Tau method approximate solution of highorder differential eigenvalue problems defined in the complex plane, with an application to orr-sommerfeld stability equation

1987 ◽  
Vol 3 (3) ◽  
pp. 187-194 ◽  
Author(s):  
K. M. Liu ◽  
E. L. Ortiz
Keyword(s):  
Approximate Solution ◽  
Complex Plane ◽  
Eigenvalue Problems ◽  
Tau Method ◽  
Stability Equation

The approximate solution of nonlinear Fredholm implicit integro-differential equation in the complex plane

2021 ◽  
Author(s):  
Samir Lemita ◽  
Sami Touati ◽  
Kheireddine Derbal
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Approximate Solution ◽  
Integral Operator ◽  
Complex Plane ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Numerical Examples ◽  
Integro Differential Equation ◽  
Numerical Process

This paper’s purpose is to study the nonlinear Fredholm implicit integro-differential equation in the complex plane, where the term implicit integro-differential means that the derivative of unknown function is founded inside of the integral operator. Initially, according to Banach fixed point theory, we ensure that the equation has a unique solution under particular conditions. However, we exhibit a numerical process based on the conjunction between Nyström and Picard methods, for the sake of approximating solutions of this equation. In addition to that, the convergence analysis of this numerical process is demonstrated, and some illustrated numerical examples are presented.

On the eigenvalue problem Tu -λ Su =0 with unbounded and nonsymetric operators T and S

1968 ◽  
Vol 262 (1130) ◽  
pp. 413-458 ◽  
Keyword(s):  
Approximate Solution ◽  
Iterative Method ◽  
Eigenvalue Problems ◽  
Generalized Method Of Moments ◽  
Elastic Stability ◽  
Theoretical Problem ◽  
Bounded Operators ◽  
Eigenvalues And Eigenvectors ◽  
Complete Section ◽  
Symmetric Operators

In this paper we study both the theoretical problem of the existence and the practical problem of the approximate calculation of eigenvalues and eigenvectors of (i) = 0, where T and S are some linear (in general unbounded and nonhermitian) operators in a Hilbert space. After a short discussion of a class of K -symmetric operators, in section 2 the author proves the existence of eigenvalues and eigenvectors of (i) under various conditions on T and S and investigates conditions under which the set of eigenvectors of (i) is complete. Section 3 indicates briefly the applicability and the unifying property of the generalized method of moments to the approximate solution of (i). Section 4 presents and thoroughly studies a very general new iterative method for the approximate solution of (i). The advantage of this method is that it does not require the practically inconvenient preliminary reduction of (i) to an equivalent problem with bounded operators and that under certain rather general conditions the convergence is mono tonic. Furthermore, by specializing operators and parameters, our iterative method contains as a special case, almost every known iterative method for the calculation of eigenvalues (mostly proved previously only for symmetric matrices and bounded operators). Finally the applicability and the numerical effectiveness of the iterative method is illustrated by calculating the smallest eigenvalue for a selfadjoint and nonselfadjoint eigenvalue problems arising in the problems of elastic stability.

scholarly journals Canonical polynomials in the problem about the bend of circular plate of variable thickness

2021 ◽  
pp. 105
Author(s):  
O.V. Bugrim ◽  
Ye.S. Sinaiskii
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Approximate Solution ◽  
Boundary Conditions ◽  
Circular Plate ◽  
Variable Thickness ◽  
Variable Coefficients ◽  
Tau Method ◽  
Canonical Polynomials ◽  
Plate Of Variable Thickness

The problem about the bend of circular plate of variable thickness under specifically selected laws of rigidity change is reduced to the ordinary differential equation with variable coefficients of polynomial kind. The construction of the approximate solution of equation that satisfies boundary conditions is realized by means of canonical polynomials and $\tau$-method of Lantzosh.

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