Reduction of two-point boundary value problems in a vector space to initial value problems by projection

1966 ◽  
Vol 8 (3) ◽  
pp. 270-289 ◽  
Author(s):  
Karl G. Guderley ◽  
Paul J. Nikolai
Keyword(s):  
Vector Space ◽  
Boundary Value Problems ◽  
Initial Value Problems ◽  
Boundary Value ◽  
Point Boundary ◽  
Initial Value

Existence and uniqueness for two-point boundary value problems

1981 ◽  
Vol 88 (3-4) ◽  
pp. 219-234 ◽  
Author(s):  
John V. Baxley ◽  
Sarah E. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Shooting Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Conditions ◽  
Point Boundary ◽  
Initial Value

SynopsisBoundary value problems associated with y″ = f(x, y, y′) for 0 ≦ x ≦ 1 are considered. Using techniques based on the shooting method, conditions are given on f(x, y,y′) which guarantee the existence on [0, 1] of solutions of some initial value problems. Working within the class of such solutions, conditions are then given on nonlinear boundary conditions of the form g(y(0), y′(0)) = 0, h(y(0), y′(0), y(1), y′(1)) = 0 which guarantee the existence of a unique solution of the resulting boundary value problem.

scholarly journals Spectral approximations to the fractional integral and derivative

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Changpin Li ◽  
Fanhai Zeng ◽  
Fawang Liu
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Caputo Derivative ◽  
Initial Value Problems ◽  
Fractional Integral ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Initial Value ◽  
Numerical Examples ◽  
Spectral Approximations

AbstractIn this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

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