Fibonacci Differential Equation and Associated Spiral Curves

The Fibonacci differential equation is defined with analogy from the Fibonacci difference equation. The linear second order differential equation is solved for suitable initial conditions. The solutions constitute spirals in the polar coordinates. The properties of the spirals with respect to the Fibonacci numbers and the differences between the new spirals and classical spirals are discussed

Elastik Yatak Üzerine Yerleştirilmiş Eğri Mikro Kirişin 2:1 İç Rezonansları

Konferans Bildirisi -- Teorik ve Uygulamalı Mekanik Türk Milli Komitesi, 2013Conference Paper -- Theoretical and Applied Mechanical Turkish National Committee, 2013Bu çalışmada, ideal olmayan sınır şartlarına sahip hafif eğrilikli rezonans mikro kirişin dinamik davranışı araştırılmıştır. Eğri mikro kirişin uçları basit basit mesnetlenmiştir ve kiriş nonlineer elastik yatak üzerine yerleştirilmiştir. Küçük AC yükünden dolayı eğri mikro kirişin zorlamalı titreşim tepkisi çok ölçekli metodun (perturbasyon metodu) direk uygulanması ile analitik olarak elde edilmiştir. Titreşimin iki modu arasında 2:1 iç rezonansları çalışılmıştır. Genlik ve faz denklemleri elde edilmiştir. Düzgün rejim çözümleri ve çözümlerin kararlılığı tartışılmış ve genlik-faz modülasyon denklemlerinin bifürkasyon analizi sunulmuştur. İdeal olmayan sınır şartlarının sistemin titreşimine etkileri araştırılmıştır.In this study, the dynamic behaviour of a slightly curved resonant microbeam having non-ideal boundary conditions is investigated. The ends of the curved microbeam are on immovable simple supports and the microbeam is resting on a non-linear elastic foundation. The forced vibration response of curved microbeam due to the small AC load is obtained analytically by means of direct application of the method of multiple scales (a perturbation method). Two-to-one internal resonances between any two modes of vibration are studied. Amplitude and phase modulation equations are obtained. Steady state solutions and stability are discussed, and a bifurcation analysis of the amplitude and phase modulation equations are presented. The effects of the non ideal boundary conditions on the vibrations of the microbeam are examined

Kuvvetli Nonlineer Sistemler İçin Çok Ölçekli Lindstedt Poincare Tekniği

Konferans Bildirisi -- Teorik ve Uygulamalı Mekanik Türk Milli Komitesi, 2013Conference Paper -- Theoretical and Applied Mechanical Turkish National Committee, 2013Çok ölçekli metot ve Lindstedt-Poincare tekniğinin birleştirilmesi esasına dayanan yeni bir perturbasyon metodu ortaya atılmıştır. Yeni metot lineer sönümlü osilatör, Duffing denklemi, sönümlü kübik nonlineer denklem, kuadratik ve kübik nonlineer denklem ve zorlamalı Duffing denklemine uygulanmıştır. Klasik çok ölçekli metot ve yeni metodu kullanarak yaklaşık analitik çözümler elde edilmiştir. Bu çözümler ana denklemin sayısal çözümü ile karşılaştırılmıştır. Yeni metot kuvvetli nonlineer sistemler için çok iyi sonuçlar vermiştir.A new perturbation method combining the Method of Multiple Scales and Lindstedt Poincare techniques is proposed. The new method is applied to Linear damped oscillator, Duffing equation, damped cubic nonlinear equation, an equation with quadratic and cubic nonlinearities and forced Duffing equation. Approximate analytical solutions are obtained using the classical Multiple scales method and the new method. Both solutions are contrasted with the direct numerical solutions of the original equation. The new method produces much better results for strong nonlinearities

Boundary Layer Equations and Lie Group Analysis of a Sisko Fluid

Boundary layer equations are derived for the Sisko fluid. Using Lie group theory, a symmetry analysis of the equations is performed. A partial differential system is transferred to an ordinary differential system via symmetries. Resulting equations are numerically solved. Effects of non-Newtonian parameters on the solutions are discussed

Application of the perturbation iteration method to boundary layer type problems

The recently developed perturbation iteration method is applied to boundary layer type singular problems for the first time. As a preliminary work on the topic, the simplest algorithm of PIA(1,1) is employed in the calculations. Linear and nonlinear problems are solved to outline the basic ideas of the new solution technique. The inner and outer solutions are determined with the iteration algorithm and matched to construct a composite expansion valid within all parts of the domain. The solutions are contrasted with the available exact or numerical solutions. It is shown that the perturbation–iteration algorithm can be effectively used for solving boundary layer type problems

REVIEW OF THE NEW PERTURBATION-ITERATION METHOD

The new perturbation iteration method developed by Pakdemirli and coworkers are reviewed. First, applications of the method to algebraic equations are discussed and some new root-finding algorithms developed by this method are given. Next, the applications of the new method to first order, second order and systems of first order ordinary differential equations are discussed. Three sample problems are selected. Results are compared with analytical solutions, solutions by other methods and numerical solutions. The new perturbation iteration method is an effective method that does not require small parameter assumption as in classical perturbation methods

A New Perturbation Approach to Optimal Polynomial Regression

A new approach to polynomial regression is presented using the concepts of orders of magnitudes of perturbations. The data set is normalized with the maximum values of the data first. The polynomial regression of arbitrary order is then applied to the normalized data. Theorems for special properties of the regression coefficients as well as some criteria for determining the optimum degrees of the regression polynomials are posed and proven. The new approach is numerically tested, and the criteria for determining the best degree of the polynomial for regression are discussed

Group Classification and Some Similarity Solutions for a Nonlinear Filtration Equation

A nonlinear filtration equation in which the filter coefficient is an arbitrary function of the specific deposit is considered. Lie Group theory is applied to the coupled system of partial differential equations. Group classification is performed with respect to the arbitrary filter coefficient. Some similarity solutions are constructed using the symmetries