scholarly journals Inverse properties of a class of seven-diagonal (near) Toeplitz matrices

2021 ◽  
Vol 10 (1) ◽  
pp. 67-86
Author(s):  
Bakytzhan Kurmanbek ◽  
Yogi Erlangga ◽  
Yerlan Amanbek
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Order Differential Equation ◽  
Toeplitz Matrices ◽  
Inverse Matrix ◽  
Fourth Order Differential Equation

Abstract This paper presents the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence explicit inverse formula is derived using the Sherman-Morrison formula. Related to the fixed-point iteration used to solve the differential equation, we show the positivity of the inverse matrix and construct an upper bound for the norms of the inverse matrix, which can be used to predict the convergence of the method.

scholarly journals Numerical solutions of the flexure equation

2021 ◽  
Author(s):  
David Hindle ◽  
Olivier Besson
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Order Differential Equation ◽  
Foreland Basin ◽  
Isostatic Rebound ◽  
Continuous Plate ◽  
Thickness Parameter ◽  
Fourth Order Elliptic Equations

Abstract. The 4th order differential equation describing elastic flexure of the lithosphere is one of the cornerstones of geodynamics, key to understanding topography, gravity, glacial isostatic rebound, foreland basin evolution and a host of other phenomena. Despite being fully formulated in the 1940’s, a number of significant issues concerning the basic equation have remained overlooked to this day. We first explain the different fundamental forms the equation can take and their difference in meaning and solution procedures. We then show how numerical solutions to flexure problems in general as they are currently formulated, are potentially unreliable in an unpredictable manner for cases where the coefficient of rigidity varies in space due to variations of the elastic thickness parameter. This is due to fundamental issues related to the numerical discretisation scheme employed. We demonstrate an alternative discretisation that is stable and accurate across the broadest conceivable range of conditions and variations of elastic thickness, and show how such a scheme can simulate conditions up to and including a completely broken lithosphere more usually modelled as an end loaded, single, continuous plate. Importantly, our scheme will allow breaks in plate interiors, allowing for instance, the creation of separate blocks of lithosphere which can also share the support of loads. The scheme we use has been known for many years, but remains rarely applied or discussed. We show that it is generally the most suitable finite difference discretisation of fourth order, elliptic equations of the kind describing many phenomena in elasticity, including the problem of bending of elastic beams. We compare the earlier discretisation scheme to the new one in 1 dimensional form, and also give the 2 dimensional discretisation based on the new scheme.We also describe a general issue concerning the numerical stability of any second order finite difference discretisation of a fourth order differential equation like that describing flexure where contrasting magnitudes of coefficients of different summed terms lead to round off problems which in turn destroy matrix positivity. We explain the use of 128 bit, floating point storage for variables to mitigate this issue.

scholarly journals Fixed point results under generalized c-distance with application to nonlinear fourth-order differential equation

2019 ◽  
Vol 20 (2) ◽  
pp. 635-648 ◽  
Author(s):  
Ghasem Soleimani Rad ◽  
◽  
Hamidreza Rahimi ◽  
Calogero Vetro ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fourth Order ◽  
Order Differential Equation ◽  
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.

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