We introduce nabla type Laplace transform and Sumudu transform on general time scale. We investigate the properties and the applicability of these integral transforms and their efficiency in solving fractional dynamic equations on time scales.

The main aim of this work is to present a user friendly numerical algorithm based on homotopy perturbation Sumudu transform method for nonlinear fractional partial differential arising in spatial diffusion of biological populations in animals. The movements are made generally either by mature animals driven out by invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. The homotopy perturbation Sumudu transform method is a combined form of the Sumudu transform method and homotopy perturbation method. The obtained results are compared with Sumudu decomposition method. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally very attractive.

We introduce some vector-valued sequence spaces defined by a Musielak-Orlicz function and the concepts of lacunary convergence and strong ($A$)-convergence, where $A=(a_{ik})$ is an infinite matrix of complex numbers. We also make an effort to study some topological properties and some inclusion relations between these spaces.

We are interested in the following fractional boundary value problem: $D^{\alpha }u(t)+a\left(t\right)u^{\sigma }=0$, $t\in (0,\infty )$, ${\lim }_{t\to 0}{t}^{2-\alpha }u(t)=0$, ${\lim }_{t\to \infty }{t}^{1-\alpha }u(t)=0$, where , $\sigma \in (-\mathrm{1,1})$, $D^{\alpha }$ is the standard Riemann-Liouville fractional derivative, and $a$ is a nonnegative continuous function on $(0,\infty )$ satisfying some appropriate assumptions related to Karamata regular variation theory. Using the Schauder fixed point theorem, we prove the existence and the uniqueness of a positive solution. We also give a global behavior of such solution.

This work introduces new functions based on the spherical harmonics and the solid harmonics which have been used to construct a multipole development for the 3D Stokes problem in order to reduce the operations costs in the BEM method. We show that the major properties of those functions are inherited from the solid harmonics. The contribution of this paper is the introduction of new formulas that serve to calculate the multipole moments and the transfer functions that are necessary for the schemes of order $O(N\text{log}N)$. Moreover, new translation formulas are introduced to obtain an $O(N)$ scheme. The error truncation of the resulting scheme is discussed. In comparison to the BEM that attains a limit storage at $O(10^4)$, we present here a method based on FMM-BEM that attains a storage at a limit of $O(10^6)$. The implementation of the method achieves a high accuracy level at a reasonable cost.

The purpose of this paper is to introduce new classes of generalized seminormed difference sequence spaces defined by a Musielak-Orlicz function. We also study some topological properties and prove some inclusion relations between resulting sequence spaces.

The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.

We develop a method to obtain approximate solutions for nonlinear systems of Volterra integrodifferential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled Volterra integrodifferential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples and results of the present technique have close agreement with approximate solutions which were obtained with the help of Adomian decomposition method (ADM).

A numerical method for the modified time fractional Fokker-Planck equation is proposed. Stability and convergence of the method are rigorously discussed by means of the Fourier method. We prove that the difference scheme is unconditionally stable, and convergence order is $O(\tau +h^{\mathrm{4}})$, where $\tau$ and $h$ are the temporal and spatial step sizes, respectively. Finally, numerical results are given to confirm the theoretical analysis.

We obtain the necessary and sufficient conditions for an almost conservative matrix to define a compact operator. We also establish some necessary and sufficient (or only sufficient) conditions for operators to be compact for matrix classes $(f,X)$, where $X=c,c_0,l_{\infty }$. These results are achieved by applying the Hausdorff measure of noncompactness.