Recently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called the Laplace homotopy analysis method (LHAM) using the Caputo–Fabrizio (CF) fractional derivative operator. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). We have used the fractional operator suggested by Caputo and Fabrizio in 2015 based on the exponential kernel. We have considered the LHAM with this derivative in order to obtain the solutions of the fractional Black–Scholes equations (FBSEs) with the initial conditions. In addition to this, the convergence and stability analysis of the model have been constructed. According to the results of this study, it can be concluded that the LHAM in the sense of the CF fractional derivative is an effective and accurate method, which is computable in the series easily in a short time.

In this paper, we study the mathematical model of interaction cancer cells and immune system cells presented Castiglione and Piccoli. As the interaction between cancer cells and the immune system is weak, when the immune system of the body begins to decrease, the cancer cells get stronger and increase rapidly. Helper CD4+ T and cytotoxic CD8+ T cells, cancer cells, dendritic cells and cytokine interleukin-2 (IL-2) cells are involved in the mathematical model of this competition in the living body. As can be seen in the literature, since the cancer cells have memory structure, fractional models describe the struggle between the cancer cells and immune system give more meaningful results than classical models as closer to the reality. The main motivation of the present work is to generalize the model in Castiglione and Piccoli [J. Theor. Biol.247(2007) 723–732] by using Caputo fractional derivative. The main aim is to analyze the behaviors of system cells by changing of the fractional parameter. In this sense, we study on the stability analysis of treatment free and the fixed points of the prescribed model. To get the numerical solutions, we apply the Adam-Bashforth-Moulton (ABM) algorithm and also illustrate the results by the graphics held by Matlab program. We have reached the excellent result that cancer cells decrease asθdiminishes in this process.

In this paper, we have aimed the numerical inverse Laplace homotopy technique for solving some interesting 1-D time-fractional heat equations. This method is based on the Laplace homotopy perturbation method, which is combined form of the Laplace transform and the homotopy perturbation method. Firstly, we have applied to the fractional 1-D PDE by using He?s polynomials. Then we have used Laplace transform method and discussed how to solve these PDE by using Laplace homotopy perturbation method. We have declared that the proposed model is very efficient and powerful technique in finding approximate solutions to the fractional PDE.

In this manuscript, we have proposed a comparison based on newly defined fractional derivative operators which are called as Caputo-Fabrizio (CF) and Atangana-Baleanu (AB). In 2015, Caputo and Fabrizio established a new fractional operator by using exponential kernel. After one year, Atangana and Baleanu recommended a different-type fractional operator that uses the generalized Mittag-Leffler function (MLF). Many real-life problems can be modelled and can be solved by numerical-analytical solution methods which are derived with these operators. In this paper, we suggest an approximate solution method for PDEs of fractional order by using the mentioned operators. We consider the Laplace homotopy transformation method (LHTM) which is the combination of standard homotopy technique (SHT) and Laplace transformation method (LTM). In this study, we aim to demonstrate the effectiveness of the aforementioned method by comparing the solutions we have achieved with the exact solutions. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

Cancer that is difficult to treat, is a very common disease today and there are many types of cancer such as lung, colon, stomach. When cancer settles in the body, the immune system tries to resist it. In this study, the mathematical model of the interaction between immune system components and cancer is discussed and is modified by using Atangana-Baleanu derivative. After investigating the existence and uniqueness of the solution of the fractional immune system-cancer model, numerical simulations are given via predictor-corrector scheme.

The objective of this manuscript is to present a novel approach to modeling influenza A disease dynamics by incorporating the Caputo-Fabrizio (CF) fractional derivative operator into the model. Particularly distinct contact rates between exposed and infected individuals are taken into account in the model under study, and the fractional derivative concept is explored with respect to this component. We demonstrate the existence and uniqueness of the solution and obtain the series solution for all compartments using the Laplace transform method. The reproduction number of the Influenza A model, which was created to show the effectiveness of different contact rates, was obtained and examined in detail in this sense. To validate our approach, we applied the predictor-corrector method in the sense of the Caputo-Fabrizio fractional derivative and demonstrate the effectiveness of the fractional derivative in accurately predicting disease dynamics. Our findings suggest that the use of the Caputo-Fabrizio fractional derivative can provide valuable insights into the mechanisms underlying influenza A disease and enhance the accuracy of disease models.