A New Method for Nonlinear Two-Point Boundary Value Problems in Solid Mechanics

2001 ◽  
Vol 68 (5) ◽  
pp. 776-786 ◽  
Author(s):  
L. S. Ramachandra ◽  
D. Roy
Keyword(s):  
Boundary Value Problems ◽  
Vector Fields ◽  
Solid Mechanics ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Post Buckling ◽  
Independent Variable ◽  
Solution Parameters

A local and conditional linearization of vector fields, referred to as locally transversal linearization (LTL), is developed for accurately solving nonlinear and/or nonintegrable boundary value problems governed by ordinary differential equations. The locally linearized vector field is such that solution manifolds of the linearized equation transversally intersect those of the nonlinear BVP at a set of chosen points along the axis of the only independent variable. Within the framework of the LTL method, a BVP is treated as a constrained dynamical system, which in turn is posed as an initial value problem. (IVP) In the process, the LTL method replaces the discretized solution of a given system of nonlinear ODEs by that of a system of coupled nonlinear algebraic equations in terms of certain unknown solution parameters at these chosen points. A higher order version of the LTL method, with improved path sensitivity, is also considered wherein the dimension of the linearized equation needs to be increased. Finally, the procedure is used to determine post-buckling equilibrium paths of a geometrically nonlinear column with and without imperfections. Moreover, deflections of a tip-loaded nonlinear cantilever beam are also obtained. Comparisons with exact solutions, whenever available, and other approximate solutions demonstrate the remarkable accuracy of the proposed LTL method.

scholarly journals The Solutions of Second Order Nonlinear Two Point Boundary Value Problems

2019 ◽  
Vol 4 (8) ◽  
pp. 49-54
Author(s):  
Abdurkadir Edeo Gemeda
Keyword(s):  
Nonlinear Systems ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuation Method ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Second Order ◽  
Point Boundary ◽  
Algebraic Equations ◽  
Operational Matrix Of Differentiation

In this paper, generalized shifted Legendre polynomial approximation on a given arbitrary interval has been designed to find an approximate solution of a given second order nonlinear two point boundary value problems of ordinary differential equations. Here an approach using Tau method based on Legendre operational matrix of differentiation [2] & [5] has been addressed to generate the nonlinear systems of algebraic equations. The unknown Legendre coefficients of these nonlinear systems are the solutions of the system and they have been solved by continuation method. These unknown Legendre coefficients are then used to write the approximate solutions to the second order nonlinear two point boundary value problems. The validity and efficiency of the method has also been illustrated with numerical examples and graphs assisted by MATLAB.

scholarly journals Analytic Solution of a Class of Singular Second-Order Boundary Value Problems with Applications

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 172
Author(s):  
Hoda Ali ◽  
Elham Alali ◽  
Abdelhalim Ebaid ◽  
Fahad Alharbi
Keyword(s):  
Exact Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Second Order ◽  
Governing Equations ◽  
Special Cases ◽  
Independent Variable ◽  
Straightforward Approach ◽  
Nano Fluids

Recently, it was observed that the concentration/heat transfer of pure/nano fluids are finally governed by singular second-order boundary value problems with exponential coefficients. These coefficients were transformed into polynomials and therefore the governing equations become singular in a new independent variable. Unfortunately, the published approximate solutions in the literature suffer from some weaknesses as showed by one of the present coauthors. Hence, the exact solution for such types of problems becomes a challenge. In this paper, a straightforward approach is presented to obtaining the exact solution for such class of singular second-order boundary value problems. The results are also applied to some selected problems within the literature. Accordingly, the published solutions are recovered as special cases of the present ones.

scholarly journals An elegant operational matrix based on harmonic numbers: Effective solutions for linear and nonlinear fourth-order two point boundary value problems

2016 ◽  
Vol 21 (4) ◽  
pp. 448-464 ◽  
Author(s):  
Waleed M. Abd-Elhameed
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Collocation Methods ◽  
Point Boundary ◽  
Algebraic Equations ◽  
Harmonic Numbers ◽  
Linear And Nonlinear

This paper analyzes the solution of fourth-order linear and nonlinear two point boundary value problems. The suggested method is quite innovative and it is completely different from all previous methods used for solving such kind of boundary value problems. The method is based on employing an elegant operational matrix of derivatives expressed in terms of the well-known harmonic numbers. Two algorithms are presented and implemented for obtaining new approximate solutions of linear and nonlinear fourth-order boundary value problems. The two algorithms rely on employing the new introduced operational matrix for reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. For this purpose, the two spectral methods namely, Petrov-Galerkin and collocation methods are applied. Some illustrative examples are considered aiming to ascertain the wide applicability, validity, and efficiency of the two proposed algorithms. The obtained numerical results are satisfactory and the approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature.

scholarly journals A numerical scheme for problems in fractional calculus

2018 ◽  
Vol 20 ◽  
pp. 02001
Author(s):  
M. Razzaghi
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Boundary Value ◽  
Bernoulli Polynomials ◽  
Algebraic Equations ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Fractional Differential

In this paper, a new numerical method for solving the fractional differential equations with boundary value problems is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the boundary value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Quasilinearization and boundary value problems at resonance

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Kareem Alanazi ◽  
Meshal Alshammari ◽  
Paul Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Approximate Solutions ◽  
Monotonicity Condition ◽  
Uniqueness Of Solutions ◽  
Shift Method ◽  
At Resonance ◽  
Problem At Resonance

Abstract A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

Multidimensional Manifold Continuation for Adaptive Boundary-Value Problems

2020 ◽  
Vol 15 (5) ◽  
Author(s):  
Harry Dankowicz ◽  
Yuqing Wang ◽  
Frank Schilder ◽  
Michael E. Henderson
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
General Purpose ◽  
Solution Set ◽  
Benchmark Problems ◽  
Finite Dimensional ◽  
Predictor Corrector ◽  
Parameter Continuation ◽  
Autonomous Dynamical Systems

Abstract Parameter continuation of finitely parameterized, approximate solutions to integro-differential boundary-value problems typically involves regular adaptive updates to the number and meaning of the unknowns and/or the associated constraints. Different continuation steps produce solutions with different discretizations or to formally different sets of equations. Existing general-purpose, multidimensional continuation algorithms fail to account for such differences without significant additional coding and are therefore prone to redundant coverage of the set of solutions. We describe a new algorithm, implemented in the software package coco, which overcomes this problem by characterizing the solution set in an invariant, finite dimensional, projected geometry rather than in the space of unknowns corresponding to any particular discretization. It is in this geometry that distances between solutions and angles between tangent spaces are quantified and used to construct possible directions of outward expansion. A pointwise lift identifies such directions in the projected geometry with directions of continuation in the full set of unknowns, used by a nonlinear predictor-corrector algorithm to expand into uncharted parts of the solution set. Several benchmark problems from the analysis of periodic orbits in autonomous dynamical systems are used to illustrate the theory.

A Discretization Approach to Modeling Capacitive MEMS Filters

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Boundary Value ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Wide Range

We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.

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