This paper examines the solution of a damped beam equation whose damping characteristics are well-defined by the fractional derivative (FD). Homotopy Analysis Method (HAM) is applied for calculating the dynamic response (DR). Unit step and unit impulse functions are deliberated for this analysis. Acquired results are illustrated to show the movement of the beam under various sets of parameters ...

Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adju...

The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotop...

A construction of various immersed Klein bottles that belong to different regular homotopy classes, and which thus cannot be smoothly transformed into one another, is introduced. It is shown how these shapes can be partitioned into two Möbius bands and how the twistedness of these bands defines the homotopy type. Some wild and artistic variants of Klein bottles are presented for their aesthetic...

We introduce two powerful methods to solve the Davey-Stewartson equations: one is the homotopy perturbation method (HPM) and the other is the homotopy analysis method (HAM). HAM is a strong and easy to use analytic tool for nonlinear problems. Comparison of the HPM results with the HAM results, and compute the absolute errors between the exact solutions of the DS equations with the HPM solution...

The construction of various of Klein bottles that belong to different regular homotopy classes, and which thus cannot be smoothly transformed into one another, is formally introduced. For all cases it is shown how these shapes can be partitioned into two Moebius bands and how the twistedness of these bands defines the homotopy type. Some wild and artistic variants of Klein bottles are presented...

We compare Friedlander’s definition of étale homotopy for simplicial schemes to another definition involving homotopy colimits of pro-simplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the étale site of schemes over S to the category of pro-spaces....

We examine the “homotopy coniveau tower” for a general cohomology theory on smooth k-schemes, satisfying some natural axioms, and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. We show how these constructions lead to a tower of functors on the Morel-Voevodsky stable homotopy category, and identify this stable homotopy coniveau tower with Voevodsky’s s...