, ” Applied Mathematics Letters, vol 12, no 3, pp 11–17, 1999 [7] S J Yang, B Shi, and M J Gai, “Boundary value problems for functional differential systems,” Indian Journal...

Ngày tải lên: 22/06/2014, 19:20

... forms of the phase function for some typical cases in both SAR and ISAR systems (1) For nonmoving targets in SAR systems and for constant velocity targets in ISAR systems, the phase function can ... (2) So, the received signal is a 2D complex sinusoid (2) Phase function of moving targets in the SAR systems is analyzed in [1] Similar results are observed in some ISAR systems with uniform ... ISAR targets with fast and 3D maneuvers can produce phase function of the form SIGNAL MODEL In both SAR and ISAR systems series of signals is transmitted toward radar target Commonly, these signals...

Ngày tải lên: 22/06/2014, 19:20

... see, for example, [4–6] and the references therein A mapping S of C into itself is called nonexpansive if Su − Sv ≤ u − v , (1.4) for all u,v ∈ C We denote by F (S) the set of fixed points of S ... of S For finding an element of F (S) ∩ Ω under the assumption that a set C ⊂ H is closed and convex, a mapping S of C into itself is nonexpansive and a mapping A of C into H is α-inverse strongly ... − xn − as n −→ ∞, Syn − xn+1 ≤ Syn − Stn + Stn − xn+1 ≤ yn − tn + αn Stn − u + βn Stn − xn ≤ yn − tn + αn Stn − u + βn Stn − Sxn + βn Sxn − xn ≤ yn − tn + αn Stn − u + βn tn − xn + βn Sxn − xn...

Ngày tải lên: 22/06/2014, 22:20

... corresponding to μ Travis [7] established some comparison results for the smallest eigenvalues of two eigenvalue problems for boundary value problems of 2nth-order linear differential equations, ... papers mentioned above, the comparison results are for the smallest eigenvalues only The purpose of this paper is to establish the existence and comparison theorems for all the eigenvalues of ... eigenvalues of the problems (1.3) and (1.4) We will also prove the existence of positive eigenvectors corresponding to the smallest eigenvalues of the problems Eigenvalue comparisons In this section, we...

Ngày tải lên: 22/06/2014, 22:20

... pressures Most of the numerical methods developed for fluid dynamics problems can be classified into the mesh-based and meshless methods Among these methods, a Lagrangian meshless method (also ... reviewed These methods can be broadly categorized into mesh-based methods and meshless methods 1.2 Mesh-based methods for fluid-mechanics problems A common feature of mesh-based methods is that the ... are some other SPH studies that impose the incompressible condition strictly by solving a pressure Poisson equation (PPE) This category of SPH is termed as the incompressible SPH (i.e ISPH) First...

Ngày tải lên: 09/09/2015, 08:11

... Acknowledgements First of all, I would like to express my sincere gratitude to my supervisor, Professor Sun Defeng for all of his guidance, encouragement and support In the past two years, Professor Sun has ... its convergence properties Chapter concentrates on the semismooth Newton-CG method for solving the inner problems and the convergence analysis In Chapter 5, we discuss some implementation issues ... following key issues: i) the target matrix in (1.2) is positive semidefinite; ii) the target matrix in (1.2) is asked to satisfy some prescribed constraints on its components Thus, the problem...

Ngày tải lên: 26/09/2015, 09:39

... Dang Duc Trong for his most helpful comments and discussion on this paper References [1] K A Ames, R J Hughes; Structural Stability for Ill-Posed Problems in Banach Space, Semigroup Forum, Vol 70 ... Stabilized quasi-reversibility and other nearly-best-possible methods for non-well posed problems, Symposium on Non-Well Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, ... obtained Some numerical tests illustrate that the proposed method is feasible and effective Keywords and phrases: Backward heat problem, Ill-posed problem, Quasi-boundary value methods, Quasi-Reversibility...

Ngày tải lên: 17/12/2017, 16:43

... , h ω · As ∞ sds M h s sds h s ∞ M sds h M ∞, 3.4 so h is a Nagumo function Furthermore, there exists a constant N depending on α, β, and Nagumo function h such that M sds h s ≥ N sds > max β ... homogeneous problem associated with 1.1 - 1.2 has only the trivial solution Therefore, by Green s function method, the solution of 1.1 - 1.2 can be written as m−2 G t, s f s, x s , x s ds x t τi ... Boundary Value Problems References G S Ladde, V Lakshmikantham, and A S Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure...

Ngày tải lên: 21/06/2014, 11:20

... tau methods Spectral methods have been applied successfully to numerical simulations of many problems in science and engineering, see [12–15] Spectral tau method is similar to Galerkin methods ... upon its current state but also upon all of its historical states This makes studying fractional order systems an active area of research Spectral methods are a widely used tool in the solution ... representation of the solution must be chosen This may be done by polynomials interpolation of the solution based on some suitable nodes such as the well known Gauss or Gauss-Lobatto nodes The second...

Ngày tải lên: 21/06/2014, 17:20

... |Γ dt 2.57 fT1 un − Tm un dxdt Q s Setting S1 s T t − Tm t dt, then ≤ S1 s ≤ |s| sign0 |s| − m , for all s ∈ R, where 1 if s > 0, sign0 s if s ≤ Thus we get sign0 s N aij x, un Dj un Di un dxdt ... 1–7 For linear equations, the existence, uniqueness of solution to the corresponding problem are well understood cf 1– , for the purpose, the Galerkin method was used For semilinear equations, ... York, NY, USA, 1990 Boundary Value Problems 23 14 R Landes, “On the existence of weak solutions for quasilinear parabolic initial- boundary value problems, ” Proceedings of the Royal Society of...

Ngày tải lên: 21/06/2014, 20:20

... possesses at least a nontrivial solution Proof It is easy to see that α ≡ is a subsolution of I Denote 1− β r r | s| 1/ p s −1 −1 s ds, 3.46 where μ is a positive constant satisfying β R see ... results on this case are rare There are many papers on the existence of solutions for p-Laplacian boundary value problems via subsuper solution method see 20–24 But results on the sub-super-solution ... sub-super-solution method for p x -Laplacian equations and systems are rare In this paper, when p r is a general function, we establish several sub-super-solution theorems for the existence of solutions for...

Ngày tải lên: 22/06/2014, 03:20