A New Approach Using Integral Transform to Solve Cancer Models

Abstract

The objective of this work is to investigate analytical solutions of some models of cancer tumors using the Laplace residual power series method (LRPSM). The proposed method was effective and required simple calculations to find the analytic series solution, utilizing computer software such as the Mathematica package. Figures and graphs of the attained analytical Maclaurin solutions are presented to depict the procedure. The outcomes we obtained in this research showed the applicability and strength of the proposed approach in studying numerical series solutions of differential equations of fractional orders.

1. Introduction

Malignant diseases are viewed as active systems, characterized by the continuous growth and spread of malignant cells with respect to healthy cells by competing for both oxygen and nutrients provided by the blood. The spread of malignant cells close to their location usually ends in their death due to the lack of basic vital elements in the blood, such as nutrients and oxygen. Such a transfer is considered a firing step [1]. If the malignancy is to be treated effectively, the proposed treatment must be faster than the spread of the malignancy, bearing in mind that the growth of the malignancy is being controlled rapidly. The growth of the malignancy was assumed to be monotonous in [2], and it was assumed that treatment based on this assumption could be used. This is because, when dealing with such topics, there are numerous aspects to consider. All that remains is to focus on treating the problem. Mathematical modelling is important in the treatment of malignancies, because it provides analytical tools that consider the components of the immune system in the treatment of malignancies.

Growth factors, including cytokines and hormones; stromal elements, such as immune system cells; signaling activity molecules, such as chemokines; and fibroblasts create the tumor microenvironment [3,4,5,6,7]. These interactions are crucial in the search for efficient immunotherapies for malignancies. The use of mathematical models is crucial in this case, since new models of the existing data must be built as accurately as possible. Using fractions is believed to be the most efficient representation tool used to manipulate the existing data [8,9]. Unlike integer-order derivatives, fractional derivatives are successful in virtually all biological systems. With the development of science, various phenomena have memory and inheritance properties that cannot be well expressed by the standard differential equations. To address these problems, some of these phenomena have been described by fractional differential equations (FDEs) [10,11,12,13,14,15,16,17,18].

El-Ajou and others established the LRPSM for the first time in [19,20] to solve fractional differential equations, which is a novel combination between the Laplace transformation and he residual power series method.

LRPSM [21,22,23,24] is a simple and fast calculation for constructing analytical solutions, using computer software, such as Mathematica. Furthermore, in contrast to other power series methods, such as the homotopy analysis method, residual power series method, finite difference method, and others [25,26,27,28,29,30], LRPSM needs less time and provides simple calculations with high accuracy in finding the solution. The Laplace integral transformation is usually only used to solve linear differential equations, but in the LRPSM combination of the residual power series method and Laplace transformation (LT), we can overcome this disadvantage, and thus adapt it, to get analytical series solutions of nonlinear equations of different types. The LRPSM presents a series solution using the concept of the Maclurin series and the LT. What distinguishes LRPSM from the residual power series method (RPSM) is the use of the idea of the limit at infinity when finding the coefficients of a series solution rather than the concept of a fractional derivative, as in RPSM. There are many researchers who have used the presented method to solve different types of fractional order differential equations [25,26,27,28,29,30,31]. The novelty of this work is evident in the method used, LRPSM, which is simple, fast, and applicable to solving fractional partial differential equations and others. The method does not require hard calculations or linearization like other numerical methods; it essentially depends on finding the limit at infinity that represents the solution in a series form that quickly converges to the solution in the integer case of the target problem.

In this study, we examined some diffusion models of cancer models with fractional orders, such as

$$D_t^{\alpha }f\left(x,t\right)=\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}-k\left(x,t\right)f\left(x,t\right),t>0,0<\alpha \le 1,$$

where $D_t^{\alpha }$ denotes the Caputo fractional derivative that presents the diffusivity factor, $k\left(x,t\right)$ refers to the therapy-dependent killing ratio, $f\left(x,t\right)$ denotes the the concentration of tumor cells at the position $x$, and $t$ denotes the time. k could be expressed in three cases, as a constant, a function of time, or a function that does not depend on time.

Similar models were also studied in [32], and the authors investigated fractional diffusion models by using the estimates of time and the spatial dependency of the concentration of tumor cells, as well as that of the killing ratio. In [33], Burgess et al. presented a diffusion model, which considered a globular tumor, which had the reproduction ratio $f$ and therapy-dependent killing ratio $k$.

The basic goal of this article is to analyze the procedure of LRPSM using the non-local fractional Caputo’s derivative to get analytical solutions of some tumor models. We introduce graphical expressions and figures to prove the reliability, efficiency, and accuracy of the method.

This study is organized as follows: in the next section, we describe definitions and numerous theorems that are required for our work with fractional calculus. We also illustrate the convergence assessment of LRPSM. In Section 3, we present the process of LRPSM in dealing with tumor models. Section 4 presents some numerical applications of LRPSM.

2. Basic Preliminaries

This section presents basic preliminaries of fractional calculus. Moreover, we introduce the fractional power series and some convergence analysis theorems.

2.1. Fractional Operators

Definition 1.

[5] The time fractional Caputo operator with order$\alpha$of the function$f\left(x,t\right)$is defined as

$$D_t^{\alpha }f\left(x,t\right)=\left\{\begin{array}{cc}\frac{1}{\Gamma \left(n-\alpha \right)}\underset{0}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\alpha -1}\frac{{\partial }^{n}f\left(x,\tau \right)}{\partial {\tau }^n}\partial \tau ,& n-1<\alpha <n,n\in \mathbb{N},\\ \frac{{\partial }^{n}f\left(x,t\right)}{\partial t^n}& \alpha =n.\end{array}\right.$$

The LT [19,20] of the function $f\left(x,t\right)$ with respect to the variable $t$ is defined as

$$\mathcal{L}\left[f\left(x,t\right)\right]=F\left(x,s\right)=\underset{0}{\overset{\infty }{{\displaystyle \int }}}f\left(x,t\right)e^{-st}dt,s>0,$$

and the inverse LT is given by

$$f\left(x,t\right)={\mathcal{L}}^{-1}\left[F\left(x,s\right)\right]=\underset{c-i\infty }{\overset{c+i\infty }{{\displaystyle \int }}}F\left(x,s\right)e^{st}ds,c=\mathrm{Re}\left(s\right)>0.$$

In addition, if $\mathcal{L}\left[f\left(x,t\right)\right]=F\left(x,s\right)$, then we have the following properties of LT:

$$\begin{gathered} \mathcal{L}\left[t^{\beta }\right]=\frac{\Gamma \left(\beta +1\right)}{s^{\beta +1}},\beta >-1. \\ \mathcal{L}\left[D_t^{\alpha }f\left(x,t\right)\right]=s^{\alpha }F\left(x,s\right)-{\displaystyle \sum }_{k=0}^{n-1}{s}^{\alpha -k-1}{D}_t^{k}f\left(x,0\right),n-1<\alpha \le n,n\in \mathbb{N}. \end{gathered}$$

Theorem 1.

[20] Let$f\left(x,t\right)$be a polynomial of fractional order that has the series representation

$$f\left(x,t\right)={{\displaystyle \sum }}_{m=0}^{\infty }{a}_m\left(x\right)t^{m\alpha },0\le t<T,T\in ℝ, 0<\alpha <1.$$

(1)

If $D_t^{\alpha }f\left(x,t\right)$ is continuous on the region $I\times \left[0,R\right]$, then the coefficients $a_n\left(x\right)$ can be expressed as

$$a_m\left(x\right)=\frac{D_t^{m\alpha }f\left(x,0\right)}{\Gamma \left(m\alpha +1\right)},m=0,1,2,\cdots ,$$

where $D_t^{m\alpha }=D_t^{\alpha }{D}_t^{\alpha }\cdots D_t^{\alpha }$ (m-times) and the radius of convergence of the fractional power series (1) is $T$; for proof see [24].

The fractional Taylor’s expansion (1), when putting $\alpha =1$, expresses the classical Taylor’s expansion of the function $f\left(x,t\right)$ around $t=0$ as

$$f\left(x,t\right)={\displaystyle \sum }_{m=0}^{\infty }\frac{{\partial }^{m}f\left(x,0\right)}{\partial t^m}\frac{t^m}{m!},x\in I,0\le t<T.$$

2.2. Convergence Analysis of LRPSM

In this section, we demonstrate the fractional power series in the Laplace space and the conditions of convergence for the obtained expansion.

Theorem 2.

[20] If the function$F\left(x,s\right)=\mathcal{L}\left[f\left(x,t\right)\right]$has the fractional power series

$$F\left(x,s\right)={\displaystyle \sum }_{m=0}^{\infty }\frac{a_m\left(x\right)}{s^{m\alpha +1}},0〈\alpha \le 1, s〉0.$$

(2)

Then, the coefficients $a_m\left(x\right)$ have the following form: $a_m\left(x\right)=D_t^{m\alpha }f\left(x,0\right)$, where $D_t^{m\alpha }=D_t^{\alpha }{D}_t^{\alpha }\cdots D_t^{\alpha }$ (m-times).

The inverse LT of the series representation (2) is defined by

$$f\left(x,t\right)={\displaystyle \sum }_{m=0}^{\infty }\frac{D_t^{m\alpha }f\left(x,0\right)}{\Gamma \left(m\alpha +1\right)}{t}^{m\alpha },t\ge 0,0<\alpha \le 1.$$

Theorem 3.

[20] Assume that

$$\left|s\mathcal{L}[D_t^{\left(m+1\right)\alpha }f\left(x,t\right)\right|\le M,$$

on $0\le s<q$, $0<\alpha \le 1,$ and $M=M\left(x\right)$ for some $x\in I$. Then, the remainder $R_m\left(x,s\right)$ of the new fractional Taylor’s expansion (2) satisfies the following inequality

$$\left|R_m\left(x,s\right)\right|\le \frac{M}{s^{\left(m+1\right)\alpha +1}}.$$

Theorem 4.

[26,31] Assume that$\Vert f_{n+1}\left(x,t\right)\Vert \le \delta \Vert f_n\left(x,t\right)\Vert ,\forall n\in \mathbb{N}$for some$\delta \in \left(0,1\right),$and$0<t<T<1$; then, the obtained approximate series solution converges to the exact solution,

where$f_n\left(x,t\right)={\displaystyle \sum }_{m=0}^n\frac{D_t^{m\alpha }f\left(x,0\right)}{\Gamma \left(m\alpha +1\right)}{t}^{m\alpha }.$

3. Algorithm of LRPSM

In this section, we present the algorithm of LRPSM for solving cancer tumor models. Consider the time FPDE of the form

$$D_t^{\alpha }f\left(x,t\right)=\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}-k\left(x,t\right)f\left(x,t\right),t>0,0<\alpha \le 1,$$

(3)

with the initial condition (IC):

$$f\left(x,0\right)=g\left(x\right).$$

(4)

To apply LRPSM, we first apply the LT to Equation (3)

$$\mathcal{L}\left[D_t^{\alpha }f\left(x,t\right)\right]=\mathcal{L}\left[\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}\right]-\mathcal{L}\left[k\left(x,t\right)f\left(x,t\right)\right].$$

(5)

Using the property that $\mathcal{L}\left[D_t^{\alpha }f\left(x,t\right)\right]=s^{\alpha }\mathcal{L}\left[f\left(x,t\right)\right]-s^{\alpha -1}f\left(x,0\right),0<\alpha \le 1,$ and the IC (4), Equation (5) can be written as

$$F\left(x,s\right)-\frac{g\left(x\right)}{s}-\frac{1}{s^{\alpha }}\frac{{\partial }^{2}F\left(x,s\right)}{\partial x^2}+\frac{1}{s^{\alpha }}\mathcal{L}\left[k\left(x,t\right){\mathcal{L}}^{-1}\left[F\left(x,s\right)\right]\right]=0,$$

(6)

where $F\left(x,s\right)=\mathcal{L}\left[f\left(x,t\right)\right]$.

Consequently, we assume that $F\left(x,s\right),$ which is the transformed function, has the following series representation

$$F\left(x,s\right)={\displaystyle \sum }_{n=0}^{\infty }\frac{a_n\left(x\right)}{s^{n\alpha +1}},$$

(7)

and the kth truncated series of the series (7) has the form

$$F_k\left(x,s\right)={\displaystyle \sum }_{n=0}^k\frac{a_n\left(x\right)}{s^{n\alpha +1}}=\frac{a_0}{s}+{\displaystyle \sum }_{n=1}^k\frac{a_n\left(x\right)}{s^{n\alpha +1}}.$$

(8)

Now, we define the Laplace residual function of Equation (6) as

$$\mathcal{L}\mathrm{Res}\left(x,s\right)=F\left(x,s\right)-\frac{g\left(x\right)}{s}-\frac{1}{s^{\alpha }}\frac{{\partial }^{2}F\left(x,s\right)}{\partial x^2}+\frac{1}{s^{\alpha }}\mathcal{L}\left[k\left(x,t\right){\mathcal{L}}^{-1}\left[F\left(x,s\right)\right]\right], k=1,2,\cdots ,$$

(9)

and the kth Laplace residual function of Equation (9) is given by

$$\mathcal{L}{\mathrm{Res}}_k\left(x,s\right)=F_k\left(x,s\right)-\frac{g\left(x\right)}{s}-\frac{1}{s^{\alpha }}\frac{{\partial }^2{F}_k\left(x,s\right)}{\partial x^2}+\frac{1}{s^{\alpha }}\mathcal{L}\left[k\left(x,t\right){\mathcal{L}}^{-1}\left[F_k\left(x,s\right)\right]\right],k=1,2,\cdots .$$

(10)

Thirdly, we recall some facts of the residual power series method [23]; here in, we mention that all of these facts are illustrated and proved by El-Ajou in [23,24]. These facts are basics in constructing the LRPSM.

$$\begin{gathered} \begin{array}{c}\mathcal{L}\mathrm{Res}\left(x,s\right)=0\mathrm{and}\underset{j\to \infty }{\lim }\mathcal{L}{\mathrm{Res}}_j\left(x,s\right)=\mathcal{L}\mathrm{Res}\left(x,,,s\right)\mathrm{for\; all}s>0 \\ \\ \underset{s\to \infty }{\lim }s\mathcal{L}\mathrm{Res}\left(x,s\right)=0\mathrm{which\; implies}\underset{s\to \infty }{\lim }s\mathcal{L}{\mathrm{Res}}_j\left(x,s\right)=0. \\ \\ \underset{s\to \infty }{\lim }{s}^{j\alpha +1}\mathcal{L}\mathrm{Res}\left(x,s\right)=\underset{s\to \infty }{\lim }{s}^{j\alpha +1}\mathcal{L}{\mathrm{Res}}_j\left(x,s\right)=0,0<\alpha \le 1,j=1,2,\cdots .\end{array} \end{gathered}$$

Now, to determine the coefficient functions $a_n\left(x\right)$, we solve the recurrence relations of the following system

$$\underset{s\to \infty }{\lim }{s}^{k\alpha +1}\mathcal{L}{\mathrm{Res}}_k\left(x,s\right)=0,0<\alpha \le 1,k=1,2,\cdots .$$

(11)

Fourthly, we operate the inverse LT to the kth Laplace approximate $F_k\left(x,s\right)$ to get the kth approximate solution $f_k\left(x,t\right)$.

4. Applications of LRPSM

In this section, three interesting examples are solved by the proposed method. The results show that LRPSM is applicable and could be a powerful technique in solving fractional cancer tumor models.

Example 1. [32] Consider the time fractional equation of the clear killing ratio of the cancer cells

$$D_t^{\alpha }f\left(x,t\right)=D_x^{2}f\left(x,t\right)-t^{2}f\left(x,t\right),t>0,0\le x\le 1,0<\alpha \le 1,$$

(12)

subject to the IC:

$$f\left(x,0\right)=e^{rx}.$$

(13)

Solution. To get the solution by the LRPSM in the series form about $t=0$. We first apply the LT on both sides of Equation (12) to get

$$\mathcal{L}\left[D_t^{\alpha }f\left(x,t\right)\right]=\mathcal{L}\left[D_x^{2}f\left(x,t\right)\left(x,t\right)\right]-\mathcal{L}\left[t^{2}f\left(x,t\right)\right].$$

Using the IC (13), we have

$$F\left(x,s\right)=\frac{e^{rx}}{s}+\frac{D_x^{2}F\left(x,s\right)}{s^{\alpha }}-\frac{\mathcal{L}[t^2{\mathcal{L}}^{-1}\left[F\left(x,s\right)\right]}{s^{\alpha }}.$$

(14)

We define the kth truncated series of Equation (14) as

$$F_k\left(x,s\right)=\frac{e^{rx}}{s}+{\displaystyle \sum }_{m=1}^k\frac{a_m\left(x\right)}{s^{m\alpha +1}},k=1,2,\cdots ,$$

(15)

and the kth Laplace residual function of Equation (14) is defined as

$$\mathcal{L}{\mathrm{Res}}_k\left(x,s\right)=F_k\left(x,s\right)-\frac{e^{rs}}{s}-\frac{D_x^{2}F\left(x,s\right)}{s^{\alpha }}+\frac{\mathcal{L}\left[t^2{\mathcal{L}}^{-1}\left[F_k\left(x,s\right)\right]\right]}{s^{\alpha }}.$$

(16)

Hence, to get the values of the coefficients functions $a_k\left(x\right),k=1,2,\cdots$, we substitute the kth truncated series $F_k\left(x,s\right)$ in Equation (15) into the kth Laplace residual function (16); then, we multiply the obtained equation by $s^{k\alpha +1}$ and solve the recurrence relations $\underset{s\to \infty }{\lim }{s}^{k\alpha +1}\mathcal{L}{\mathrm{Res}}_k\left(x,s\right)=0,k=1,2,\cdots$ for the unknown coefficients $a_k\left(x\right),$ $k=1,2,\cdots$. Now, following few terms of the sequence $\left\{a_k\left(x\right)\right\}$, we get

$$\begin{gathered} a_1\left(x\right)=r^2{e}^{rx}, \\ {a}_2\left(x\right)=r^4{e}^{rx}, \\ {a}_3\left(x\right)=\frac{r^6-2}{2}{e}^{rx}, \\ {a}_4\left(x\right)=\frac{r^2\left(r^6-8\right)}{6}{e}^{kx}, \\ {a}_5\left(x\right)=\frac{r^4\left(r^6-20\right)}{24}{e}^{rx}. \end{gathered}$$

Hence the Laplace residual solution of Equations (12) and (13) can be expressed in the series expansion

$$F\left(x,s\right)=\frac{1}{s}{e}^{rx}+\frac{r^2}{s^{\alpha +1}}{e}^{rs}+\frac{r^4}{s^{2\alpha +1}}{e}^{rs}+\frac{1}{2}\frac{\left(r^6-2\right)}{s^{3\alpha +1}}{e}^{rx}+\cdots .$$

(17)

Finally, to get the solution of Equations (12) and (13) in the original space, we run the inverse LT on Equation (17), to get

$$f\left(x,t\right)=e^{rx}+\frac{r^2{e}^{rx}{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}+\frac{r^4{e}^{rx}{t}^{2\alpha }}{\Gamma \left(2\alpha +1\right)}+\frac{\left(r^6-2\right)e^{rx}{t}^{3\alpha }}{2\Gamma \left(3\alpha +1\right)}+\frac{r^2\left(r^6-8\right)e^{rx}{t}^{4\alpha }}{6\Gamma \left(4\alpha +1\right)}+\frac{r^4\left(r^6-20\right)e^{rx}{t}^{5\alpha }}{24\Gamma \left(4\alpha +1\right)}+\cdots .$$

Moreover, we mention that our results coincide with those obtained by RPSM in [32].

The 3D graphics and the contour graphics of the fifth approximate series solution $f\left(x,t\right)$ of Example 1 are illustrated below in Figure 1 and Figure 2. The graphics show that the concentration of cancer cells decreases over time and finally approaches zero.

Example 2. [32] Consider the IVP

$$D_t^{\alpha }f\left(x,t\right)=D_x^{2}f\left(x,t\right)-\frac{2}{x^2}f\left(x,t\right), t>0, 0\le x\le 1, 0<\alpha \le 1,$$

(18)

subject to the IC

$$f\left(x,0\right)=\frac{1}{x}+x^2.$$

(19)

Solution. By similar arguments to Example 1, one can obtain

$$a_1\left(x\right)=a_2\left(x\right)=a_3\left(x\right)=a_4\left(x\right)=a_5\left(x\right)=0.$$

Thus, the Laplace residual solution of Equation (18) is

$$F_5\left(x,s\right)=\frac{1}{s}\left(\frac{1}{x}+x^2\right).$$

(20)

Hence, to get the fifth approximate solution in the original space, we apply the inverse LT on Equation (20), to get

$$f_5\left(x,t\right)=f_6\left(x,t\right)=\dots =\frac{1}{x}+x^2.$$

Here, we get the exact solution of the IVP (18) and (19), when putting $\alpha =1$.

The 3D graphics of the fifth approximate series solution $f\left(x,t\right)$ of Example 2 is illustrated below in Figure 3. Here in, we note that the fractional order is not effective.

Example 3. [32] Consider the following nonlinear IVP

$$D^{\alpha }f\left(x,t\right)=D_x^{2}f\left(x,t\right)-\frac{2}{x}{D}_{x}f\left(x,t\right)-f^2\left(x,t\right),t>0,\hspace{1em}\hspace{1em}0\le x\le 1,\hspace{1em}\hspace{1em}0<\alpha \le 1,$$

(21)

subject to the IC:

$$f\left(x,0\right)=x^p,\hspace{1em}\hspace{1em}p>0.$$

(22)

Solution. Applying the procedure of the LRPSM, we get the fifth approximate solution:

$$\begin{array}{cc}\hfill f_5(x,t)=x^p+& \frac{x^{-2+p}{t}^{\alpha }}{\Gamma (\alpha +1)}\left(-3p+p^2-x^{p+2}\right)\hfill \\ \hfill & +\frac{x^{p-4}{t}^{2\alpha }}{\Gamma (2\alpha +1)}\left(-10p^3+p^4+2x^{2p+4}+p^2\left(31-6x^{p+2}\right)\right.\hfill \\ \hfill & \left.+6p\left(2x^{p+2}-5\right)\right)\hfill \\ \hfill & -\frac{x^{-6+p}{t}^{3\alpha }}{2\Gamma (3\alpha +1)}\left(840p-1198p^2+651p^3-169p^4+21p^5-p^6\right.\hfill \\ \hfill & -180p{x}^{2+p}+308p^2{x}^{2+p}-164p^3{x}^{2+p}+28p^4{x}^{2+p}+54p{x}^{4+2p}\hfill \\ \hfill & \left.-34p^2{x}^{4+2p}+6x^{6+3p}\right)\hfill \\ \hfill & +\frac{x^{-8+p}{t}^{4\alpha }}{6\Gamma (4\alpha +1)}\left(-45360p+77292p^2-53964p^3+20089p^4\right.\hfill \\ \hfill & -4320p^5+538p^6-36p^7+p^8+6720p{x}^{2+p}-15520p^2{x}^{2+p}\hfill \\ \hfill & +14128p^3{x}^{2+p}-6328p^4{x}^{2+p}+1392p^5{x}^{2+p}-120p^6{x}^{2+p}\hfill \\ \hfill & -1080p{x}^{4+2p}+2492p^2{x}^{4+2p}+404p^4{x}^{4+2p}+288p{x}^{6+3p}\hfill \\ \hfill & \left.-212p^2{x}^{6+3p}+24x^{8+4p}\right)\hfill \\ \hfill & -\frac{x^{-10+p}{t}^5}{24\Gamma (5\alpha +1)}\left(3991680p-7663536p^2+6262740p^3-2870440p^4\right.\hfill \\ \hfill & +815815p^5-149513p^6+17710p^7-1310p^8+55p^9-p^{10}\hfill \\ \hfill & -453600p{x}^{2+p}+1219824p^2{x}^{2+p}-1409952p^3{x}^{2+p}\hfill \\ \hfill & +902384p^4{x}^{2+p}-343504p^5{x}^{2+p}+77456p^6{x}^{2+p}-9552p^7{x}^{2+p}\hfill \\ \hfill & \left.+496p^8{x}^{2+p}+50400p{x}^{4+2p}+\cdots \right).x\hfill \end{array}$$

Moreover, we mention that our results coincide with those obtained by RPSM in [32].

The 3D graphics and contour graphics of the fifth approximate series solution $f\left(x,t\right)$ of Example 3 is illustrated below in Figure 4 and Figure 5. The graphic shows that the concentricity of cancer tumor cells decreases over time, and in the end, it approaches zero.

5. Mathematical Simulations

In this section, we present some figures to study the behaviour of the proposed solutions in Examples 1 and 3. We examined the effects of the fractional order time derivative on the concentricity of tumor cells.

In Figure 6, we sketch 2D graphics of the fifth numerical approximate solution of Example 1 with different values of $\alpha$ in (a); we note that as $\alpha$ approaches 1, we get good results, with a close form to the solution in the integer case. In (b), we sketch some approximates of the series expansion of the solution of Example 1, and we can see that as much as we take terms of the series solution, we get good agreement with the solution in the integer case.

Furthermore, one can see that as time goes by, the concentricity of the cancer cells decreases eventually reaches zero.

In Figure 7, we sketch 2D graphics of the fifth numerical approximate solution of Example 3 with different values of $\alpha$ in (a), and we note that as $\alpha$ approaches 1, we get good results, with a close form to the solution in the integer case. In (b), we sketch some approximates of the series expansion of the solution of Example 3, and we can see that as much as we take terms of the series solution, we get good agreement with the solution in the integer case.

In addition, one can see that with increasing time, the concentricity of the cancer cells decreases and eventually reaches zero.

6. Conclusions

In this paper, we examined the concentration of tumor cells over time as a function of the fractional derivatives. The concentration of cancer cells can also affect the percentage of apparent cell death. The importance of this research is due to the proposed technique, which can be considered as an alternative technique to obtain analytical solutions for some types of tumor equations. LRPSM is a powerful technique for providing a close approximation to exact solutions for FPDEs. Furthermore, the results demonstrated the simplicity of the method and its applicability without the need for linearity, estimation, or additional constraints on the target problems. In the following arguments, we illustrate the advantages of the proposed procedure:

LRPSM is easy to use in solving fractional differential equations and systems.

The results obtained are presented in the form of a convergent infinite series, which is easy to handle and use.

The method does not require any linearization or differentiation to apply, and it is simple and easy compared to other numerical methods.

LRPSM can smoothly handle nonlinear problems and differential equations with variable coefficients.

All numerical results in this article are obtained using Mathematic software 13.