simple two point boundary value problems

... Two Point Boundary Value Problems desired boundary value required boundary value x Figure 17.0.1 Shooting method (schematic) Trial integrations that satisfy the boundary condition at one endpoint ... There are two distinct classes of numerical methods for solving two point boundary value problems In the shooting method (§17.1) we choose values for all of the dependent variables at one boundary ... 756 Chapter 17 Two Point Boundary Value Problems Problems Reducible to the Standard Boundary Problem dyi (x) = gi (x, y1 , , yN , λ) dx (17.0.4) and one has to satisfy N + boundary conditions...

Ngày tải lên: 24/12/2013, 12:16

... 758 Chapter 17 Two Point Boundary Value Problems Now, as far as Newton-Raphson is concerned, we are nearly in business We want to find a vector value of V that zeros the vector value of F We this ... integrations This illustrates the enormous extra effort involved in solving two point boundary value problems compared with intial value problems If the differential equations are linear, then only one ... odeint(y,nvar,x1,x2,EPS,h1,hmin,&nok,&nbad,derivs,rkqs); score(x2,y,f); free_vector(y,1,nvar); 760 Chapter 17 Two Point Boundary Value Problems 17.2 Shooting to a Fitting Point yi (x1 ) = yi (x1 ; V(1)1 , , V(1)n2 ) i = 1, , N (17.2.1)...

Ngày tải lên: 24/12/2013, 12:16

... f[]) Routine for use with newt to solve a two point boundary value problem for nvar coupled ODEs by shooting from x1 and x2 to a fitting point xf Initial values for the nvar ODEs at x1 (x2) are ... n1] Once again a sample program illustrating shooting to a fitting point is given in §17.4 762 Chapter 17 Two Point Boundary Value Problems for (i=1;i

Ngày tải lên: 24/12/2013, 12:16

... N unknowns The remaining N equations come from the boundary conditions At the first boundary we have 764 Chapter 17 Two Point Boundary Value Problems where Sj,n = ∂Ej,M +1 , ∂yn,M n = 1, 2, ... a value for the average correction err by summing the absolute value of all corrections, weighted by a scale factor appropriate to each type of variable: 768 Chapter 17 Two Point Boundary Value ... email to trade@cup.cam.ac.uk (outside North America) X X X X X X X X 766 Chapter 17 Two Point Boundary Value Problems V V V A A A (b ) 0 S S S S 0 S S V V V S S S Figure 17.3.3 Reduction process...

Ngày tải lên: 24/12/2013, 12:16

... Chapter 17 Two Point Boundary Value Problems x = −1) Note that the boundary condition is actually applied a distance dx from the boundary to avoid having to evaluate y2 right on the boundary The ... solution near the singular points x = ±1 Substituting a power series expansion of the form 774 Chapter 17 Two Point Boundary Value Problems and similarly for Smn The boundary conditions on (17.4.4) ... email to trade@cup.cam.ac.uk (outside North America) At x = we have two conditions: 776 Chapter 17 Two Point Boundary Value Problems Now recall that the matrix of partial derivatives Si,j of equation...

Ngày tải lên: 21/01/2014, 18:20

... 129ff 17.6 Handling Internal Boundary Conditions or Singular Points Singularities can occur in the interiors of two point boundary value problems Typically, there is a point xs at which a derivative ... 784 Chapter 17 Two Point Boundary Value Problems where φ(x) is chosen by us Written in terms of the mesh variable q, this equation ... term forces more grid points to be used where y is changing rapidly The constants act to make every logarithmic change in y of an amount δ about as “attractive” to a grid point as a change in...

Ngày tải lên: 21/01/2014, 18:20

... Two Point Boundary Value Problems where xs is the unknown location of the singularity, and change the independent variable to t by setting x − x1 = tz, 0≤t≤1 (17.6.4) D(x, y) = (17.6.5) The boundary ... possible to integrate away from the point Both the relaxation method and the method of “shooting” to a fitting point handle such problems easily Also, in those problems the presence of singular ... overall problem contains equations with boundary conditions at the first point, one “internal” boundary condition, and two final boundary conditions The figure shows the structure of the overall matrix...

Ngày tải lên: 21/01/2014, 18:20

... m -point boundary value problems, ” Boundary Value Problems, vol 2008, Article ID 254593, 10 pages, 2008 14 Z Yang, “Existence and uniqueness of positive solutions for higher order boundary value ... of a fourth-order boundary value problem,” Applied Mathematics and Computation, vol 168, no 2, pp 1219–1231, 2005 R Ma, “Nodal solutions for a fourth-order two- point boundary value problem,” Journal ... order boundary value problem,” Journal of Computational and Applied Mathematics, vol 10, no 2, pp 203–217, 1984 R Ma and H P Wu, “Positive solutions of a fourth-order two- point boundary value...

Ngày tải lên: 21/06/2014, 16:20

... second-order two- point boundary value problems with nonlinearities across several eigenvalues J Appl Math Lett 18, 587–595 (2005) [4] Rabinowitz, PH: Some global results for nonlinear eigenvalue problems ... study the existence and multiplicity of nodal solutions of the nonlinear second-order two- point boundary value problems, u + f (t, u) = 0, t ∈ (0, 1), u(0) = u(1) = The proof of our main results ... Nodal solutions of second-order two- point boundary value problems Ruyun Ma, Bianxia Yang and Guowei Dai∗ Department of Mathematics, Northwest...

Ngày tải lên: 21/06/2014, 17:20

... Stochastics and Stochastics Reports, vol 37, no 1-2, pp 61–74, 1991 [4] Z Wu, “One kind of two- point boundary value problems associated with ordinary equations and application,” Journal of Shandong University, ... has a unique fixed point (x(·), p(·)) that is a solution of (3.2) Therefore, (3.2) has a solution for any β ∈ [β0 ,β0 + δ] The proof of the lemma is complete 8 Boundary Value Problems Lemma 3.2 ... that such a δ > can be chosen independently of β0 Then J is a contraction in this case 6 Boundary Value Problems Case (α1 = and α2 > 0) Apply the variation of constants formula to the equation...

Ngày tải lên: 21/06/2014, 22:20

... singular boundary value problems for ordinary differential equations, Tbilisi University Press, Tbilisi, 1975 , Some optimal conditions for the solvability of two- point singular boundary value problems, ... T Kiguradze, On a singular boundary value problem, Journal of Mathematical Analysis and Applications 30 (1970), no 3, 475–489 , On a singular multi -point boundary value problem, Annali di Matematica ... 259–281, Functional differential equations and applications (Beer-Sheva, 2002) , On two- point boundary value problems for higher order singular ordinary differential [7] equations, Georgian Academy...

Ngày tải lên: 23/06/2014, 00:20

... of m -point boundary value problems with nonlinear growth,” Journal of Mathematical Analysis and Applications, vol 212, no 2, pp 467–480, 1997 C P Gupta, “A generalized multi -point boundary value ... nonlinear boundary value problems, ” Journal of Mathematical Analysis and Applications, vol 294, no 1, pp 104–112, 2004 Z Du, X Lin, and W Ge, “On a third-order multi -point boundary value problem ... “Solvability of multi -point boundary value problem at resonance III,” Applied Mathematics and Computation, vol 129, no 1, pp 119–143, 2002 B Liu, “Solvability of multi -point boundary value problem at...

Ngày tải lên: 21/06/2014, 22:20

... supported points, which leads into a standard twopoint boundary value condition and large size bridges are sometimes contrived with multipoint supports, which corresponds to a multi -point boundary value ... multi -point boundary value problem [3] Many problems in the theory of elastic stability can be handled by the method of multi -point problems [4] Small size bridges are often designed with two supported ... solutions of boundary value problems for p-Laplacian dynamic equations on time scales J Math Anal Appl 321, 911–920 (2006) 16 Hong, S: Triple positive solutions of three -point boundary value problems...

Ngày tải lên: 20/06/2014, 20:20

... article as: Alsaedi and Aqlan: On nonlocal three -point boundary value problems of Duffing equation with mixed nonlinear forcing terms Boundary Value Problems 2011 2011:47 Page 11 of 11 ... x) − H(t, y) ≥ −L(x − y), x ≥ y, x, y ∈ R, L > Alsaedi and Aqlan Boundary Value Problems 2011, 2011:47 http://www.boundaryvalueproblems.com/content/2011/1/47 Page of 11 A quasilinearization technique ... and upper solutions of (1.1-1.2), respectively Alsaedi and Aqlan Boundary Value Problems 2011, 2011:47 http://www.boundaryvalueproblems.com/content/2011/1/47 Page of 11 (A2) N Ỵ C[J × ℝ, ℝ] be...

Ngày tải lên: 20/06/2014, 22:20

... a superlinear two point boundary value problem Topol Methods Nonlinear Anal 6, 175–188 (1995) Ma, R, Thompson, B: Multiplicity results for second-order two- point boundary value problems with ... Rynne pointed out that Ma et al Boundary Value Problems 2011, 2011:14 http://www.boundaryvalueproblems.com/content/2011/1/14 Page of 11 ν a If u ∈ Tk , then u has exactly one zero between each two ... nonlinear eigenvalue problems J Funct Anal 7(3), 487–513 (1971) doi:10.1016/ 0022-1236(71)90030-9 Page 10 of 11 Ma et al Boundary Value Problems 2011, 2011:14 http://www.boundaryvalueproblems.com/content/2011/1/14...

Ngày tải lên: 21/06/2014, 01:20

... m -point boundary- value problems, ” Electronic Journal of Differential Equations, vol 2004, no 89, pp 1–14, 2004 S Jingxian, X Xian, and D O’Regan, “Nodal solutions for m -point boundary value problems ... m -point boundary value problems, ” Boundary Value Problems, vol 2008, Article ID 254593, 10 pages, 2008 R Ma and Y An, “Global structure of positive solutions for superlinear second order m -point ... m -point, boundary value problems, ” Nonlinear Analysis: Theory, Methods & Applications, vol 67, no 12, pp 3318–3327, 2007 R Ma and D O’Regan, “Nodal solutions for second-order m -point boundary value...

Ngày tải lên: 21/06/2014, 11:20

... three point boundary value problems on time scales,” Applied Mathematics Letters, vol 18, no 5, pp 577–585, 2005 21 A C Peterson, Y N Raffoul, and C C Tisdell, “Three point boundary value problems ... to boundary value problems on time scales,” Advances in Difference Equations, vol 2004, no 2, pp 93–109, 2004 12 C C Tisdell and H B Thompson, “On the existence of solutions to boundary value problems ... solutions of boundary value problems BVPs on time scales has attracted the attention of many authors; see, for example, 5–12 and the references therein for the existence theory of some two- point BVPs,...

Ngày tải lên: 21/06/2014, 16:20

... solutions for fourth-order multi -point boundary value problems Boundary Value Problems 2011, 21 (2011) 32 Ma, R: Existence of solutions of nonlinear m -point boundary value problem J Math Anal Appl ... involving two fractional orders with boundary value conditions Boundary Value Problems 2011, 17 (2011) doi:10.1155/2011/516481 29 Chen, SH, Hu, T, Chen, L: Existence results for n -point boundary value ... this technique is similar to that of the two- step procedure proposed in [18,20] for the initial boundary value problem and in [21] for the two- point boundary value problem To the best of the our...

Ngày tải lên: 21/06/2014, 17:20

... some second-order four -point boundary- value problems Nonlinear Anal 60, 491–500 (2004) [4] Yang, L, Liu, X, Jia, M: Multiplicity results for second-order m -point boundary value problem J Math Anal ... development of boundary value problems for differential equations [1–5], dif- ference equations [6, 7], and the theory of time scales [8–12], the existence of solutions for boundary value problems on ... solutions for a nonlinear three -point boundary value problem Electron J Diff Equ 34, 1–8 (1999) [2] Ma, R, Cataneda, N: Existence of solution for nonlinear m -point boundary value problem J Math Anal...

Ngày tải lên: 21/06/2014, 17:20