scholarly journals Lyapunov-Schmidt Method in Bifurcation Solutions of Nonlinear Fourth Order Differential Equation

2019 ◽  
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Schmidt Method ◽  
Fourth Order Differential Equation

scholarly journals Asymptotic theory for a critical case for a general fourth-order differential equation

1998 ◽  
Vol 21 (3) ◽  
pp. 479-488
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Critical Case ◽  
Asymptotic Theory ◽  
Order Differential Equation ◽  
Fourth Order Equations ◽  
Fourth Order Differential Equation

In this paper we identify a relation between the coefficients that represents a critical case for general fourth-order equations. We obtained the forms of solutions under this critical case.

25.—On the Eigenfunction Expansion associated with a Singular Complex-valued Fourth-order Differential Equation

1976 ◽  
Vol 75 (4) ◽  
pp. 325-332
Author(s):  
Jyoti Chaudhuri ◽  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Representation Theorem ◽  
Eigenfunction Expansion ◽  
Order Differential Equation ◽  
Convergence Theory ◽  
Infinite Intervals ◽  
Complex Valued ◽  
Class Of Functions ◽  
Fourth Order Differential Equation

SynopsisThe direct convergence theory of eigenfunction expansions associated with boundry value problems, not necessarily self-adjoint, generated from complex-valued fourth-order symmetric ordinary differential expressions on semi-infinite intervals, is discussed. An admissible class of functions for the expansion is characterised. Also a generalisation of Stieltjes representation theorem for analytic functions discussed in [13, §§ 22.23 and 24] is obtained.

XXIII.—On an Extension to an Integro-differential Inequality of Hardy, Littlewood and Polya

1972 ◽  
Vol 69 (4) ◽  
pp. 295-333 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Differential Inequality ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisThis paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) andThe extension consists in replacing f′ by M[f] wherechoosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the formwhere K is a positive number independent of f.The analysis involves a fourth-order differential equation and the second-order equation associated with M.A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

scholarly journals The Solution of Nonlinear Fourth-Order Differential Equation with Integral Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yanli Fu ◽  
Huanmin Yao
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Reproducing Kernel ◽  
Order Differential Equation ◽  
Integral Boundary Conditions ◽  
Reproducing Kernel Space ◽  
Integral Boundary ◽  
Derivatives Of ◽  
Fourth Order Differential Equation

An iterative algorithm is proposed for solving the solution of a nonlinear fourth-order differential equation with integral boundary conditions. Its approximate solutionun(x)is represented in the reproducing kernel space. It is proved thatun(x)converges uniformly to the exact solutionu(x). Moreover, the derivatives ofun(x)are also convergent to the derivatives ofu(x). Numerical results show that the method employed in the paper is valid.

scholarly journals Euler case for a general fourth-order differential equation

2004 ◽  
Vol 2004 (51) ◽  
pp. 2705-2717
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
General Formula ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Fourth Order Differential Equation

We deal with an Euler case for a general fourth-order equation and under this case, we obtain the general formula for the asymptotic form of the solutions.

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