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

. Tis article aims to propose an efcient Fibonacci wavelet-based collocation method for solving the nonlinear reaction-difusion equation of Fisher-type. Te 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 efciency of the proposed wavelet-based numerical method, we compare the obtained absolute, L ∞ , L 2 , and root mean square (RMS) error norms with the existing Lie symmetry method and cubic trigonometric B -spline (CTB) diferential quadrature method in tabular form. From the numerical outcomes, it is ascertained that the proposed numerical technique is computationally more efective and yields precise outcomes in comparison to the existing ones.


Introduction
Te 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 difcult of all, in which signifcant work has been conducted by Rayleigh, Stokes, Korteweg, de Vries, Benard, Boussinesque, and Fisher.Tis 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.Tis occurrence is frst eradicated in the vicinity of the mutation and then throughout the remainder of its range [1].Te most interesting practical application of genetic modeling involve animal breeding and treatment of their chromosomal disorders.Te one-dimensional reaction-difusion equation of Fisher's type plays a signifcant role in characterizing diverse viewpoints in genetic engineering.Te primary purpose of Fisher's equation is to investigate the wave-like spread of benefcial 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.Te most generic form of the nonlinear reaction-difusion equation of Fisher-type is expressed as follows: subjected to the conditions, u(x, 0) � y 0 (x), zu(x, 0) zt � y 1 (x) where K, μ, F, τ, and x denote the difusion coefcient, 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, diferent analytical and numerical methods have been employed by many researchers for obtaining the solution of nonlinear reactiondifusion equation of Fisher-type as given in equation (1).For instance, Al-Khaled numerically studied the equilibrium between linear difusion and nonlinear reaction via the Sinc collocation method [7].Olmos and Shizgal developed pseudospectral method for obtaining an approximate solution of the prototypical reaction-difusion 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 difusion term numerically precedes the reaction term [9].Later on, Verma et al. employed the classical Lie symmetry methodology together with the polynomial diferential quadrature method for obtaining the approximate solutions of nonlinear difusion equations of Fisher-type.Te symmetries and invariant solutions pertaining to these equations were examined using the Lie symmetry approach, that converts the governing equations into ordinary diferential equations [10].Tamsir et al. presented a hybrid method, which is a combination of the base functions (cubic trigonometric B-splines) and diferential quadrature procedure for obtaining the approximate solution of the reactiondifusion equation of Fisher-type [11].Recently, Jebreen presented an efcient algorithm relying upon the method of fnite diferences 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 diferential equation.Subsequently, in order to solve the resulting ordinary diferential 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.Tis versatile applicability is owed to their lucid numerical procedure, computational reliability, increased convergence rate, and ability to localize the smallscale variations of solutions [13,14].In the wide spectrum of diferent wavelet families, the most widely employed wavelets in diferent physical as well as biological problems encompass the Haar, Gaugenbaur, Bernoulli, and Chebyshev wavelets [15][16][17].Nonetheless, the Fibonacci wavelets are a recent wavelet family that has received considerable interest from researchers working in diferent 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 signifcance to formulate an efcient Fibonacci wavelet-based numerical scheme for the numerical treatment of Fisher's equation given in equation (1).Te obtained results demonstrate that the proposed numerical scheme is highly reliable, particularly for approximating the smooth and piecewise smooth functions.Te 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: Section 2 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-difusion equation Fisher-type (1).To ascertain the numerical error and the convergence of the proposed procedure a detailed analysis is given in Section 3. Te 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.Te discussion is wrapped up with certain concluding remarks and an impetus to the future research work in Section 6.

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 diferential equation is converted into a linear diferential equation.

Fibonacci Wavelets and Function Approximation.
For any x ∈ R + , the Fibonacci polynomials are defned by the following recurrence formula: with  P 0 (x) � 0,  P 1 (x) � 1 [19].Without loss of generality, the Fibonacci polynomials are also defned as follows:

Journal of Mathematics
For any x ∈ [0, 1), the Fibonacci wavelets on the closed interval [0, 1] are mathematically defned by the following equation [20]: where k and n, respectively, represents the level of resolution and translation parameters with k � 1, 2, . .., n � 1, 2, . . ., 2 k− 1 , and  P m (x) is the m th degree Fibonacci polynomial as follows [21]: with ⌊•⌋ representing the usual foor function.In equation (5), the entity 1/ �� � S m  is a normalization factor, that is explicitly obtained via the following equation: For the case k � 2, M � 3, the corresponding collection of Fibonacci wavelets takes the form as follows: Any function u(x) ∈ L 2 (R) can be approximated via Fibonacci wavelets as follows: where h n,m denotes the Fibonacci wavelet coefcients given by the following equation: Te corresponding matrix form of equation ( 9) is given by the following equation: where H is the row vector of the form.
Te matrix Ψ(x) appearing in equation (11) is the Fibonacci wavelet matrix of order 1 × 2 k− 1 M and is given by the following equation: To facilitate the intended approximation via the Fibonacci wavelets, the collocation points are chosen as follows:

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: where P denotes the Fibonacci wavelet operational matrix of order 2 k− 1 M × 2 k− 1 M. Let the Fibonacci wavelets ψ n,m belongs to a family of Ψ given by equation ( 13), then ψ n,m can recast as follows: where χ(x) denotes the characteristic function.
Invoking the power series representation of Fibonacci polynomials (6), relation ( 16) becomes as follows: Integrating equation ( 17) both sides, we obtain the following equation: where Te function F j (x) can now be represented as follows: Plugging equation (20) in equation ( 18), we have the following equation: where Consequently, we obtain the operational matrix of integration P as follows: On choosing k � 2, M � 3, and integrating equation ( 8) about the collocations points given in equation ( 14) yields as follows: Terefore, equation ( 15) takes the form as follows: Journal of Mathematics where 2.3.Quasi-Linearization. Bellman and Kalaba [22] frst introduced the notion of quasi-linearization technique for obtaining an approximation to the solution of the m th -degree nonlinear diferential equations.To proceed the argument, we consider the following diferential equation: Application of quasi-linearization technique to equation (27) yields as follows: Consequently, the recurrence relation for the nonlinear diferential equations of the order n th is given by the following equation: with v r+1 (α) � a, v r+1 (β) � b.Relation (30) is the desired linear diferential equation, which can be recursively solved for v r+1 (x) from v r (x).

Error Estimation and Convergence Analysis
Tis section deals with estimation of the discretization errors and convergence results associated with the Fibonacci wavelet method for the nonlinear reaction-difusion equation of Fisher-type.

Theorem 1. Given a sufciently smooth real valued function
Ten, the norm of the truncated error e(x) is given by the following equation: Proof.[23] □ Theorem 2. Given a function u(x) ∈ L 2 [0, 1), which is bounded by  M. Ten, the Fibonacci wavelet expansion given by equation (9), which converges uniformly to u(x) with the following equation: (32) Proof.[19].

Method of Solution
Tis 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-difusion equation given by equation (1).For solving equation ( 1), it is imperative to recall the nonlinear reaction-difusion equation of the following Fisher-type: with the initial and boundary conditions.
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: where h ℓ denotes the unknown Fibonacci wavelet coefcients row vector given by equation (10).Integrating equation ( 35) with respect to τ over the interval spanning from τ to τ s yields as follows: Integrating equation (36) two times with respect to x between the limits 0 to x, we obtain the following equation: Utilizing the boundary constraints (34) yields as follows: Taking derivative on both sides of equation ( 51) with respect to τ, we obtain the following equation: Putting x � 1 in equations ( 38) and (39) yields as follows and which is expressed as follows: Substituting equations ( 54)-(42) into equations (51)-(39) and discretizing the equations by assuming x ⟶ x ℓ , τ ⟶ τ s+1 , we have the following equation: Implementing equations ( 36) and (44) in equation (33) yields a set of algebraic equations involving the Fibonacci wavelets as follows: where Δτ � (τ s+1 − τ s ).Subsequently, we solve solving the system of algebraic equations obtained in (45) for the unknown coefcients h ℓ by using the Newton method.In the sequel, implementing these values of h ℓ in (43), we obtain the approximate solution via Fibonacci wavelet of the nonlinear reaction-difusion equation of Fisher-type (33).

Numerical Examples and Discussion
To section is meant to explore the efectiveness of the above formulated numerical scheme in the process of achieving an approximate solution for the Fisher-type equation given by equation (33).Te efciency 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.Te precision of the newly formulated procedure is measured by absolute, L ∞ , L 2 , and root mean square (RMS) error norms given as follows: where u(x, τ) and u n,m (x, τ) denotes exact and approximate solutions, respectively.
Example 1.Consider the nonlinear reaction-difusion equation Fisher-type [11]: with the initial conditions: and the boundary conditions: Te exact solution corresponding to equation ( 47) is as follows: Now, by implementing the already discussed quasilinearization methodology, the nonlinear equation ( 47) is converted into the linear diferential equation: with initial guess: Next, by adopting the procedure as in Section 4, the system of algebraic equations of (51) is obtained as follows: For demonstrating the precision of the numerical scheme based on the Fibonacci wavelets, the obtained L ∞ , L 2 and RMS error norms have been compared to that of Lie symmetry method [10] over diferent values of τ, whereas the same results have also been compared to CTB diferential quadrature method [11] for diferent values of τ as tabulated in Table 1.In view of Table 1, it is quite clear that the approximate solution is more efective in lieu of the Lie symmetry and CTB diferential 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.
Example 2. Consider the following nonlinear reaction-diffusion equation Fisher-type [11]: with the initial conditions: and the boundary conditions: Journal of Mathematics exact solution of problem ( 54) is as follows: Applying the quasi-linearization technique as discussed in Section 2 to the nonlinear problem (54), we get a linearized diferential equation given by the following equation: with initial guess: Now, in consonance to the strategy pointed-out in Section 4, the system of algebraic equations of ( 58) is as follows: In order to examine the efcacy of the newly proposed numerical technique, the approximate solution together with its comparison to the exact solution is plotted graphically in Figures 4 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.Te comparison of L ∞ , L 2 , and RMS error norms of the proposed method are compared with the existing Lie symmetry 10 Journal of Mathematics method [10] and the CTB diferential method [11] in Tables 2 and 3. From Tables 2 and 3, we conclude that the Fibonacci wavelet method is more accurate and efcient than the existing ones.
Example 3. Consider the following nonlinear reaction-diffusion equation of the Fisher-type [12]: subject to initial conditions: and boundary conditions: Te exact solution pertaining to equation ( 61) is as follows: By using the quasi-linearization technique as discussed in Section 2, nonlinear equation ( 61) reduces to the following form: with initial guess: Adopting the same methodology as discussed in Section 4, the system of algebraic equations corresponding to equation ( 65) is obtained as follows: To demonstrate the efciency and accuracy of the proposed method, we have compared the obtained results with the analytic solution in Figures 7 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 Journal of Mathematics obtained via Fibonacci wavelet method is good agreement with the analytic solution.
Example 4. Finally, consider the following nonlinear reaction-difusion equation of Fisher's type [9]: Te exact solution of the Example 4 is as follows: Te 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: with initial guess: (71) Te system of algebraic equations corresponding to equation ( 71) is as follows: and RMS error norms of present method with [10,11] for Example 1 at diferent τ. τ Lie symmetry [10] CTB diferential quadrature method [11] Present method 12 Journal of Mathematics Journal of Mathematics 13 show the efectiveness of the proposed method, we have compared the approximate and exact solutions to the problem via diferent graphical illustrations.In   5 and 6 compare the L ∞ and L 2 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].[10,11] for Example 2 at diferent τ and fxed δ � 1.5.

Conclusion and Future Work
In the present article, we have introduced an efcient Fibonacci wavelet-based collocation method for solving the nonlinear reaction-difusion equation Fisher-type.To demonstrate the precision of the proposed wavelet-based numerical method, we have compared the obtained absolute L ∞ , L 2 , and RMS error norms with the existing Lie symmetry and CTB diferential quadrature methods.Moreover, we have demonstrated that the proposed method is computationally more efective and yields more precise outcomes than the existing ones.Tis strategy of obtaining the numerical solution of diferential equations is novel to the literature and stands for its efciency 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.Te present work also stimulates interest in solving the (n + 1) dimensional models arising in the transport of air, difusion of neutrons, adsorption of pollutants in soil, and oil reservoir fow transport.Tis is precisely our future research objective.
Figures 10 and 11, we have compared the approximate solution with the exact solution for diferent values of α 2 via three-dimensional and two-dimensional plots.
Figure 10 depicts the comparison of approximate solution and exact solution for diferent values of α 2 .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

Figure 4 :
Figure 4: Approximate and exact solution for diferent τ of Example 2.

Figure 7 :
Figure 7: Approximate and exact solution for diferent τ of Example 3.

Table 3 :
Comparison of L ∞ , L 2 , and RMS error norms of present method with