# Fibonacci Wavelet Method for the Numerical Solution of Nonlinear Reaction-Diffusion Equations of Fisher-Type

Authors: Naied A. Nayied; Firdous A. Shah; M. A. Khanday
Journal: Journal of Mathematics
 (2023)

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## Abstract

This article aims to propose an efficient Fibonacci wavelet-based collocation method for solving the nonlinear reaction-diffusion equation of Fisher-type. The underlying numerical scheme starts by formulating operational matrices of integration corresponding to the Fibonacci wavelets. Besides, we study the error analysis and convergence theorem of the proposed technique. Subsequently, a set of algebraic equations are formed corresponding to the given problem, which could be handled via any conventional method, for instance, the Newton iteration technique. To demonstrate the efficiency of the proposed wavelet-based numerical method, we compare the obtained absolute,L∞, L2, and root mean square (RMS) error norms with the existing Lie symmetry method and cubic trigonometric B-spline (CTB) differential quadrature method in tabular form. From the numerical outcomes, it is ascertained that the proposed numerical technique is computationally more effective and yields precise outcomes in comparison to the existing ones.


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## Body

## 1. Introduction

The challenges of the propagation of nonlinear waves have captivated researchers and scientists for nearly two centuries, especially the propagation of waves in water, which is the most difficult of all, in which significant work has been conducted by Rayleigh, Stokes, Korteweg, de Vries, Benard, Boussinesque, and Fisher. This article is aimed at solving Fisher’s equation, which describes how a dominant gene grows and spreads on its own. Fisher studies a population that is spread linearly in a uniformly dense environment. If the mutation occurs anywhere in the ecosystem, the mutant gene is predicted to spread, posing a threat to allelomorphs who previously occupied the same spot. This occurrence is first eradicated in the vicinity of the mutation and then throughout the remainder of its range [1]. The most interesting practical application of genetic modeling involve animal breeding and treatment of their chromosomal disorders. The one-dimensional reaction-diffusion equation of Fisher’s type plays a significant role in characterizing diverse viewpoints in genetic engineering. The primary purpose of Fisher’s equation is to investigate the wave-like spread of beneficial quality gene in a population [2]. Moreover, Fisher’s equation is a nonlinear partial differential equation of second-order that arises in heat and mass transfer, ecology, and biology. Also, Fisher’s equation is of tremendous use in physical, biological, and chemical sciences such as gene propagation [3], autocatalytic chemical reactions [4], neurophysiology [5], tissue engineering [6], and population dynamics that comprise problems such as nonlinear population evolution in a one-dimensional ecosystem and neutron population in a nuclear reaction. The most generic form of the nonlinear reaction-diffusion equation of Fisher-type is expressed as follows:(1)∂ux,τ∂τ=K∂2ux,τ∂x2+μFu,subjected to the conditions,(2)ux,0=y0x,∂ux,0∂t=y1xu0,τ=v0τ,u1,τ=v1τ,where K, μ, F, τ, and x denote the diffusion coefficient, constant of reaction factor, nonlinear reaction term, time, and spatial coordinate, respectively. Originally, the Fisher’s equation appeared in the model governing the transmission of a mutant gene, wherein the parameter μ is meant for the density of advantageous mutation.Over the last couple of decades, different analytical and numerical methods have been employed by many researchers for obtaining the solution of nonlinear reaction-diffusion equation of Fisher-type as given in equation (1). For instance, Al-Khaled numerically studied the equilibrium between linear diffusion and nonlinear reaction via the Sinc collocation method [7]. Olmos and Shizgal developed pseudospectral method for obtaining an approximate solution of the prototypical reaction-diffusion Fisher’s equation [8]. In addition, Mittal and Arora have applied the method of cubic B-splines for the numerical solution of the Fisher’s equation under the constraint that the diffusion term numerically precedes the reaction term [9]. Later on, Verma et al. employed the classical Lie symmetry methodology together with the polynomial differential quadrature method for obtaining the approximate solutions of nonlinear diffusion equations of Fisher-type. The symmetries and invariant solutions pertaining to these equations were examined using the Lie symmetry approach, that converts the governing equations into ordinary differential equations [10]. Tamsir et al. presented a hybrid method, which is a combination of the base functions (cubic trigonometric B-splines) and differential quadrature procedure for obtaining the approximate solution of the reaction-diffusion equation of Fisher-type [11]. Recently, Jebreen presented an efficient algorithm relying upon the method of finite differences and the wavelet Galerkin method for solving Fisher’s equation. It is imperative to point out that, for discretizing the given interval of time into certain time steps, the authors have employed the Crank–Nicolson scheme, so that at each time step, the equation reduces to an ordinary differential equation. Subsequently, in order to solve the resulting ordinary differential equations, they have implemented the multi-wavelet Galerkin scheme [12].However, wavelet-based numerical methods have time and proved to be promising alternatives to all the existing numerical approaches. This versatile applicability is owed to their lucid numerical procedure, computational reliability, increased convergence rate, and ability to localize the small-scale variations of solutions [13, 14]. In the wide spectrum of different wavelet families, the most widely employed wavelets in different physical as well as biological problems encompass the Haar, Gaugenbaur, Bernoulli, and Chebyshev wavelets [15–17]. Nonetheless, the Fibonacci wavelets are a recent wavelet family that has received considerable interest from researchers working in different aspects of science and engineering, primarily for the reason that they are based on the well-known class of Fibonacci polynomials.Keeping in view the pleasant features of the family of Fibonacci wavelets, it is of critical significance to formulate an efficient Fibonacci wavelet-based numerical scheme for the numerical treatment of Fisher’s equation given in equation (1). The obtained results demonstrate that the proposed numerical scheme is highly reliable, particularly for approximating the smooth and piecewise smooth functions. The proposed numerical scheme starts with the construction of Fibonacci wavelet-based operational matrices of integration via the celebrated approach of Chen and Haiso [18]. Subsequently, the given problem is reduced to a conventional set of algebraic equations which could be handled conveniently via any traditional method, for instance, the Newton iteration scheme.A brief account of the section-wise breakdown of the rest of the content is as follows: Section2 is completely concerned with an exposition on the Fibonacci wavelets together with the corresponding operational matrices of integration, which are to be used in solving the nonlinear reaction-diffusion equation Fisher-type (1). To ascertain the numerical error and the convergence of the proposed procedure a detailed analysis is given in Section 3. The formal solution corresponding to the equation (1) is presented in Section 4, wherein the description of the Fibonacci wavelet-based numerical method is interpreted in the section entitled “Method of solution.” In Section 5, the validity of the proposed numerical scheme is illustrated via a couple of examples, which lucidly demonstrate the supremacy of the newly formulated Fibonacci wavelet-based numerical method. The discussion is wrapped up with certain concluding remarks and an impetus to the future research work in Section 6.


## 2. Fibonacci Wavelets

Here, we shall illustrate the construction of a new family of wavelets via the well-known Fibonacci polynomials and then proceed to the formulation of Fibonacci wavelet-based operational matrices of integration by adopting the approach of Chen and Haiso [18]. Nonetheless, the prominent quasi-linearization technique is also revisited so that the given nonlinear differential equation is converted into a linear differential equation.


### 2.1. Fibonacci Wavelets and Function Approximation

For anyx∈ℝ+, the Fibonacci polynomials are defined by the following recurrence formula:(3)P˜m+2x=xP˜m+1x+P˜mx,m∈ℕ,with P˜0x=0,P˜1x=1 [19]. Without loss of generality, the Fibonacci polynomials are also defined as follows:(4)P˜m+1x=1,m=0,x,m=1,xP˜mx+P˜m−1x,m≥2.For anyx∈0,1, the Fibonacci wavelets on the closed interval [0, 1] are mathematically defined by the following equation [20]:(5)ψn,mx=2k−1/2SmP˜m2k−1x−n+1,n−12k−1≤x<n2k−1,0,otherwise,where k and n, respectively, represents the level of resolution and translation parameters with k=1,2,…, n=1,2,…,2k−1, and P˜mx is the mth degree Fibonacci polynomial as follows [21]:(6)P˜mx=∑i=0m/2m−iixm−2i,m≥0,with ⋅ representing the usual floor function. In equation (5), the entity 1/Sm is a normalization factor, that is explicitly obtained via the following equation:(7)Sm=∫01P˜mxP˜mxdx,m=0,1,…,M−1,=∑i=0m/2∑j=0m/2m−iim−jj2m−i−j+1−1.For the casek=2,M=3, the corresponding collection of Fibonacci wavelets takes the form as follows:(8)ψ1,0x=2,ψ1,1x=26x,ψ1,2x=15141+4x2,x∈0,12,ψ2,0x=2,ψ2,1x=62x−1,ψ2,2x=3072x2−2x+1.x∈12,1..Any functionux∈L2ℝ can be approximated via Fibonacci wavelets as follows:(9)ux≈uk,Mx=∑n=12k−1∑m=0M−1hn,mψn,mx,where hn,m denotes the Fibonacci wavelet coefficients given by the following equation:(10)hn,m=u,ψn,m=∫01uxψn,mxdx.The corresponding matrix form of equation (9) is given by the following equation:(11)F=HTΨx,where H is the row vector of the form.(12)H=h1,0,h1,1,…,h1,M−1,h2,0,h2,1,…,h2,M−1,…,h2k−1,0,h2k−1,1,…,h2k−1,M−1T.The matrixΨx appearing in equation (11) is the Fibonacci wavelet matrix of order 1×2k−1M and is given by the following equation:(13)Ψx=ψ1,0,ψ1,1,…,ψ1,M−1,ψ2,0,ψ2,1,…,ψ2,M−1,…,ψ2k−1,0,ψ2k−1,1,…,ψ2k−1,M−1T.To facilitate the intended approximation via the Fibonacci wavelets, the collocation points are chosen as follows:(14)xℓ=2ℓ−12kM,ℓ=1,2,…,2k−1M.


### 2.2. Operational Matrices of Integration via Fibonacci Wavelets

Here, our aim is to initiate the formal construction of operational matrices of integration corresponding to the Fibonacci wavelets (5). We reiterate that the strategy is adopted from the approach of Chen and Hsiao [18], as follows:(15)∫0xΨn,msds≅PΨn,mx,where P denotes the Fibonacci wavelet operational matrix of order 2k−1M×2k−1M. Let the Fibonacci wavelets ψn,m belongs to a family of Ψ given by equation (13), then ψn,m can recast as follows:(16)ψn,mx=2k−1/2P˜m2k−1x−n+1χn−1/2k−1,n/2k−1x,n=1,2,…,2k−1,m=0,1,…,M−1,where χx denotes the characteristic function.Invoking the power series representation of Fibonacci polynomials (6), relation (16) becomes as follows:(17)ψn,mx=2k−1/2Sm∑i=0m/2m−ii2k−1x−n+1m−2iχn−1/2k−1,n/2k−1x=2k−1/2Sm∑i=0m/2∑j=0m/2m−iim−jj2kj−jxj1−nm−2i−jχn−1/2k−1,n/2k−1x.Integrating equation (17) both sides, we obtain the following equation:(18)∫0xψn,msds=2k−1/2Sm∑i=0m/2∑j=0m/2m−iim−jj2kj−j1−nm−2i−jFjx,where(19)Fjx=∫n−1/2k−1xsjχn−1/2k−1,n/2k−1xds+∫n−1/2k−1n/2k−1sjχn−1/2k−1,n/2k−1xds,0≤x≤m−2i.The functionFjx can now be represented as follows:(20)Fjx=∑r=12k−1∑s=0M−1αp,qψp,qx.Plugging equation (20) in equation (18), we have the following equation:(21)∫0xψn,msds=∑r=12k−1∑s=0M−1ϕp,qn,mψp,qx,where(22)ϕp,qn,m=2k−1/2Sm∑i=0m/2∑j=0m−2m−ii2kj−jxj1−nm−2i−jαp,q.Consequently, we obtain the operational matrix of integrationP as follows:(23)P=ϕ1,01,0ϕ1,01,1⋯ϕ2k−1,M−11,0ϕ1,01,1ϕ1,11,1⋯ϕ2k−1,M−11,1⋮⋮⋱⋮ϕ1,02k−1,M−1ϕ1,12k−1,M−1⋯ϕ2k−1,M−12k−1,M−1.On choosingk=2,M=3, and integrating equation (8) about the collocations points given in equation (14) yields as follows:(24)∫0xψ1,0sds=0,36,0,12,0,0TΨ6×6x,∫0xψ1,1sds=−34,0,3510,34,0,0TΨ6×6x,∫0xψ1,2sds=−291051680,3535,14,521,0,0TΨ6×6x,∫0xψ2,0sds=0,0,0,0,36,0TΨ6×6x,∫0xψ2,1sds=0,0,0,−34,0,3510TΨ6×6x,∫0xψ2,2sds=0,0,0,−291051680,3535,14TΨ6×6x.Therefore, equation (15) takes the form as follows:(25)∫0xΨ6×1sds≅P6×6Ψ6x,where(26)P6×6=03601200−34035103400−291051680353514521000000360000−3403510000−291051680353514.


### 2.3. Quasi-Linearization

Bellman and Kalaba [22] first introduced the notion of quasi-linearization technique for obtaining an approximation to the solution of the mth-degree nonlinear differential equations. To proceed the argument, we consider the following differential equation:(27)v″x=fv′x,vx,x,subject to(28)vα=a,vβ=b,α≤x≤β.Application of quasi-linearization technique to equation (27) yields as follows:(29)vr+1″x=fvr′x,vrx,x+vr+1x−vrxfvrvr′x,vrx,x+vr+1′−vr′xfvr′vr′x,vrx,x,with condition vr+1α=a,vr+1β=b.Consequently, the recurrence relation for the nonlinear differential equations of the ordernth is given by the following equation:(30)Lnvr+1x=fvrx,vr′x,…,vrn−1x,x+∑l=0n−1vr+1lx−vrlxfvlvrx,vr′x,…,vrn−1x,x,with vr+1α=a,vr+1β=b. Relation (30) is the desired linear differential equation, which can be recursively solved for vr+1x from vrx.


## 2.1. Fibonacci Wavelets and Function Approximation

For anyx∈ℝ+, the Fibonacci polynomials are defined by the following recurrence formula:(3)P˜m+2x=xP˜m+1x+P˜mx,m∈ℕ,with P˜0x=0,P˜1x=1 [19]. Without loss of generality, the Fibonacci polynomials are also defined as follows:(4)P˜m+1x=1,m=0,x,m=1,xP˜mx+P˜m−1x,m≥2.For anyx∈0,1, the Fibonacci wavelets on the closed interval [0, 1] are mathematically defined by the following equation [20]:(5)ψn,mx=2k−1/2SmP˜m2k−1x−n+1,n−12k−1≤x<n2k−1,0,otherwise,where k and n, respectively, represents the level of resolution and translation parameters with k=1,2,…, n=1,2,…,2k−1, and P˜mx is the mth degree Fibonacci polynomial as follows [21]:(6)P˜mx=∑i=0m/2m−iixm−2i,m≥0,with ⋅ representing the usual floor function. In equation (5), the entity 1/Sm is a normalization factor, that is explicitly obtained via the following equation:(7)Sm=∫01P˜mxP˜mxdx,m=0,1,…,M−1,=∑i=0m/2∑j=0m/2m−iim−jj2m−i−j+1−1.For the casek=2,M=3, the corresponding collection of Fibonacci wavelets takes the form as follows:(8)ψ1,0x=2,ψ1,1x=26x,ψ1,2x=15141+4x2,x∈0,12,ψ2,0x=2,ψ2,1x=62x−1,ψ2,2x=3072x2−2x+1.x∈12,1..Any functionux∈L2ℝ can be approximated via Fibonacci wavelets as follows:(9)ux≈uk,Mx=∑n=12k−1∑m=0M−1hn,mψn,mx,where hn,m denotes the Fibonacci wavelet coefficients given by the following equation:(10)hn,m=u,ψn,m=∫01uxψn,mxdx.The corresponding matrix form of equation (9) is given by the following equation:(11)F=HTΨx,where H is the row vector of the form.(12)H=h1,0,h1,1,…,h1,M−1,h2,0,h2,1,…,h2,M−1,…,h2k−1,0,h2k−1,1,…,h2k−1,M−1T.The matrixΨx appearing in equation (11) is the Fibonacci wavelet matrix of order 1×2k−1M and is given by the following equation:(13)Ψx=ψ1,0,ψ1,1,…,ψ1,M−1,ψ2,0,ψ2,1,…,ψ2,M−1,…,ψ2k−1,0,ψ2k−1,1,…,ψ2k−1,M−1T.To facilitate the intended approximation via the Fibonacci wavelets, the collocation points are chosen as follows:(14)xℓ=2ℓ−12kM,ℓ=1,2,…,2k−1M.


## 2.2. Operational Matrices of Integration via Fibonacci Wavelets

Here, our aim is to initiate the formal construction of operational matrices of integration corresponding to the Fibonacci wavelets (5). We reiterate that the strategy is adopted from the approach of Chen and Hsiao [18], as follows:(15)∫0xΨn,msds≅PΨn,mx,where P denotes the Fibonacci wavelet operational matrix of order 2k−1M×2k−1M. Let the Fibonacci wavelets ψn,m belongs to a family of Ψ given by equation (13), then ψn,m can recast as follows:(16)ψn,mx=2k−1/2P˜m2k−1x−n+1χn−1/2k−1,n/2k−1x,n=1,2,…,2k−1,m=0,1,…,M−1,where χx denotes the characteristic function.Invoking the power series representation of Fibonacci polynomials (6), relation (16) becomes as follows:(17)ψn,mx=2k−1/2Sm∑i=0m/2m−ii2k−1x−n+1m−2iχn−1/2k−1,n/2k−1x=2k−1/2Sm∑i=0m/2∑j=0m/2m−iim−jj2kj−jxj1−nm−2i−jχn−1/2k−1,n/2k−1x.Integrating equation (17) both sides, we obtain the following equation:(18)∫0xψn,msds=2k−1/2Sm∑i=0m/2∑j=0m/2m−iim−jj2kj−j1−nm−2i−jFjx,where(19)Fjx=∫n−1/2k−1xsjχn−1/2k−1,n/2k−1xds+∫n−1/2k−1n/2k−1sjχn−1/2k−1,n/2k−1xds,0≤x≤m−2i.The functionFjx can now be represented as follows:(20)Fjx=∑r=12k−1∑s=0M−1αp,qψp,qx.Plugging equation (20) in equation (18), we have the following equation:(21)∫0xψn,msds=∑r=12k−1∑s=0M−1ϕp,qn,mψp,qx,where(22)ϕp,qn,m=2k−1/2Sm∑i=0m/2∑j=0m−2m−ii2kj−jxj1−nm−2i−jαp,q.Consequently, we obtain the operational matrix of integrationP as follows:(23)P=ϕ1,01,0ϕ1,01,1⋯ϕ2k−1,M−11,0ϕ1,01,1ϕ1,11,1⋯ϕ2k−1,M−11,1⋮⋮⋱⋮ϕ1,02k−1,M−1ϕ1,12k−1,M−1⋯ϕ2k−1,M−12k−1,M−1.On choosingk=2,M=3, and integrating equation (8) about the collocations points given in equation (14) yields as follows:(24)∫0xψ1,0sds=0,36,0,12,0,0TΨ6×6x,∫0xψ1,1sds=−34,0,3510,34,0,0TΨ6×6x,∫0xψ1,2sds=−291051680,3535,14,521,0,0TΨ6×6x,∫0xψ2,0sds=0,0,0,0,36,0TΨ6×6x,∫0xψ2,1sds=0,0,0,−34,0,3510TΨ6×6x,∫0xψ2,2sds=0,0,0,−291051680,3535,14TΨ6×6x.Therefore, equation (15) takes the form as follows:(25)∫0xΨ6×1sds≅P6×6Ψ6x,where(26)P6×6=03601200−34035103400−291051680353514521000000360000−3403510000−291051680353514.


## 2.3. Quasi-Linearization

Bellman and Kalaba [22] first introduced the notion of quasi-linearization technique for obtaining an approximation to the solution of the mth-degree nonlinear differential equations. To proceed the argument, we consider the following differential equation:(27)v″x=fv′x,vx,x,subject to(28)vα=a,vβ=b,α≤x≤β.Application of quasi-linearization technique to equation (27) yields as follows:(29)vr+1″x=fvr′x,vrx,x+vr+1x−vrxfvrvr′x,vrx,x+vr+1′−vr′xfvr′vr′x,vrx,x,with condition vr+1α=a,vr+1β=b.Consequently, the recurrence relation for the nonlinear differential equations of the ordernth is given by the following equation:(30)Lnvr+1x=fvrx,vr′x,…,vrn−1x,x+∑l=0n−1vr+1lx−vrlxfvlvrx,vr′x,…,vrn−1x,x,with vr+1α=a,vr+1β=b. Relation (30) is the desired linear differential equation, which can be recursively solved for vr+1x from vrx.


## 3. Error Estimation and Convergence Analysis

This section deals with estimation of the discretization errors and convergence results associated with the Fibonacci wavelet method for the nonlinear reaction-diffusion equation of Fisher-type.Theorem 1.
Given a sufficiently smooth real valued functionux∈CM0,1, suppose that u∗x=∑n=12k−1∑m=0M−1hn,mψn,mx is the Fibonacci wavelet expansion of ux∈0,1. Then, the norm of the truncated error ex is given by the following equation:(31)ex2=u−u∗2≤RM!2M+1,with R=maxx∈0,1uMx.Proof.
[23]Theorem 2.
Given a functionux∈L20,1, which is bounded by M˜. Then, the Fibonacci wavelet expansion given by equation (9), which converges uniformly to ux with the following equation:(32)hn,m<M˜δs2k−1/22m,δ=1Wm.Proof.
[19].Theorem 3.
The approximate solutionux,τ=HTΨx obtained through Fibonacci wavelets converge to the exact solutions of the examples 1, 2, 3, and 4, respectively as 2k−1M⟶∞.Proof.
[23].


## 4. Method of Solution

This section constitutes the centre piece of this article and here we jointly employ the Fibonacci wavelet expansion and the already obtained operational matrices of integration for the numerical treatment of the nonlinear reaction-diffusion equation given by equation (1). For solving equation (1), it is imperative to recall the nonlinear reaction-diffusion equation of the following Fisher-type:(33)∂ux,τ∂τ=∂2ux,τ∂x2+Fu,with the initial and boundary conditions.(34)ux,0=y0x,∂∂τux,0=y1xu0,τ=v0τ,u1,τ=v1τ.For the purpose of solving equation (33), we shall approximate the highest-order partial derivative of equation (33) via the basic functions determined by the Fibonacci wavelet as follows:(35)∂3ux,τ∂x2∂τ=∑ℓ=12k−1Mhℓψℓx,where hℓ denotes the unknown Fibonacci wavelet coefficients row vector given by equation (10). Integrating equation (35) with respect to τ over the interval spanning from τ to τs yields as follows:(36)∂2ux,τ∂x2=τ−τs∑ℓ=12k−1Mhℓψx+∂2ux,τs∂x2.Integrating equation (36) two times with respect to x between the limits 0 to x, we obtain the following equation:(37)ux,τ=τ−τs∑ℓ=12k−1MhℓPℓ2x+ux,τs−u0,τs+x∂u0,τ∂x−∂u0,τs∂x+u0,τ.Utilizing the boundary constraints (34) yields as follows:(38)ux,τ=τ−τs∑ℓ=12k−1MhℓPℓ2x+ux,τs−v0τs+xv0′τ−v0′τs+v0τ.Taking derivative on both sides of equation (51) with respect to τ, we obtain the following equation:(39)∂ux,τ∂τ=∑ℓ=12k−1MhℓPℓ2x+x∂v0′τ∂τ+∂v0τ∂τ.Puttingx=1 in equations (38) and (39) yields as follows(40)v0′τ−v0′τs=−τ−τs∑ℓ=12k−1MhℓPℓ21+v1τ−v0τ−v1τs+v0τs,and(41)∂v0′τ∂τ=∂u1,τ∂τ−∑ℓ=12k−1MhℓPℓ21−∂v0τ∂τ,which is expressed as follows:(42)∂v0′τ∂τ=∂v1τ∂τ−∑i=02k−1MhℓPℓ21−∂v0τ∂τ.Substituting equations (54)–(42) into equations (51)–(39) and discretizing the equations by assuming x⟶xℓ, τ⟶τs+1, we have the following equation:(43)uxℓ,τs+1=τs+1−τs∑ℓ=12k−1MhℓPℓ2xℓ−xℓτs+1−τs∑ℓ=12k−1MhℓPℓ21+xℓuxℓ,τs+xℓv1τ−xℓv0τs+1−xℓv1τs+xℓv0τs−v0τs,(44)∂uxℓ,τs+1∂τ=∑ℓ=12k−1MhℓPℓ2xℓ−xℓ∑ℓ=12k−1MhℓPℓ21+xℓ∂v1τs+1∂x−xℓ∂v0τs+1∂τ.Implementing equations (36) and (44) in equation (33) yields a set of algebraic equations involving the Fibonacci wavelets as follows:(45)∑i=02k−1MhℓPℓ2xℓ−xℓ∑ℓ=12k−1MhℓPℓ21−Δτ∑ℓ=12k−1Mhℓψxℓ=μFxℓ,τs+1+∂2uxℓ,τs∂x2−xℓ∂g1τs+1∂x+xℓ∂g0τs+1∂τ,where Δτ=τs+1−τs. Subsequently, we solve solving the system of algebraic equations obtained in (45) for the unknown coefficients hℓ by using the Newton method. In the sequel, implementing these values of hℓ in (43), we obtain the approximate solution via Fibonacci wavelet of the nonlinear reaction-diffusion equation of Fisher-type (33).


## 5. Numerical Examples and Discussion

To section is meant to explore the effectiveness of the above formulated numerical scheme in the process of achieving an approximate solution for the Fisher-type equation given by equation (33). The efficiency and validity of the proposed numerical technique is exhibited via certain test problems which have been already solved via some other numerical techniques available in the literature. The precision of the newly formulated procedure is measured by absolute, L∞, L2, and root mean square (RMS) error norms given as follows:(46)Eabs=ux,τ−un,mx,τL∞=max1≤ℓ≤2k−1Muxℓ,τℓ−un,mxℓ,τℓL2=∑ℓ=12k−1Muxℓ,τℓ−un,mxℓ,τℓ21/2RMS=12k−1M∑ℓ=12k−1Muxℓ,τℓ−un,mxℓ,τℓ21/2,where ux,τ and un,mx,τ denotes exact and approximate solutions, respectively.Example 1.
Consider the nonlinear reaction-diffusion equation Fisher-type [11]:(47)∂ux,τ∂τ=∂2ux,τ∂x2+αu2x,τ1−ux,τ,0<x<1and0≤τ≤1,with the initial conditions:(48)ux,0=12−12tanh1+2x4,∂ux,0∂τ=18sech22x+44,and the boundary conditions:(49)u0,τ=12−12tanh1−τ4,u1,τ=12−12tanh1+241−τ2.
The exact solution corresponding to equation (47) is as follows:(50)ux,τ=12−12tanh1+24x−τ2.
Now, by implementing the already discussed quasi-linearization methodology, the nonlinear equation (47) is converted into the linear differential equation:(51)∂ur+1∂τ−∂2ur+1∂x2−2αurur+1+3αur2ur+1=2αur3+αur2,with initial guess:(52)u0x,τ=y0x+τy1x=12−12tanh1+2x4+τ8sech22x+44.
Next, by adopting the procedure as in Section4, the system of algebraic equations of (51) is obtained as follows:(53)∑ℓ=12k−1MhℓPℓ2xℓ−xℓ∑ℓ=12k−1MhℓPℓ21−Δτ∑ℓ=12k−1Mhℓψℓxℓ−2Δτur∑ℓ=12k−1MhℓPℓ2xℓ+2Δτurxℓ∑ℓ=12k−1MhℓPℓ21+3Δτur2∑ℓ=12k−1MhℓPℓ2xℓ−3Δur2xℓ∑ℓ=12k−1MhℓPℓ21=∂2uxℓ,τs∂x2+2uruxℓ,τs−2uru0,τs−2xℓuru1,τs+2xℓuru0,τs−2xℓuru0,τs+1+2xℓuru1,τs+1+2uru0,τs+1−ur2−3ur2uxℓ,τs+3ur2u0,τs+3xℓur2u1,τs−3xℓur2×u0,τs+3xur2u0,τs+1−3ur2u1,τs+1−3ur2u0,τs+1+2ur3−x∂u1,τs+1∂τ+x∂u0,τs+1∂τ.
For demonstrating the precision of the numerical scheme based on the Fibonacci wavelets, the obtainedL∞, L2 and RMS error norms have been compared to that of Lie symmetry method [10] over different values of τ, whereas the same results have also been compared to CTB differential quadrature method [11] for different values of τ as tabulated in Table 1. In view of Table 1, it is quite clear that the approximate solution is more effective in lieu of the Lie symmetry and CTB differential quadrature methods [10, 11]. Moreover, Figures 1 and 2 also demonstrate that the obtained numerical solutions are in good agreement with the exact solutions. Also, Figure 3 lucidly depicts that with an increase in the values of k and M, the absolute error diminishes.Table 1
Comparison ofL∞, L2, and RMS error norms of present method with [10, 11] for Example 1 at different τ.
τLie symmetry [10]CTB differential quadrature method [11]Present methodL∞L2RMSL∞L2RMSL∞L2RMS0.22.105×10−54.475×10−33.633×10−31.050×10−67.022×10−71.604×10−71.2781×10−82.2769×10−73.6001×10−80.51.736×10−54.444×10−33.821×10−31.124×10−68.470×10−71.848×10−71.4413×10−82.5663×10−74.0576×10−81.01.103×10−52.690×10−32.464×10−31.203×10−69.499×10−72.154×10−71.7190×10−83.0590×10−74.8367×10−83.03.788×10−52.277×10−52.254×10−51.215×10−69.285×10−73.160×10−72.4894×10−84.3704×10−76.9102×10−85.02.777×10−71.823×10−71.818×10−77.057×10−72.997×10−73.160×10−71.9290×10−83.4215×10−75.4099×10−8Figure 1
Approximate and exact solution for differentτ of Example 1.Figure 2
(a) Approximate solution and (b) exact solution of Example1 at k=2, M=3.
(a)(b)Figure 3
(a) Absolute error atk=4, M=5 and (b) absolute error at k=4, M=5 of Example 1.
(a)(b)Example 2.
Consider the following nonlinear reaction-diffusion equation Fisher-type [11]:(54)∂ux,τ∂τ=∂2ux,τ∂x2+αux,τux,τ−δ1−ux,τ,0<x<1and0≤τ≤1,with the initial conditions:(55)ux,0=12+δ2+12−δ2tanh1+24δ−1x∂ux,0∂τ=δδ2−18sech22δx−2x+44,and the boundary conditions:(56)u0,τ=12+δ2+12−δ2tanh1+24δ−1−1+δτ2u1,τ=12+δ2+12−δ2tanh1+24δ−11−1+δτ2.
The exact solution of problem (54) is as follows:(57)ux,τ=12+δ2+12−δ2tanh1+24δ−1x−1+δτ2.
Applying the quasi-linearization technique as discussed in Section2 to the nonlinear problem (54), we get a linearized differential equation given by the following equation:(58)∂ur+1∂t−∂2ur+1∂x2+δur+1−2δ+1urur+1+3ur2ur+1=2ur3−δ+1ur2,with initial guess:(59)u0x,τ=12+δ2+12−δ2tanh1+24δ−1x+δδ2−1τ8sech22δx−2x+44.
Now, in consonance to the strategy pointed-out in Section4, the system of algebraic equations of (58) is as follows:(60)∑ℓ=12k−1MhℓPℓ2xℓ−xℓ∑ℓ=12k−1MhℓPℓ21−Δτ∑ℓ=12k−1Mhℓψxℓ+δΔτ∑ℓ=12k−1MhℓPℓ2xℓ−δxℓ×∑ℓ=12k−1MhℓPℓ21−2δ+1Δτur∑ℓ=12k−1MhℓPℓ2xℓ+2δ+1Δτxur∑ℓ=12k−1MhℓPℓ21+3Δτur2∑ℓ=12k−1MhℓPℓ2xℓ−3Δτxur2∑ℓ=12k−1MhℓPℓ21=∂2uxℓ,τs∂x2+δuxℓ,τs−δu0,τsδxℓu1,τs+δxℓu0,τs−δxℓu0,τs+1+δxℓu1,τs+1+δu0,τs+1+2δ+1uruxℓ,τs−2δ+1uru0,τs−2δ+1xℓuru1,τs+2δ+1xℓuru0,τs−2δ+1uru0,τs+1+2δ+1xℓuru1,τs+1+2δ+1xℓuru0,τs+1−δ+1ur2−3ur2uxℓ,τs+3ur2u0,τs+3xur2u1,τs−3xℓur2u0,ts+3xℓur2u0,τs+1−3xℓur2u1,τs+1−3ur2u0,τs+1−3ur2u0,τs+1+2ur2−xℓ∂u1,τs+1∂τ+xℓ∂u0,τs+1∂τ.
In order to examine the efficacy of the newly proposed numerical technique, the approximate solution together with its comparison to the exact solution is plotted graphically in Figures4 and 5. It is quite evident that the numerical outcomes are in good agreement with the analytic solution. From Figure 6, it is clear that with an increase in the values of k and M, the absolute error diminishes. The comparison of L∞, L2, and RMS error norms of the proposed method are compared with the existing Lie symmetry method [10] and the CTB differential quadrature method [11] in Tables 2 and 3. From Tables 2 and 3, we conclude that the Fibonacci wavelet method is more accurate and efficient than the existing ones.Figure 4
Approximate and exact solution for differentτ of Example 2.Figure 5
(a) Approximate solution and (b) exact solution of Example2 at k=4, M=5.
(a)(b)Figure 6
(a) Absolute error atk=4, M=5 and (b) absolute error at k=2, M=3 of Example 2.
(a)(b)Table 2
Comparison ofL∞, L2, and RMS error norms of present method with [10, 11] for Example 2 at different τ and fixed δ=0.5.
τLie symmetry [10]CTB differential quadrature method [11]Present methodL∞L2RMSL∞L2RMSL∞L2RMS0.22.412×10−67.903×10−73.045×10−61.410×10−71.042×10−76.144×10−82.1656×10−92.4644×10−83.8966×10−90.52.227×10−67.225×10−72.802×10−61.364×10−71.106×10−75.705×10−82.2130×10−92.5631×10−84.0526×10−91.01.936×10−66.193×10−72.425×10−61.254×10−71.010×10−74.957×10−82.2858×10−92.7193×10−84.2995×10−93.01.039×10−63.196×10−71.285×10−67.741×10−86.044×10−82.655×10−82.5095×10−93.2481×10−85.1373×10−95.05.228×10−71.577×10−76.425×10−64.159×10−83.201×10−81.336×10−82.6491×10−93.6583×10−85.7843×10−9Table 3
Comparison ofL∞, L2, and RMS error norms of present method with [10, 11] for Example 2 at different τ and fixed δ=1.5.
τLie symmetry [10]CTB differential quadrature method [11]Present methodL∞L2RMSL∞L2RMSL∞L2RMS0.22.105×10−57.054×10−35.726×10−31.333×10−79.879×10−88.983×10−87.7196×10−91.2356×10−71.9537×10−80.51.736×10−56.210×10−35.340×10−31.441×10−71.209×10−81.033×10−78.0627×10−91.2982×10−72.0527×10−81.01.103×10−53.335×10−33.055×10−31.533×10−71.346×10−81.275×10−78.6418×10−91.4061×10−72.2232×10−83.06.243×10−53.767×10−53.717×10−38.807×10−86.164×10−82.105×10−71.1060×10−81.8674×10−72.9526×10−85.06.174×10−73.760×10−73.750×10−71.492×10−71.177×10−71.751×10−71.2933×10−82.2537×10−73.5634×10−8Example 3.
Consider the following nonlinear reaction-diffusion equation of the Fisher-type [12]:(61)∂ux,τ∂τ=∂2ux,τ∂x2+ux,τ1−u6x,τ0<x<1and0≤τ≤1,subject to initial conditions:(62)ux,0=12tanh−3x4+121/3,∂ux,0∂τ=1522/3xsech232x−5τ/8641−tanh32x−5τ/82/3,and boundary conditions:(63)u0,τ=12tanh15τ8+121/3,u1,τ=12tanh−34+15τ8+121/3.
The exact solution pertaining to equation (61) is as follows:(64)ux,τ=12tanh−3x4+15τ8+121/3.
By using the quasi-linearization technique as discussed in Section2, nonlinear equation (61) reduces to the following form:(65)∂ur+1∂τ−∂2ur+1∂x2+7ur6ur+1−ur+1=6ur7,with initial guess:(66)u0x,τ=12tanh−3x4+121/3+1522/3xτsech232x−5τ/8641−tanh32x−5τ/82/3.
Adopting the same methodology as discussed in Section4, the system of algebraic equations corresponding to equation (65) is obtained as follows:(67)∑ℓ=12k−1MhℓPℓ2xℓ−x∑ℓ=12k−1MhℓPℓ21−Δτ∑ℓ=12k−1MhℓPℓ2xℓ+7Δτur6∑ℓ=12K−1MhℓPℓ2xℓ−7Δτur6x∑ℓ=12k−1MhℓPℓ21−Δτ∑ℓ=12k−1MhℓPℓ2xℓ+Δτxℓ∑ℓ=12k−1MhℓPℓ21=∂2uxℓ,τs∂x2−7ur6xℓ,ts+7ur6u0,τs+7ur6xℓu1,τs−7ur6u0,τs+7ur6u0,τs+1−7ur6xℓu1,τs+1−7ur6u0,τs+1+uxℓ,τs−u0,τs−xℓu1,τs+xℓu0,τs−xℓu0,τs+1−xℓu0,τs+1+xℓu1,τs+1+u0,τs+1+6ur7−xℓ∂u1,τs+1∂τ+xℓ∂u0,τs+1∂τ.
To demonstrate the efficiency and accuracy of the proposed method, we have compared the obtained results with the analytic solution in Figures7 and 8. From Figure 7, we infer that as time τ increases, the approximate solution also increases. A graphical comparison is also established between the proposed method and the exact solution in Figure 8 graphically, which shows that the proposed technique is in reasonable with the exact solution. Moreover, from Figure 9 and Table 4, we have also shown that the obtained absolute error decreases as we increase the wavelet parameters k and M. It is quite evident that the approximate solution obtained via Fibonacci wavelet method is good agreement with the analytic solution.Figure 7
Approximate and exact solution for differentτ of Example 3.Figure 8
(a) Approximate solution and (b) exact solution of Example3.
(a)(b)Figure 9
(a) Absolute error atk=4, M=5 and (b) absolute error at k=2, M=4 of Example 3.
(a)(b)Table 4
Comparison of the absolute error by the proposed method for different values ofk, M, of Example 3.
x,τk=2,M=3k=3,M=4k=4,M=5(0.1, 0.1)3.536×10−52.425×10−58.521×10−6(0.2, 0.2)1.796×10−42.952×10−51.146×10−5(0.3, 0.3)—2.949×10−51.807×10−5(0.4, 0.4)9.085×10−53.704×10−51.555×10−5(0.5, 0.5)1.438×10−42.839×10−52.915×10−5(0.6, 0.6)—5.206×10−69.798×10−6(0.7, 0.7)4.557×10−51.868×10−59.4282×10−6(0.8, 0.8)—8.150×10−67.256×10−6(0.9, 0.9)8.588×10−51.839×10−59.086×10−6Example 4.
Finally, consider the following nonlinear reaction-diffusion equation of Fisher’s type [9]:(68)∂ux,τ∂τ=∂2ux,τ∂x2+α2ux,τ1−ux,τ,0<x<1and0≤τ≤1.
The exact solution of the Example4 is as follows:(69)ux,τ=11+expα2/6x−5α2/6τ2.
The initial and boundary conditions are obtained from exact solution (69). Now, implementing the quasi-linearization technique to problem (68), the linearized form of equation (68) is as follows:(70)∂ur+1∂τ−∂2ur+1∂x2+2α2urur+1−α2ur+1=α2ur2,with initial guess:(71)ux,τ=11+expα2/6x2+τ5α2expα2/6x+131+expα2/6x3.
The system of algebraic equations corresponding to equation (71) is as follows:(72)∑ℓ=12k−1MhℓPℓ2xℓ−x∑ℓ=12k−1MhℓPℓ21−Δτ∑ℓ=12k−1Mhℓψxℓ−α2Δτ∑ℓ=12K−1MhℓPℓ2xℓ+α2Δτx×∑ℓ=12k−1MhℓPℓ21+2α2Δτur∑ℓ=12k−1MhℓPℓ2xℓ+2α2urΔτxℓ∑ℓ=12k−1MhℓPℓ2xℓ=∂2uxℓ,τs∂x2−α2ur2xℓ,ts+2α2uru0,τs+2α2urxℓu1,τs−2α2uru0,τs+2α2xℓuru0,τs+1−2α2urxℓu1,τs+1−2α2uru0,τs+1+α2uxℓ,τs−α2u0,τs−α2xℓu1,τs+α2u0,τs−α2xℓu0,τs+1−α2xℓu0,τs+1+α2xℓu1,τs+1+α2u0,τs+1−xℓ∂u1,τs+1∂τ+xℓ∂u0,τs+1∂τ−∂u0,τs+1∂τ.
To show the effectiveness of the proposed method, we have compared the approximate and exact solutions to the problem via different graphical illustrations. In Figures10 and 11, we have compared the approximate solution with the exact solution for different values of α2 via three-dimensional and two-dimensional plots. Figure 10 depicts the comparison of approximate solution and exact solution for different values of α2. Moreover, Figure 11 illustrates the graphical comparison of the approximate solution and exact solution, along with the absolute error which shows that the approximate solution is in good agreement with the exact solution. Furthermore, Tables 5 and 6 compare the L∞ and L2 error norms obtained through the proposed method, with the existing cubic B-spline method [24]. It is quite evident from Tables 5 and 6 that the approximate solution obtained through proposed technique are more accurate than those obtained in [24].Figure 10
(a) Comparison of approximate solution and exact solution atα2=2000. (b) Comparison of approximate solution and exact solution at α2=10000 of Example 4 at k=4,M=5.
(a)(b)Figure 11
(a) Approximate solution, (b) exact solution, and (c) absolute error of Example4 at α2=10000, k=4,andM=5.
(a)(b)(c)Table 5
Comparison ofL∞ and L2 norm errors of proposed method with cubic B-spline method [24] at α2=2000 of Example 4 at k=4,M=5.
τFWMk=4,M=5CubicB-splineL∞L2L∞L20.0012.0940×10−56.7523×10−55.1823×10−39.0884×10−40.00152.4955×10−51.0176×10−42.4526×10−34.4933×10−40.0022.7320×10−51.1606×10−41.1091×10−32.0939×10−40.00252.7349×10−51.2245×10−44.9251×10−49.5672×10−50.0032.7636×10−51.4692×10−42.1682×10−44.6126×10−50.00352.8351×10−51.5020×10−49.5010×10−53.0024×10−50.00352.9192×10−51.5030×10−47.2326×10−52.8783×10−5Table 6
Comparison ofL∞ and L2 norm errors of proposed method with cubic B-spline method [24] at α2=10000 of Example 4 at k=4,M=5.
τFWMk=4,M=5CubicB-splineL∞L2L∞L20.0011.8677×10−51.2789×10−41.9179×10−33.9350×10−40.00157.2565×10−52.5931×10−47.2132×10−31.6280×10−30.0021.8663×10−51.7135×10−41.4389×10−23.3457×10−30.00256.3456×10−52.6142×10−42.2691×10−25.3175×10−30.0032.6404×10−47.6510×10−43.1669×10−27.4283×10−30.00352.0531×10−45.4365×10−44.1033×10−29.6176×10−30.00354.3413×10−48.8186×10−49.4202×10−21.5361×10−2


## 6. Conclusion and Future Work

In the present article, we have introduced an efficient Fibonacci wavelet-based collocation method for solving the nonlinear reaction-diffusion equation Fisher-type. To demonstrate the precision of the proposed wavelet-based numerical method, we have compared the obtained absoluteL∞, L2, and RMS error norms with the existing Lie symmetry and CTB differential quadrature methods. Moreover, we have demonstrated that the proposed method is computationally more effective and yields more precise outcomes than the existing ones. This strategy of obtaining the numerical solution of differential equations is novel to the literature and stands for its efficiency and simplicity. Moreover, this technique is advantageous over the existing ones in the sense that it is computer oriented, compatible, and useful in investigating the problems in science and engineering demanding numerical solutions of nonlinear systems. The present work also stimulates interest in solving the n+1 dimensional models arising in the transport of air, diffusion of neutrons, adsorption of pollutants in soil, and oil reservoir flow transport. This is precisely our future research objective.



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