scholarly journals Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method

2008 ◽  
Vol 2008 (1) ◽  
pp. 749865 ◽  
Author(s):  
Chein-Shan Liu
Keyword(s):  
Lie Group ◽  
Liouville Problem ◽  
Group Method ◽  
Lie Group Method ◽  
Sturm Liouville

scholarly journals Solutions of Sturm-Liouville Problems

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2074
Author(s):  
Upeksha Perera ◽  
Christine Böckmann
Keyword(s):  
Boundary Conditions ◽  
Lie Group ◽  
Arbitrary Order ◽  
Liouville Problem ◽  
Higher Order ◽  
Group Method ◽  
Magnus Expansion ◽  
Arbitrary Boundary ◽  
Lie Group Method ◽  
Sturm Liouville

This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm–Liouville problems. Next, a concrete implementation to the inverse Sturm–Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm–Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm–Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.

scholarly journals Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 544 ◽  
Author(s):  
Upeksha Perera ◽  
Christine Böckmann
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Group Method ◽  
Universal Method ◽  
Magnus Expansion ◽  
Arbitrary Boundary ◽  
Lie Group Method ◽  
Present Technique ◽  
Even Order ◽  
Sturm Liouville

In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.

scholarly journals Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 224
Author(s):  
Ghaylen Laouini ◽  
Amr M. Amin ◽  
Mohamed Moustafa
Keyword(s):  
Lie Group ◽  
Traveling Wave ◽  
Invariant Solution ◽  
Traveling Wave Solutions ◽  
Group Method ◽  
Lie Group Method ◽  
Reduced Equations ◽  
Wave Solutions ◽  
Negative Order ◽  
Petviashvili Equation

A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.

scholarly journals Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Winter Sinkala
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Second Order ◽  
Point Transformation ◽  
Group Method ◽  
Lie Group Method ◽  
Generic Solution ◽  
Point Transformations

Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs). There are various characterisations of such ODEs. We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach with three examples.

scholarly journals Solving Nonlinear Differential Algebraic Equations by an Implicit Lie-Group Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Chein-Shan Liu
Keyword(s):  
Closed Form ◽  
Lie Group ◽  
Iterative Scheme ◽  
Differential Algebraic Equations ◽  
Group Method ◽  
Computational Results ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Closed Form Solutions ◽  
Lie Group Method

We derive an implicit Lie-group algorithm together with the Newton iterative scheme to solve nonlinear differential algebraic equations. Four numerical examples are given to evaluate the efficiency and accuracy of the new method when comparing the computational results with the closed-form solutions.

scholarly journals Lie-group method for unsteady flows in a semi-infinite expanding or contracting pipe with injection or suction through a porous wall

2006 ◽  
Vol 197 (2) ◽  
pp. 465-494 ◽  
Author(s):  
Youssef Z. Boutros ◽  
Mina B. Abd-el-Malek ◽  
Nagwa A. Badran ◽  
Hossam S. Hassan
Keyword(s):  
Lie Group ◽  
Unsteady Flows ◽  
Porous Wall ◽  
Group Method ◽  
Lie Group Method
Sign in / Sign up

Export Citation Format

Share Document