Homotopy Asymptotic Research Articles

Semi analytical scheme for the presentation of solution to Fractional Fokker–Planck Equation

In this work, we have also proposed new semi analytical scheme combining least-squares and asymptotic homotopy perturbation methods AHPM for the solution of Fractional Fokker–Planck Equation (FFPE). The Caputo version of derivative is used. The finding of this work is fast convergence of the AHPM to the solutions of linear and nonlinear applied problems. AHPM is a modified form of the HPM and is particularly useful for solving problems with small or large parameters. This can be seen as a clear advantage of this new semi analytical method over existing methods because its implementation and the calculation involved in it are much simpler and easier. The proposed method solved the problems without using Ji-Huan He and Adomian polynomials. Numerical solutions obtained in this way show that this approach is simple and computationally very attractive. An error estimate for the solution is also provided. Approximate solutions of AHPM are tabulated and graphically displayed to show that AHPM is effective and accurate.

A study of heat and mass transfer flow of a variable viscosity couple stress fluid between inclined plates

In this paper, the Poiseuille flow of non-isothermal couple stress fluid of Reynolds model between two heated parallel inclined plates has been studied. The strongly nonlinear coupled ordinary differential equations have been investigated using the Asymptotic Homotopy Perturbation method (AHPM) and the Optimal Homotopy Asymptotic Method with DJ Polynomials (OHAM-DJ). Employing both these techniques, the approximate solutions for velocity profile, temperature distributions, shear stress, average velocity and volumetric flow rate have been achieved. The results obtained using both these methods are in the form of infinite series, hence the results can be obtained easily. Comparison of various results, obtained through these methods, is carried out. Here, it is worth to mention that the results gained by both these approaches are in excellent resemblance. The effect of different parameters on the flow problem has been studied numerically as well as presented graphically.

Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.

Approximate solution for high order fractional integro-differential equations via an optimum parameter method

The most significant objective of this article is the adoption of a method with a free parameter known as “The Optimum Asymptotic Homotopy Method” which has been utilized in order to obtain answers for integral differential equations of high-order non integer derivative.The process in this method is more favorable than “Homotopy Perturbation Method” as it has a more rapid convergence compared to the mentioned method or even the similar methods. Another advantage of this method is that the convergence rate is recognized as control area. It is worth mentioning that Caputo derivative is adopted in this article.A number of instances are provided to better understand the method and its level of efficiency compared to other same methods.

Analysis of couple stress fluid flow with variable viscosity using two homotopy-based methods

Abstract In this article, the generalized plane Couette flow of Vogel’s model of incompressible, non-isothermal, couple stress fluid flowing steadily between two parallel walls is investigated. The governing equations are reduced to ordinary differential equations. To investigate the non-linear coupled system of differential equations, the optimal homotopy asymptotic method with DJ polynomial and asymptotic homotopy perturbation method have been used. Important flow properties are presented and discussed. We have obtained expressions for velocity, average velocity, shear stress, volume flux and temperature. The results gained employing these techniques are in the form of infinite series; thus, the results can be easily calculated. Comparison of various results, obtained through the suggested approaches, is carried out and an excellent agreement is achieved.

Approximate solutions to nonlinear factional order partial differential equations arising in ion-acoustic waves

An approximative procedure named asymptotic homotopy perturbation method (AHPM) is introduced to obtain solutions of the non-linear fractional order models. The two special cases, FZK$(3, 3, 3)$ and FZK$(2, 2, 2)$ of fractional Zakharov-Kuznetsov equations are chosen for the illustrative purpose of our method. AHPM is a very recent new procedure as compare with other existing homotopy perturbation procedures. A new auxiliary function has been introduced in AHPM. The AHPM solutions are compared with solutions of fractional complex transform FCT using variational iteration method VIM and exact solutions. Further, the surface graph of AHPM solutions are compared with surface graph of solutions of homotopy perturbation transform method (HPTM) solutions. In comparison, the solutions computed by AHPM are in agreement with exact solutions of the problems. The simulation section reveals that our new developed procedure is effective and explicit.

An optimal method for approximating the delay differential equations of noninteger order

The main purpose of this paper is to use a method with a free parameter, named the optimal asymptotic homotopy method (OHAM), in order to obtain the solution of delay differential equations, delay partial differential equations, and a system of coupled delay equations featuring fractional derivative. This method is preferable to others since it has faster convergence toward homotopy perturbation method, and the convergence rate can be set as a controlled area. Various examples are given to better understand the use of this method. The approximate solutions are compared with exact solutions as well.

On the asymptotic homotopy type of inductive limitC *-algebras

Let X, Y be compact, connected, metrisable spaces with base points Xo, Yo and let denote the compact operators. It is shown that Co(X\xo)| is asymptotically homotopic (or shape equivalent) to Co(Y~yo) @ Yi r if and only if X and Y have isomorphic K-groups. Similar results are obtained for certain inductive limits of nuclear C*-algebras. Let ~r denote the category whose objects are all the separable C*-algebras and whose set of morphisms from A to B, denoted [[A, B]], consists of homotopy classes of asymptotic morphisms. The construction of this category is due to Connes and Higson [CH], who defined a bivariant homology theory E(A, B) = lISA, SB | ~ff]] and have shown how to define the intersection product for arbitrary extensions of separable C*-algebras. If A is K-nuclear then E-theory agrees with Kasparov's bivariant K-theory [K]. On the other hand the asymptotic homotopy category d appears to be the "right" framework for the homotopy theory of separable C*-algebras. This point of view is supported by results in [H; CH; D; CHI; CK; D1]. For instance we have shown in [D1] that asymptotic homotopy is equivalent to a strong shape theory and hence is intimately related to the shape theories of [EK1] and [B] which were intended as homotopy theories for noncommutative singular spaces. In particular it turned out that two separable C*-algebras are shape equivalent if and only if they are asymptotically homotopic i.e. isomorphic in ~r The isomorphism class in sr of a separable C*-algebra A is called the asymptotic homotopy type of A. In this note we exhibit large classes of (projeetionless) stable, nuclear C*algebras whose asymptotic homotopy type is determined by K-theoretical data (Theorem 6). This is done via a suspension isomorphism

Asymptotic homotopy cycles for flows and Π₁ de Rham theory

We define the asymptotic homotopy of trajectories of flows on closed manifolds. These homotopy cycles take values in the 2 2 -step nilpotent Lie group which is associated to the fundamental group by means of Malcev completion. The cycles are an asymptotic limit along the orbit of the product integral of a Lie algebra valued 1 1 -form. Propositions 5.1-5.7 show how the formal properties of our theory parallel the properties of the asymptotic homology cycles of Sol Schwartzman. In particular, asymptotic homotopy is an invariant of topological conjugacy, and, in certain cases, of topological equivalence. We compute the asymptotic homotopy of those measure-preserving flows on Heisenberg manifolds which lift from the torus T 2 {T^2} (Theorem 8.1), and then show how this invariant distinguishes up to topological equivalence certain of these flows which are indistinguishable homologically (Theorem 9.1). We also compute the asymptotic homotopy of those geodesic flows for Heisenberg manifolds which come from left invariant metrics on the Heisenberg group (Example 8.1), and then show how this invariant distinguishes up to topological conjugacy certain of these flows which are indistinguishable homologically.