Sadia Arshad

Fuzzy fractional integral equations under compactness type condition

In this paper we study a fuzzy fractional integral equation. The fractional derivative is considered in the sense of Riemann-Liouville and we establish existence of the solutions of fuzzy fractional integral equations using the Hausdorff measure of noncompactness.

A Schauder fixed point theorem in semilinear spaces and applications

In this paper we present existence and uniqueness results for a class of fuzzy fractional integral equations. To prove the existence result, we give a variant of the Schauder fixed point theorem in semilinear Banach spaces.MSC:34A07, 34A08.

Dynamical analysis of fractional order model of immunogenic tumors

In this article, we examine the fractional order model of the cytotoxic T lymphocyte response to a growing tumor cell population. We investigate the long-term behavior of tumor growth and explore the conditions of tumor elimination analytically. We establish the conditions for the tumor-free equilibrium and tumor-infection equilibrium to be asymptotically stable and provide the expression of the basic reproduction number. Existence of physical significant tumor-infection equilibrium points is investigated analytically. We show that tumor growth rate, source rate of immune cells, and death rate of immune cells play vital role in tumor dynamics and system undergoes saddle-node and transcritical bifurcation based on these parameters. Furthermore, the effect of cancer treatment is discussed by varying the values of relevant parameters. Numerical simulations are presented to illustrate the analytical results.

LP-solutions for fractional integral equations

In this article, we examine Lp-solutions of fractional integral equations in Banach spaces involving the Riemann-Liouville integral operator. Using a compactness type condition, we obtain local and global existence of solutions. Also other types of existence and uniqueness results are established. At the end, an application is given to illustrate the main result.

Trapezoidal scheme for time–space fractional diffusion equation with Riesz derivative

Abstract In this paper, a finite difference scheme is proposed to solve time–space fractional diffusion equation which has second-order accuracy in both time and space direction. The time and space fractional derivatives are considered in the senses of Caputo and Riesz, respectively. First, the centered difference approach is used to approximate the Riesz fractional derivative in space. Then, the obtained fractional ordinary differential equations are transformed into equivalent Volterra integral equations. And then, the trapezoidal rule is utilized to approximate the Volterra integral equations. The stability and convergence of our scheme are proved via mathematical induction method. Finally, numerical experiments are performed to confirm the high accuracy and efficiency of our scheme.

Dynamical Study of Fractional Order Tumor Model

In this paper, we have studied the fractional order model of tumor cells growth and their interactions with general immune effector cells, by using multi-step generalized differential transform method (MSGDTM). We discussed this nonlinear model because it differs from most others in the literature. It takes into account (i) the infiltration of the tumor by CTLs (cytotoxic T lymphocytes) and (ii) the possible effects of effector cell inactivation. The approximate solutions obtained by MSGDTM are highly accurate and valid for a longer period of time.

Numerical analysis of fractional-order tumor model

In this paper, the tumor-immune dynamics are simulated by solving a nonlinear system of differential equations. The fractional-order mathematical model incorporated with three Michaelis–Menten terms to indicate the saturated effect of immune response, the limited immune response to the tumor and to account the self-limiting production of cytokine interleukin-2. Two types of treatments were considered in the mathematical model to demonstrate the importance of immunotherapy. The limiting values of these treatments were considered, satisfying the stability criteria for fractional differential system. A graphical analysis is made to highlight the effects of antigenicity of the tumor and the fractional-order derivative on the tumor mass.

Finite difference method for time–space linear and nonlinear fractional diffusion equations

ABSTRACT In this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the Riesz fractional derivative in space. A trapezoidal formula is used to solve a system of Volterra integral equations transformed from spatial discretization. Stability and convergence of the proposed scheme is discussed which shows second-order accuracy both in temporal and spatial directions. Finally, examples are presented to show the accuracy and effectiveness of the schemes.

Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative

In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald–Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis.