# An Analytical Investigation of Fractional-Order Biological Model Using an Innovative Technique

Authors: Hassan Khan; Adnan Khan; Maysaa Al Qurashi; Dumitru Baleanu; Rasool Shah
Journal: Complexity
 (2020)

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## Abstract

In this paper, a new so-called iterative Laplace transform method is implemented to investigate the solution of certain important population models of noninteger order. The iterative procedure is combined effectively with Laplace transformation to develop the suggested methodology. The Caputo operator is applied to express the noninteger derivative of fractional order. The series form solution is obtained having components of convergent behavior toward the exact solution. For justification and verification of the present method, some illustrative examples are discussed. The closed contact is observed between the obtained and exact solutions. Moreover, the suggested method has a small volume of calculations; therefore, it can be applied to handle the solutions of various problems with fractional-order derivatives.


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## Body

## 1. Introduction

Over the last century, fractional differential equations (FDEs) have attracted a great deal of attention from scientists due to their ability to raise real-world issues in numerous engineering fields and physics. FDEs are broadly used in certain fields of science [1–5]. Several phenomena in chemistry, physics, engineering, and other sciences can be effectively described using fractional calculus. Acoustics, the nonlinear oscillation of earthquake, electrochemistry, electromagnetism, signal processing, and diffusion processes can be modeled by fractional equations [6, 7]. In modern times, it is difficult to imagine the modeling of several real-world issues without the use of fractional partial differential equations (FPDEs). Indeed, this century’s calculus [8] can be called a fractional calculus because of the diversity of implementations in different fields of science and technology. The researchers have used several analytical and numerical techniques such as variational iteration method (VIM) [9], homotopy analysis method (HAM) [10], and generalized fractional Taylor series method [11] to solve linear and nonlinear FPDEs.Any phenomena in the areas of engineering and science may be alternatively modeled via fractional-order derivatives. It is due to their nonlocal properties, which are inherent in some complex structures. They are used as modeling devices in financial, viscoelasticity, transport phenomenon, nanotechnology, control theory, and biological modeling [12–15]. In biology, biological cell membranes are shown to have electrical conductance in a fractional order and are classified among noninteger order systems [16–18]. Anomalous diffusion concepts in nonhomogeneous media can be discussed by noninteger, derivative-based diffusion equations [19–21]. Another example of a fractional-order element is practice, which is a noninteger-ordered electrical circuit with resistance and capacitance properties [10]. In this connection, certain important techniques have been used including the fractional operational matrix method (FOMM) [22, 23], fractional wavelet method (FWM) [24–27], homotopy analysis method (HAM) [28], homotopy perturbation method (HPM) [29], homotopy perturbation transform method (HPTM) [30], Laplace Adomian decomposition method (LADM) [31], and fractional variational iteration method (FVIM) [32].Daftardar-Gejji and Jafari in 2006 have developed the iterative technique to solve nonlinear functional equations [33, 34]. Later on, the iterative technique is applied to solve noninteger differential equations (DEs) [35]. In recent time, Jafari et al. have used Laplace transform together with an iterative technique for the first time which is nowadays becoming an effective technique named as the iterative Laplace transform method (ILTM) [36]. ILTM is implemented to solve partial differential equations (PDEs) and Fokker–Plank problems [37]. Recently, many other FPDEs have been solved by using ILTM such as time-fractional Schrodinger equations [38], fractional telegraph equations [39], fractional heat and wave-like equations [40], and time-fractional Fisher equation [41].The main theme of the present research work is to use ILTM for obtaining the analytical solution of the noninteger biological population model [42].(1)∂ρψμ,ν,τ∂τρ=∂2ψ2μ,ν,τ∂μ2+∂2ψ2μ,ν,τ∂ν2+gψμ,ν,τ,τ>0,0<ρ≤1,subject to the starting values: ψμ,ν,0=g0μ,ν.Equation (1) is identified as the time fractional-order biological population model, used by Gurney and Nisbet as a unique situation for modeling the animal population. In general, movements are made both by mature animals driven by mature invaders or by young animals just reaching maturity moving out of their parental territory to establish their breeding territory. In both cases, the assumption that they would be driven towards neighboring vacant land is much more likely. Hence, movement in this model occurs almost entirely down the gradient of population density and will be faster at high population densities than at low ones, where the population density is represented by ψμ,ν,τ and the population rate is expressed by gψμ,ν,τ. For ρ⟶1, Various properties such like Holder estimates for its solution are discussed in [43, 44]. The three consecutive cases for gu are as follows:gψ=c, for any constant c that reduces to Malthusian lawgψ=ψd1−d2ψ, for positive constants d1 and d2 that reduce to Verhulst lawgψ=−dψk,d≥0,0<k<1, for positive d that reduces to porous mediaThe ILTM solutions are found to be in good contact with the exact solutions of the problems. The solutions at different fractional orders are also calculated. It is investigated that the fractional-order solutions are convergent towards integer-order solution of the problems as fractional order approaches to an integer order. It is also observed that the suggested method is very simple and effective and required small number of calculation. Moreover, the present method provides the series form solution with easily computable components. It is also shown that the series form solution has the desire rate of convergence towards the exact solution of the problem and the closed form solution is achieved.


## 2. Definitions and Preliminaries

In this part of the paper, some important definitions related to FC and Laplace transform are briefly discussed. These preliminaries are important to continue and complete the present research work.Definition 1.
The fractional derivative in terms of Caputo operator is expressed as(2)Dτρψμ,τ=1Γn−ρ∫0ττ−ζn−ρ−1ψnμ,ζdζ,n−1<ρ≤n,n∈N=Jτn−ρDnuμ,τ.
Here,(3)Dn=dndτn.Definition 2.
The fractional integral in terms of Riemann–Liouville integral is expressed as(4)jτρψμ,τ=1Γρ∫0ττ−ζρ−1ψμ,ζdζ,ζ>0,n−1<ρ≤n,n∈N,where jτρ represents the fractional integral operator.Definition 3.
The Laplace transform is describe as(5)Lgτ=Gτ=∫0∞e−sτgτdτ.Definition 4.
The Laplace transform of the fractional derivativeDτρψμ,τ is defined as(6)LDτρψμ,τ=sρLψμ,τ−∑k=0n−1ψkμ,0sρ−k−1,n−1<ρ≤n,n∈N.Definition 5.
The Mittag-Leffler function is given by(7)Eρz=∑q=0∞zqΓρq+1,ρ∈C,Reρ>0.


## 3. The Basic Concept of ILTM

In this section, we will briefly discuss ILTM to solve fractional-order nonlinear PDEs.(8)Dτρψμ,ν,τ+Rψμ,ν,τ+Nψμ,ν,τ=gμ,ν,τ,n−1<ρ≤n,n∈N,(9)ψkμ,ν,0=hkμ,ν,k=0,1,2,…,n−1,where Dτρψμ,ν,τ is the fractional Caputo operator of order ρ, n−1<ρ≤n, denoted by equation (8), and R and N are linear and nonlinear operators. gμ,ν,τ is source function.Using Laplace transform of equation (8), we get(10)LDτρψμ,ν,τ+LRψμ,ν,τ+Nψμ,ν,τ=Lgμ,ν,τ.Applying the property of Laplace differentiation,(11)Lψμ,ν,τ=1sρ∑k=0m−1sρ−1−kψkμ,ν,0+1sρLgμ,ν,τ−1sρLRψμ,ν,τ+Nψμ,ν,τ.By using inverse Laplace transform of equation (11), we obtain(12)ψμ,ν,τ=L−11sρ∑k=0m−1sρ−1−kψkμ,ν,0+Lgμ,ν,τ−L−11sρLRψμ,ν,τ+Nψμ,ν,τ.From iterative technique,(13)ψμ,ν,τ=∑i=0∞ψiμ,ν,τ.SinceR is a linear operator,(14)R∑i=0∞ψiμ,ν,τ=∑i=0∞Rψiμ,ν,τ,and the nonlinear operator N is split as(15)N∑i=0∞ψiμ,ν,τ=Nψ0μ,ν,τ+∑i=1∞N∑k=0iψkμ,ν,τ−N∑k=0i−1ψkμ,ν,τ.Substituting equations (13)–(15) into equation (12), we obtain(16)∑i=0∞ψiμ,ν,τ=L−11sρ∑k=0m−1sρ−1−kψkμ,ν,0+Lgμ,ν,τ−L−11sρL∑i=0∞Rψiμ,ν,τ+Nψ0μ,ν,τ+∑i=1∞N∑k=0iψkμ,ν,τ−N∑k=0i−1ψkμ,ν,τ.Using equation (16), we define the following iterative formula:(17)ψ0μ,ν,τ=L−11sρ∑k=0m−1sρ−1−kψkμ,ν,0+1sρLgμ,ν,τ,(18)ψ1μ,ν,τ=−L−11sρLRψ0μ,ν,τ+Nψ0μ,ν,τ,(19)ψm+1μ,ν,τ=−L−11sρLRψmμ,ν,τ−N∑k=0mψkμ,ν,τ−N∑k=0m−1ψkμ,ν,τ,m≥1.The approximate m-term solution of equations (18) and (19) in form of series is as follows:(20)ψμ,ν,τ≅ψ0μ,ν,τ+ψ1μ,ν,τ+ψ2μ,ν,τ+⋯+ψmμ,ν,τ,m=1,2,….


## 4. Implementation of ILTM

In this section, ILTM is applied to determine the exact solution of some special cases of equation (1). It has been shown that the ILTM is an accurate and appropriate analytical technique to solve nonlinear FPDEs.Example 1.
The biological population model with time noninteger derivative is express as(21)∂ρψ∂τρ=∂2∂μ2ψ2+∂2∂ν2ψ2+hψ−11−rψ,0<ρ≤1,μ,ν∈ℜ,τ>0,with starting values(22)ψμ,ν,0=hr4μ2+hr4ν2+ν+5.
The Laplace transform to equation (21) is expressed as(23)sρLψμ,ν,τ−∑k=0m−1ψkμ,ν,0sρ−k−1=L∂2∂μ2ψ2+∂2∂ν2ψ2+hψ−11−rψ,(24)Lψμ,ν,τ=1shr4μ2+hr4ν2+ν+5+1sρL∂2∂μ2ψ2+∂2∂ν2ψ2+hψ−11−rψ.
Using inverse Laplace transform of equation (24),(25)ψμ,ν,τ=hr4μ2+hr4ν2+ν+5+L−11sρL∂2∂μ2ψ2+∂2∂ν2ψ2+hψ−11−rψ.
Using the iterative technique described in equations (14) and (15), we obtain the following solution components of Example 1:(26)ψ0μ,ν,τ=hr4μ2+hr4ν2+ν+5,ψ1μ,ν,τ=L−11sρL∂2∂μ2ψ02+∂2∂ν2ψ02+hψ0−11−rψ0=hhr4μ2+hr4ν2+ν+5−1/2τρΓρ+1,ψ2μ,ν,τ=L−11sρL∂2∂μ2ψ12+∂2∂ν2ψ12+hψ1−11−rψ1=−2h2hr4μ2+hr4ν2+ν+5−3/2τ2ρΓ2ρ+1,ψ3μ,ν,τ=L−11sρL∂2∂μ2ψ22+∂2∂ν2ψ22+hψ2−11−rψ2=3h3hr4μ2+hr4ν2+ν+5−5/2τ3ρΓ3ρ+1.
The series form of analytical solution is given as(27)ψμ,ν,τ=ψ0μ,ν,τ+ψ1μ,ν,τ+ψ2μ,ν,τ+ψ3μ,ν,τ+⋯=hr4μ2+hr4ν2+ν+51/2+hhr4μ2+hr4ν2+ν+5−1/2tρΓρ+1−2h2hr4μ2+hr4ν2+ν+5−3/2τ2ρΓ2ρ+1+3h3hr4μ2+hr4ν2+ν+5−5/2τ3ρΓ3ρ+1+⋯,ψμ,ν,τ=ψ0+hτρψ0∑n=0∞n+1Γn+1ρ+1−hτρψ02n.
The exact result is given by(28)ψμ,ν,τ=hr4μ2+hr4ν2+ν+2hτ+5.Example 2.
The biological population model with time noninteger derivative is expressed as(29)∂ρψ∂τρ=∂2∂μ2ψ2+∂2∂ν2ψ2+hψ,with initial condition(30)ψμ,ν,0=μν.
The Laplace transform to equation (29) is expressed as(31)sρLψμ,ν,τ−∑k=0m−1ψkμ,ν,0sρ−k−1=L∂2∂μ2ψ2+∂2∂ν2ψ2+hψ,sρLψμ,ν,τ=ψ0μ,ν,0sρs+L∂2∂μ2ψ2+∂2∂ν2ψ2+hψ,(32)Lψμ,ν,τ=1sμν+1sρL∂2∂μ2ψ2+∂2∂ν2ψ2+hψ.
Using inverse Laplace transform of equation (32),(33)ψμ,ν,τ=μν+L−11sρL∂2∂μ2ψ2+∂2∂ν2ψ2+hψ.
Using the iterative technique described in equations (14) and (15), we obtain the following solution components of Example 2:(34)ψ0μ,ν,τ=μν,ψ1μ,ν,τ=L−11sρL∂2∂μ2ψ02+∂2∂ν2ψ02+hψ0=hμντρΓρ+1,ψ2μ,ν,τ=L−11sρL∂2∂μ2ψ12+∂2∂ν2ψ12+hψ1=h2μντ2ρΓ2ρ+1,ψ3μ,ν,τ=L−11sρL∂2∂μ2ψ22+∂2∂ν2ψ22+hψ2=h3μντ3ρΓ3ρ+1.
The series form of analytical solution is given as(35)ψμ,ν,τ=ψ0μ,ν,τ+ψ1μ,ν,τ+ψ2μ,ν,τ+ψ3μ,ν,τ+⋯=μν+hμντρΓρ+1+h2μντ2ρΓ2ρ+1+h3μντ3ρΓ3ρ+1+⋯.(36)ψμ,ν,τ=μν∑k=0∞hτρkΓkρ+1.
The exact solution is given by(37)ψμ,ν,τ=μνEρhτρ,where Eρhτρ is the Mittag-Leffler function defined as(38)Eρz=∑k=0∞zkΓρk+1.
Asρ⟶1, we have(39)ψμ,ν,τ=μν∑k=0∞hτkk!=μνehτ.Example 3.
The biological population model with time noninteger derivative is expressed as(40)∂ρψ∂τρ=∂2∂μ2ψ2+∂2∂ν2ψ2+ψ,with initial condition(41)ψμ,ν,0=sinμsinhν.
The Laplace transform to equation (40) is expressed as(42)sρLψμ,ν,τ−∑k=0m−1ψkμ,ν,0sρ−k−1=L∂2∂μ2ψ2+∂2∂ν2ψ2+ψ,sρLψμ,ν,τ=ψ0μ,ν,0sρs+L∂2∂μ2ψ2+∂2∂ν2ψ2+ψ,(43)Lψμ,ν,τ=1sμν+1sρL∂2∂μ2ψ2+∂2∂ν2ψ2+ψ.
Using inverse Laplace transform of equation (43), we obtain(44)ψμ,ν,τ=μν+L−11sρL∂2∂μ2ψ2+∂2∂ν2ψ2+ψ.
Using the iterative technique described in equations (14) and (15), we obtain the following solution components of Example 3:(45)ψ0μ,ν,τ=sinμsinhν,ψ1μ,ν,τ=L−11sρL∂2∂μ2ψ02+∂2∂ν2ψ02+ψ0=sinμsinhντρΓρ+1,ψ2μ,ν,τ=L−11sρL∂2∂μ2ψ12+∂2∂ν2ψ12+ψ1=sinμsinhντ2ρΓ2ρ+1,ψ3μ,ν,τ=L−11sρL∂2∂μ2ψ22+∂2∂ν2ψ22+ψ2=sinμsinhντ3ρΓ3ρ+1.
The series form of analytical solution is given as(46)ψμ,ν,τ=ψ0μ,ν,τ+ψ1μ,ν,τ+ψ2μ,ν,τ+ψ3μ,ν,τ+⋯=sinμsinhν+sinμsinhντρΓρ+1+sinμsinhντ2ρΓ2ρ+1+sinμsinhντ3ρΓ3ρ+1+⋯,ψμ,ν,τ=sinμsinhν∑k=0∞τkρΓρk+1,
The exact result is given by(47)ψμ,ν,τ=sinμsinhνEρhτρ.
Asρ⟶1, we have(48)ψμ,ν,τ=sinμsinhν∑k=0∞τkk!=sinμsinhνeτ.Example 4.
The biological population model with time noninteger derivative is expressed as(49)∂ρψ∂τρ=∂2∂μ2ψ2+∂2∂ν2ψ2+ψ1−rψ,with initial condition(50)ψμ,ν,0=exp1/2r/2μ+ν.
The Laplace transform to equation (49) is expressed as(51)sρLψμ,ν,τ−∑k=0m−1ψkμ,ν,0sρ−k−1=L∂2∂μ2ψ2+∂2∂ν2ψ2+ψ1−rψ,sρLψμ,ν,τ=ψ0μ,ν,0sρs+L∂2∂μ2ψ2+∂2∂ν2ψ2+ψ1−rψ,(52)Lψμ,ν,τ=1sexp1/2r/2μ+ν+1sρL∂2∂μ2ψ2+∂2∂ν2ψ2+ψ1−rψ.
Using inverse Laplace transform of equation (52),(53)ψμ,ν,τ=exp1/2r/2μ+ν+L−11sρL∂2∂μ2ψ2+∂2∂ν2ψ2+ψ1−rψ.
Using the iterative technique described in equations (14) and (15), we obtain the following solution components of Example 4:(54)ψ0μ,ν,τ=exp1/2r/2μ+ν,ψ1μ,ν,τ=L−11sρL∂2∂μ2ψ02+∂2∂ν2ψ02+ψ01−rψ0=exp1/2r/2μ+ντρΓρ+1,ψ2μ,ν,τ=L−11sρL∂2∂μ2ψ12+∂2∂ν2ψ12+ψ11−rψ1=exp1/2r/2μ+ντ2ρΓ2ρ+1,ψ3μ,ν,τ=L−11sρL∂2∂μ2ψ22+∂2∂ν2ψ22+ψ21−rψ2=exp1/2r/2μ+ντ3ρΓ3ρ+1.
The series form of analytical solution is given as(55)ψμ,ν,τ=ψ0μ,ν,τ+ψ1μ,ν,τ+ψ2μ,ν,τ+ψ3μ,ν,τ+⋯=exp1/2r/2μ+ν+exp1/2r/2μ+νtρΓρ+1+exp1/2r/2μ+ντ2ρΓ2ρ+1+exp1/2r/2μ+ντ3ρΓ3ρ+1+⋯,ψμ,ν,τ=exp1/2r/2μ+ν∑k=0∞τkρΓρk+1.
The exact result is given by(56)ψμ,ν,τ=exp1/2r/2μ+νEρhτρ.
Asρ⟶1, we have(57)ψμ,ν,τ=exp1/2r/2μ+ν∑k=0∞τkk!=exp1/2r/2μ+ν+τ.


## 5. Review Results

In the present work, ILTM is used to solve some important biological models of noninteger order. The solution of the suggested technique is to explain with the help of its graphical representation. Figure1 shows the solution graphs of exact and ILTM for Example 1 at ρ=1. It is verified that the ILTM solution is closely related to the exact solution. In Figure 2, the error analysis of ILTM for Example 1 is discussed. It is observed that the considered technique has an accuracy of a sufficient degree. Similarly, in Figure 3, the solution plot of ILTM and exact solution is shown for Example 2. These solution graphs are very close and confirmed the reliability of the suggested method. Moreover, the higher degree of accuracy is achieved as represented by Figure 4. The exact and ILTM results of Example 3 are compared in Figure 5. The solution for both exact solution and ILTM solution is identical and supports the reliability of the suggested method. In Figure 6, the solution at various fractional orders of Example 3 is calculated. It is investigated that as fractional order approaches to an integer order, fractional-order solutions are convergent to integer-order solutions. In Figure 7, the same graphical representation has been made for the exact and ILTM solution of Example 4. Figure 7 provides the graphical layout of the solution of Example 4 at different fractional orders. The convergence phenomena of the solutions at different fractional orders can be seen in Figure 8.Figure 1
The solution plot of Example 1: (a) exact solution and (b) ILTM solution atρ=1, h=0.01, and r=48.
(a)
(b)Figure 2
The absolute error for Example 1 atρ=1.Figure 3
Solution graph of Example2: (a) exact solution and (b) ILTM solution at ρ=1, h=0.01, and r=48 .
(a)
(b)Figure 4
The absolute error for Example2 at ρ=1Figure 5
The solution plot of Example3: (a) exact solution and (b) ILTM solution at ρ=1,h=0.01, and r=48.
(a)
(b)Figure 6
Different fractional orders ofρ in Example 3
(a)
(b)Figure 7
The solution plot of Example4: (a) exact solution and (b) ILTM solution at ρ=1, h=0.01, and r=48.
(a)
(b)Figure 8
Different fractional orders ofρ in Example 4.
(a)
(b)


## 6. Conclusion

The present research article is related to solve fractional-order biological population models, using an efficient analytical technique. The current method is applied for fractional and integer-order models. The solution graphs for ILTM and exact solutions to the problems are plotted. It is investigated that the ILTM results are in strong agreement within the actual solutions of the current technique. The ILTM solutions of the problems at different fractional orders are also shown with the help of their graphical representation. The phenomena of convergence fractional-order solutions toward the integer-order solution are observed. This behavior of the obtained solution has confirmed the efficiency of the suggested scheme. Due to an effective and straight forward implementation, the suggested method can be modified for the solution of other FPDEs arising in applied sciences.



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