Let M and N be differentiable manifolds of dimensions k and n respectively, kN is called an immersion if / is of class C1 and the Jacobian matrix of/ has rank k at each point of M. Such a map is also called regular. Until recently, very little was known about the existence and classification of immersions of one manifold in another. The present work addresses itsel...

We show that if the cochain complex computing Ext groups (in category of modules over Hopf algebroids) admits a cocyclic structure, then noncommutative Cartan calculus structure on Tor dualises in cyclic sense to Coext Cotor. More precisely, duals chain resp. spaces two classical derived functors lead complexes compute more exotic ones, giving opposite module an operad with multiplication induc...

In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite volume methods for scalar hyperbolic conservation laws, we first discretize the interior product (also called contraction) through integrals over Eulerian approximations of extrusions. This, along with Cartan’s homotopy formula and a discret...

In the finance market, it is well known that price change of underlying fractal transmission system can be modeled with Black-Scholes equation. This article deals finding approximate analytic solutions for time-fractional equation fractional integral boundary condition a European option pricing problem in Katugampola derivative sense. It generalizes both Riemann–Liouville and Hadamard derivativ...

This paper shows how to use the fractional Sumudu homotopy perturbation technique (SHP) with Caputo operator (CF) solve time linear and nonlinear partial differential equations. The transform (ST) (HP) are combined in this approach. In definition, derivative is defined. general, method straightforward execute yields good results. There some examples offered demonstrate technique's validity use.

This paper, for the first time ever, proposes a Laplace-like integral transform, which is called as He-Laplace its basic properties are elucidated. The homotopy perturbation method coupled with this new transform becomes much effective in solving fractal differential equations. Phi-four equation He?s derivative used an example to reveal main merits of present technology.

In the present article, we use modified homotopy perturbation method (HPM) to find approximate analytical solution of three-dimensional time fractional microscale heat transport equation. This transport equation is at first converted into the equation of two functions. First one have both, the first order fractional time derivative and second order space derivatives, while second one has only s...

In this article, the fractional derivatives in the sense of Caputo, and the homotopy analysis method (HAM) are used to construct the approximate solutions for the nonlinear fractional variant Bussinesq equations with respect to time fractional derivative. The HAM contains a certain auxiliary h parameter which provides us a simple way to adjust and control the convergence region and rate of conv...

In this work, we have applied the variational iteration method and He’s polynomials to solve partial differential equation (PDEs) with time-fractional derivative. The variational homotopy perturbation iteration method (VHPIM) is presented in two steps. Some illustrative examples are given in order to show the ability and simplicity of the approach. All numerical calculations in this manuscript ...

In this paper, we propose a simple numerical algorithm for solving multi-dimensional diffusion equations of fractional order which describes density dynamics in a material undergoing diffusion by using homotopy analysis transform method. The fractional derivative is described in the Caputo sense. This homotopy analysis transform method is an innovative adjustment in Laplace transform method and...