# Numerical Investigation of Fractional-Order Differential Equations viaφ-Haar-Wavelet Method

Authors: F. M. Alharbi; A. M. Zidan; Muhammad Naeem; Rasool Shah; Kamsing Nonlaopon
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with theφ-Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ-Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ-fractional integral of the Haar-wavelet by utilizing the φ-fractional integral operator. We also extended our method to nonlinear φ-FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ-FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.


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## Body

## 1. Introduction

Fractional differential equations are used to describe a wide range of phenomena in natural science, and because of its numerous applications in physical, chemical, and biological sciences, fractional calculus has captivated the scientific community. Several researchers have recently focused their attention on the concept of the fractional derivative. The fractional derivative is introduced in fractional calculus through the fractional integral. Riemann, Liouville, Caputo, Hadamard, Grunwald, and Letinkow are the pioneers in this field, having contributed and published extensively on the subject. The nonlinear fractional Schrodinger equations with the Riesz space and the Caputo time-fractional derivatives are studied using the finite difference/spectral-Galerkin method in [1]. For the Higgs boson equation in the de Sitter spacetime, a finite difference/Galerkin spectral scheme was introduced in [2] which retains the discrete energy dissipation property. For the two-dimensional fractional wave equation with the Weyl space-fractional operators, Ref. [3] proposes a high-order compact difference method with fourth-order precision in space and second order in time. Explicit solutions to differential equations of complex fractional orders with respect to functions and continuous variable coefficients are determined in [4]. Different types of fractional derivatives have appeared in the literature that strengthen and generalize the classical fractional operators defined by the aforementioned authors [5, 6]. Katugampola recently discovered a new type of fractional integral operator which encompasses the Riemann-Liouville and Hadamard operators in a single form [7, 8]. Moreover, several other fractional operators are being introduced to date. Due to a wide range of definitions for fractional-order integrals and derivatives [9–11], the idea of a fractional derivative of one function with respect to another function emerged. This class of fractional operators depends on a kernel function and unifies many definitions of fractional operators. Almeida used the idea of fractional derivatives in the Caputo sense and introduced the φ-Caputo fractional derivative of one function with respect to some other function [12]. The proper choice of a trial function helps in the modeling of physical phenomenon and makes the approach more suitable from the application point of view [13, 14].Wavelet analysis is a well-known and widely used mathematical method in engineering and other sciences [15, 16]. Wavelets are made up of function expressions that have been extended into a sum of basic functions. A mother wavelet function is translated and compressed to obtain these basic functions. As a result, it inherits properties of locality and smoothness, making it simple to research the properties of integer and locality during the process of expressing functions. Wavelets have sparked a lot of interest in using them to solve classical ordinary and partial differential equations numerically. Researchers have recently succeeded in extending several standard wavelet methods to numerical solutions for fractional differential equations. Numerical integration and numerical solutions of fractional ordinary and fractional partial differential equations are some of the other applications of wavelet methods in applied mathematics. So, for now, wavelets such as the Haar-wavelet, B-spline, Daubechies, and Legendre wavelet are used [17–21]. In Ref. [22], the Genocchi wavelet-like operational matrix was used together with the collocation method to solve nonlinear FDEs. For solving fractional integrodifferential equations, the Jacobi wavelet operational matrix of fractional integration is constructed and utilized in [23]. The Haar-wavelet is a simple form of orthonormal wavelets with compact support and has been used by many researchers. The Haar-wavelet family consisted of rectangular functions. It also includes the lower member of the Daubechies wavelet family, which is suitable for computer implementation. The Haar-wavelets are used to transform a fractional differential equation into an algebraic structure of finite variables [24–27].For modeling different physical problems, it is difficult to pick the right operator. Therefore, generalized operators of fractional order should be developed for which classical operators are special cases. An effective way to deal with such a variety is to merge these definitions into one by considering fractional derivatives of functionf with respect to another function φ. The Riemann-Liouville operators of fractional order are generalized by introducing the fractional-order differentiation and integration of a function by another function [28, 29]. In [12, 30], Almeida defined the φ-Caputo fractional differential and integral operators and discussed its characteristics. The contribution made by Almeida et al. plays a pivotal role in putting together a wide range of fractional operators. Moreover, recent work on the φ-Caputo derivative indicates that φ-Caputo fractional differential-based mathematical models are more flexible and provide felicitous results in many situations. In order to evaluate the growth of the world population, Almeida [12] implemented the φ-Caputo derivative and illustrated that the appropriate selection of a fractional operator determines the model’s precision. Using fixed-point theorems, Almeida et al. in [13] investigated the existence and uniqueness of a solution for nonlinear FDEs involving a φ-Caputo derivative. Almeida et al. in [31] introduced the φ-shifted Legendre polynomials for solving fractional oscillation equations containing the φ-Caputo derivative of fractional order. We therefore see the theory of φ-FDEs as a promising field for further study. In this paper, taking motivation by the work cited above, we developed a new numerical method for solving linear and nonlinear boundary value problems in φ-FDEs.The rest of the paper is organized as follows: We start Section2 with an overview of the fractional calculus followed by a discussion of the classical Haar-wavelet and an approximation of the functions by the Haar-wavelet. In Section 3, we developed the φ-Haar operational matrix of fractional-order integration of the Haar-wavelet and then utilize it for a numerical solution of the φ-FDEs. In Section 3.1, the error estimate of the developed technique is discussed in depth. Section 4 is devoted to some numerical results and figures that show the precision and effectuality of the developed technique. Finally, a conclusion is given in the last section.


## 2. Preliminaries

Here, we present some vital definitions ofφ-fractional operators and their basic properties which will be used in the subsequent sections of the paper.Let the functionf:α,β⟶ℝ be integrable, ρ a positive real number, n a natural number, and φ∈C1α,β an increasing function such that φ′ζ≠0∀ζ∈α,β.Definition 1.
The Caputo fractional derivative of a functionf is defined by(1)DCαρfζ=1Γn−ρ∫αζζ−In−ρ−1ddInfIdI,where ζ∈α,β, ρ∈ℝ+, and n=ρ.Definition 2 (see [9, 30, 32]).
Theφ-Riemann-Liouvile (φ-RL) integration operator of fractional-order ρ of a function fζ is defined as follows:(2)Iαρ,φfζ=1Γρ∫αζφ′Iφζ−φIρ−1fIdI.
Theφ-RL derivative operator of fractional-order ρ of the function fζ is defined as follows:(3)Dαρ,φfζ=1φ′ζddζnIαn−ρ,φfζ=1Γn−ρ1φ′ζddζn∫αζφ′Iφζ−φIn−ρ−1fIdI,where n=ρ+1.Definition 3 (see [12]).
Letρ be a positive real number, n a natural number, and f,φ∈Cnα,β such that φ is increasing and φ′ζ≠0∀ζ∈α,β. The φ-Caputo differential operator of fractional-order ρ is defined by(4)CDαρ,φfζ=1Γn−ρ∫αζφ′Iφζ−φIn−ρ−1Dn,φfIdI,where fφnζ=1/φ′ζd/dζnfζ, n=ρ+1 if ρ∉ℕ, whereas n=ρ if ρ∈ℕ.Remark 4.
For particular choices ofφζ, these operators are reduced to the following given operators of the fractional order:(i)
φζ=ζ refer to the classical RL and Caputo fractional operators(ii)
φζ=lnζ refer to the classical Hadamard and Caputo-Hadamard fractional operators


### 2.1. Characteristics of theφ-Fractional Operators

Some fundamental characteristics of theφ-fractional operators are listed below [12, 30].Letfζ=φζ−φαγ, where γ>n and ρ>0. Then,(5)Iαρ,φfζ=Γγ+1Γρ+γ+1φζ−φαρ+γ,Dαρ,φfζ=Γγ+1Γγ+1−ρφζ−φαγ−ρ,Iαρ,φDαρ,φfζ=fζ−∑k=0n−1Dk,φfζk!φζ−φαk.Example 5.
Letfζ=ζ−αγ, with γ>n and ρ>0. Then, the Caputo fractional derivative is given by(6)CDαρfζ=Γγ+1Γγ+1−ρζ−αγ−ρ.
The Caputo fractional derivatives ofsinζ and cosζ are given by(7)CDαρsinζ=ζ1−ρE2,2−ρ−ζ2,CDαρcosζ=ζ−ρE2,1−ρ−ζ2,where Eα,β is the two-parameter Mittag-Leffler function defined by(8)Eα,β=∑ℓ=0∞ζℓΓαℓ+β.


### 2.2. Existence and Uniqueness of Solution for Nonlinearφ-FDEs

In this section, we provide existence and uniqueness theorems for nonlinearφ-FDEs.Consider the nonlinearφ-FDE:(9)Dαρ,φyζ=fζ,yζ,t∈α,β.We have the initial conditions, namely,yα=yα and yφℓα=yαℓ, ℓ=1,⋯,n−1, where(1)
0<ρ∉N and n=ρ+1(2)
yα and yαℓ, where ℓ=1,⋯,n−1, are fixed reals(3)
y∈Cn−1α,β, such that Dαρ,φ exists and is continuous in α,β(4)
f:α,β×ℝ⟶ℝ is continuousTheorem 6.
A functiony∈Cn−1α,β is a solution to problem (9) if and only if y satisfies the following fractional integral equation:(10)yζ=fζ,yζ−∑ℓ=0n−1yαℓℓ!φζ−φαℓ.Theorem 7.
Letf be a Lipschitz continuous function with respect to the second variable, that is, ∃ is a positive constant L such that(11)fζ,x1−fζ,x2≤Lx1−x2,∀ζ∈α,β,∀x1,x2∈ℝ.
Then, there is a constanth∈ℝ+, such that there exists a unique solution to problem (9) on the interval α,α+h⊆α,β.
Proof of Theorems6 and 7 can be seen in [13].


### 2.3. Approximation of Function by the Haar-Wavelet

Theith Haar-wavelet defined on the interval α,β is given by(12)hiζ=1,forζ∈ϰ1i,ϰ2i,−1,forζ∈ϰ2i,ϰ3i,0,elsewhere,where ϰ1i=α+β−αk/m, ϰ2i=α+β−α2k+1/m, ϰ3i=α+β−αk+1/m, and m=2j, where j=0,1,2,3,⋯,J and k=0,1,2,3,⋯,m−1. Here, j and k are the wavelet’s dilation and translation parameters, whereas J is the maximum level of resolution. The relationship i=m+k+1 identifies the wavelet number i. For i≥3, equation (12) holds true.The corresponding scaling functions of the Haar-wavelet family fori=1 and i=2 are(13)h1ζ=1,forζ∈α,β,0,elsewhere,h2ζ=1,ifζ∈α,α+β2,−1,ifζ∈α+β2,β,0,elsewhere.Any functionyζ defined and square integrable over the interval 0,1 can be expressed in terms of the Haar-wavelet as follows:(14)yζ=∑i=0∞cihiζ,where the coefficients ci of the Haar-wavelet are defined by(15)ci=yζ,hiζ=∫01yζhiζdζ.In practice, only the firstm terms of the series in equation (14) are considered, where m is a power of 2, that is,(16)yζ≅ymζ=∑i=0m−1cihiζ,with vector form as(17)yζ≅ymζ=CmTHmζ,where CmT=c0,c1,c2,⋯,cm−1 and Hmζ=h0ζ,h1ζ,h2ζ,⋯,hm−1ζT.


## 2.1. Characteristics of theφ-Fractional Operators

Some fundamental characteristics of theφ-fractional operators are listed below [12, 30].Letfζ=φζ−φαγ, where γ>n and ρ>0. Then,(5)Iαρ,φfζ=Γγ+1Γρ+γ+1φζ−φαρ+γ,Dαρ,φfζ=Γγ+1Γγ+1−ρφζ−φαγ−ρ,Iαρ,φDαρ,φfζ=fζ−∑k=0n−1Dk,φfζk!φζ−φαk.Example 5.
Letfζ=ζ−αγ, with γ>n and ρ>0. Then, the Caputo fractional derivative is given by(6)CDαρfζ=Γγ+1Γγ+1−ρζ−αγ−ρ.
The Caputo fractional derivatives ofsinζ and cosζ are given by(7)CDαρsinζ=ζ1−ρE2,2−ρ−ζ2,CDαρcosζ=ζ−ρE2,1−ρ−ζ2,where Eα,β is the two-parameter Mittag-Leffler function defined by(8)Eα,β=∑ℓ=0∞ζℓΓαℓ+β.


## 2.2. Existence and Uniqueness of Solution for Nonlinearφ-FDEs

In this section, we provide existence and uniqueness theorems for nonlinearφ-FDEs.Consider the nonlinearφ-FDE:(9)Dαρ,φyζ=fζ,yζ,t∈α,β.We have the initial conditions, namely,yα=yα and yφℓα=yαℓ, ℓ=1,⋯,n−1, where(1)
0<ρ∉N and n=ρ+1(2)
yα and yαℓ, where ℓ=1,⋯,n−1, are fixed reals(3)
y∈Cn−1α,β, such that Dαρ,φ exists and is continuous in α,β(4)
f:α,β×ℝ⟶ℝ is continuousTheorem 6.
A functiony∈Cn−1α,β is a solution to problem (9) if and only if y satisfies the following fractional integral equation:(10)yζ=fζ,yζ−∑ℓ=0n−1yαℓℓ!φζ−φαℓ.Theorem 7.
Letf be a Lipschitz continuous function with respect to the second variable, that is, ∃ is a positive constant L such that(11)fζ,x1−fζ,x2≤Lx1−x2,∀ζ∈α,β,∀x1,x2∈ℝ.
Then, there is a constanth∈ℝ+, such that there exists a unique solution to problem (9) on the interval α,α+h⊆α,β.
Proof of Theorems6 and 7 can be seen in [13].


## 2.3. Approximation of Function by the Haar-Wavelet

Theith Haar-wavelet defined on the interval α,β is given by(12)hiζ=1,forζ∈ϰ1i,ϰ2i,−1,forζ∈ϰ2i,ϰ3i,0,elsewhere,where ϰ1i=α+β−αk/m, ϰ2i=α+β−α2k+1/m, ϰ3i=α+β−αk+1/m, and m=2j, where j=0,1,2,3,⋯,J and k=0,1,2,3,⋯,m−1. Here, j and k are the wavelet’s dilation and translation parameters, whereas J is the maximum level of resolution. The relationship i=m+k+1 identifies the wavelet number i. For i≥3, equation (12) holds true.The corresponding scaling functions of the Haar-wavelet family fori=1 and i=2 are(13)h1ζ=1,forζ∈α,β,0,elsewhere,h2ζ=1,ifζ∈α,α+β2,−1,ifζ∈α+β2,β,0,elsewhere.Any functionyζ defined and square integrable over the interval 0,1 can be expressed in terms of the Haar-wavelet as follows:(14)yζ=∑i=0∞cihiζ,where the coefficients ci of the Haar-wavelet are defined by(15)ci=yζ,hiζ=∫01yζhiζdζ.In practice, only the firstm terms of the series in equation (14) are considered, where m is a power of 2, that is,(16)yζ≅ymζ=∑i=0m−1cihiζ,with vector form as(17)yζ≅ymζ=CmTHmζ,where CmT=c0,c1,c2,⋯,cm−1 and Hmζ=h0ζ,h1ζ,h2ζ,⋯,hm−1ζT.


## 3. Theφ-Haar-Wavelet Operational Matrix

In this section, our endeavor is to construct theφ-Haar-wavelet operational matrix Pρ,φ of fractional-order ρ and use it to solve φ-FDEs numerically. The φ-fractional integration of the Haar-wavelet is performed using equation (2). Mathematically, the generalized fractional-order integration of the Haar-wavelet, Hm=h0,h1,h2,⋯,hm−1, is given by(18)Piρ,φζ=1Γρ∫αζφ′Iφζ−φIρ−1hiIdI.Analytically, these generalizedφ-fractional integrals can be approximated as follows:(19)Piρ,φζ=0,ifζ<ϰ1i,Φ1,ifζ∈ϰ1i,ϰ2i,Φ2,ifζ∈ϰ2i,ϰ3i,Φ3,ifζ>ϰ3i,where(20)Φ1=1Γρ+1φζ−φϰ1iρ,Φ2=1Γρ+1φζ−φϰ1iρ−2φζ−φϰ2iρ,Φ3=1Γρ+1φζ−φϰ1iρ−2φζ−φϰ2iρ+bigφζ−φϰ3iρ.Equation (19) holds for i>1; for i=1, we have(21)P1ρ,φζ=1Γρ+1φζ−φαρ.The fractional-orderφ-Haar-wavelet operational matrix Pρ,φ for the function φζ=ζ2 and ρ=0.75 is given by(22)Pm×mρ,φ=0.4342−0.2816−0.0998−0.1763−0.0356−0.0623−0.0806−0.0953−0.02100.1735−0.09980.2392−0.0356−0.06230.12970.1153−0.07390.06530.0613−0.0204−0.03560.0833−0.0173−0.00580.0653−0.065300.116700−0.10510.1635−0.02850.00220.0221−0.002910.0211−0.0066−0.0019−0.0010−0.00940.0318−0.0224−0.0090100.0435−0.0088−0.00200.0064−0.006400.06786000.0616−0.01130.0280−0.02800−0.056040000.0779.Also, the approximate and exactφ-RL fractional integration of φζ=sin5ζ for J=6 and various choices of ρ is plotted in Figure 1.Figure 1
Exact and approximateφ-RL integration of the function fζ=sin5ζ for J=6 and various choices of ρ and their maximum absolute error.


### 3.1. Convergence Analysis of theφ-Haar-Wavelet Method

In [33], the Caputo-type FDEs were recently analyzed for error. Furthermore, utilizing the Haar wavelet, [34] proves convergence for the solution of the nonlinear Fredholm integral equations. In the present work, the upper limit for the error estimate is calculated using the φ-Caputo fractional differential operator. The φ-Haar-wavelet method for FDEs is shown to be convergent.Theorem 8.
Letynζ be continuous on interval α,β, and suppose ∃K>0, such that yφnζ≤K∀ζ∈α,β, where α,β∈ℝ+, yφnζ=1/φ′ζd/dζnyζ, and Dαρ,φymζ is the approximation of Dαρ,φyζ. Then, we have(23)Dαρ,φyζ−Dαρ,φymζE≤β−αKφ′βm−ρΓm−ρ+11km−ρ11−22ρ−m1/2.Proof.
Dαρ,φy can be approximated as follows:(24)Dαρ,φyζ=∑i=α∞cihiζ,where(25)ci=Dαρ,φyζ,hiζ=∫αβDαρ,φyζhiζdζ.
Suppose thatDαρ,φym is the following approximation of Dαρ,φy:(26)Dαρ,φymζ=∑i=0m−1cihiζ,where m=2κ+1, κ=1,2,3,⋯. Then,(27)Dαρ,φyζ−Dαρ,φymζ=∑i=m∞cihiζ=∑i=2κ+1∞cihiζ,which implies that(28)Dαρ,φyζ−Dαρ,φymζE2=∫αζDαρ,φyζ−Dαρ,φymζ2dζ=∑i=2κ+1∞∑i′=2κ+1∞cici′∫αζhiζhi′ζdζ.
By orthogonality of the sequencehmζ, we have ∫αβhmζhmζdζ=Im, where Im is the identity matrix of order m. Therefore,(29)Dαρ,φyζ−Dαρ,φymζE2=∑i′=2κ+1∞ci2.
From equation (25), we have(30)ci=∫αβDαρ,φyζhiζdζ=2j/2∫α+β−αk2−jα+β−αk+1/22−jDαρ,φyζdζ−∫α+β−αk+1/22−jα+β−αk+12−jDαρ,φyζdζ.
By the mean value theorem for integration, we have∃ζ1, ζ2∈α,β, such that(31)α+β−αk2−j<ζ1<α+β−αk+122−j,α+β−αk+122−j<ζ2<α+β−αk+12−j,ci=2j/2β−αα+k+122−j−α+k2−jDαρ,φyζ1−α+k+12−j−α+k+122−jDαρ,φyζ2=2j/2β−α2−j−1Dαρ,φyζ1−Dαρ,φyζ2.
Hence,(32)ci2=2−j−2β−α2Dαρ,φyζ1−Dαρ,φyζ22.
Employing the definition of theφ-Caputo fractional derivative, the fact that φ is increasing and the condition yφnζ≤K, we arrive at(33)Dαρ,φyζ1−Dαρ,φyζ2=1Γm−ρ∫αζ1φ′ζφζ1−φζm−ρ−1yφnζdζ−∫αζ2φ′ζφζ2−φζm−ρ−1yφnζdζ≤1Γm−ρ∫αζ1φ′ζφζ1−φζm−ρ−1yφnζdζ−∫αζ1φ′ζφζ2−φζm−ρ−1yφnζdζ+∫ζ1ζ2φ′ζφζ2−φζm−ρ−1yφnζdζ≤1Γm−ρ∫αζ1φ′ζφζ1−φζm−ρ−1−φζ2−φζm−ρ−1yφnζdζ+∫ζ1ζ2φ′ζφζ2−φζm−ρ−1yφnζdζ,where(34)m−ρ−1>0=KΓm−ρ+1φζ1−φαm−ρ−φζ2−φαm−ρ+2φζ2−φζ1m−ρ.
Sinceζ1>α, ζ2>α, and ζ2>ζ1 and φζ is an increasing function, so(35)φζ1−φαm−ρ−φζ2−φαm−ρ<0.
Therefore,(36)Dαρ,φyζ1−Dαρ,φyζ2≤2KΓm−ρ+1φζ2−φζ1m−ρ.
By mean value theorem,∃ϰ∈ζ,ζ2⊆α,β, such that φζ2−φζ1≤ζ2−ζ1φ′ϰ, we get(37)Dαρ,φyζ1−Dαρ,φyζ2≤2KΓm−ρ+1ζ2−ζ1φ′ϰm−ρ≤2KΓm−ρ+12jm−ρφ′βm−ρ,which gives(38)Dαρ,φyζ1−Dαρ,φyζ22≤4K2Γ2m−ρ+122jm−ρφ′β2m−ρ.
Putting equation (38) into equation (32), we get(39)ci2≤2−j−2β−α24K2Γ2m−ρ+122jm−ρφ′β2m−ρ.
Equations (29) and (39) give(40)Dαρ,φyζ−Dαρ,φymζE2=∑i=2κ+1∞ci2=∑j=κ+1∞∑i=2j2j+1−1ci2≤∑j=κ+1∞β−α2K2Γ2m−ρ+122jm−ρ+jφ′β2m−ρ2j+1−1−2j+1=β−α2K2φ′β2m−ρΓ2m−ρ+1∑j=κ+1∞122jm−ρ=β−α2K2φ′β2m−ρΓ2m−ρ+1122κ+1m−ρ11−22ρ−m,which implies that(41)Dαρ,φyζ−Dαρ,φymζE≤β−αKφ′βm−ρΓm−ρ+112κ+1m−ρ11−22ρ−m1/2.
Letk=2κ+1; (41) can also be written as(42)Dαρ,φyζ−Dαρ,φymζE≤β−αKφ′βm−ρΓm−ρ+11km−ρ11−22ρ−m1/2.
From the value ofK, we can get an upper bound for the error.
We first estimate the value ofK. Since ynζ is continuous and bounded on α,β, so yφnζ is also continuous and bounded on α,β and is given by(43)yφnζ≅∑i=0m−1cihiζ=CmTHmζ,where Cm=c0,c1,c2,⋯,cm−1T and Hmζ=h0ζ,h1ζ,h2ζ,⋯,hm−1ζT.
Integrating equation (43), we have(44)yφn−1ζ=∫αζyφnζdζ+yφn−1α=∫αζyφnζdζ≅CmTPm×m1,φHmζ.
Similarly,(45)yφn−2ζ=∫αζyφn−1ζdζ+yφn−2α=∫αζyφn−1ζdζ≅CmTPm×m2,φHmζ.
Proceeding in the same way, we get(46)yφζ≅CmTPm×mm,φHmζ.
By defining the pointsζj=j−1/2/m, j=0,1,2,⋯,m. Substituting ζj in equation (46), we have(47)yφζj≅CmTPm×mm,φHmζj.
The matrix form of equation (47) is as follows:(48)yφ≅CmTPm×mm,φHmζj,where yφ=yφζ1,yφζ2,yφζ3,⋯,yφζm.
By using equation (48), we can investigate CmT. From equation (43), we may know the value of Dm,φζ for each ζ∈α,β.
Supposeti∈α,β, for i=1,2,3,⋯,l, ti=ti=i−1/l/l, and we calculateyφnti for i=1,2,3,⋯,l; then, ε+maxyφnti may be measured as the approximation for K.
Obviously, this approximation would have additional precision ifl increases and ε is selected as β.☐☐Theorem 9.
LetDαρ,φym, achieved by applying the φ-Haar-wavelet, be the estimation of Dαρ,φy; then, the actual upper-bound of error is given as follows:(49)yζ−ymζE≤KNΓρ+1Γm−ρ+11km−ρ11−22ρ−m1/2,where N=maxβ−αφβm−ρφζ−φ0ρ.Theorem9 can be proven simply via Theorem 8. From equation (49), we can understand that yζ−ymζE tends to 0 as m tends to ∞, which shows that the φ-Haar-wavelet technique converges.Example 10.
To demonstrate optimality of the upper bound in equation (49), we consider the following φ-FDE:(50)D0ρ,φyζ+yζ=φζ2ρ+Γ2ρ+1Γρ+1φζρ,0<ρ≤1,ζ∈0,1,with initial condition y0=0, having the exact solution yζ=φζ2ρ.Table1 shows the optimal values of the upper bound of error obtained for various options J and ρ=0.25.Table 1
Optimal value of the upper bound of error at differentJ and α=0.25.
Jyexact−yapproxxEOptimality of the upper bound of error43.5102×10−40.071452.8937×10−50.021666.8632×10−60.054273.2381×10−60.0139


## 3.1. Convergence Analysis of theφ-Haar-Wavelet Method

In [33], the Caputo-type FDEs were recently analyzed for error. Furthermore, utilizing the Haar wavelet, [34] proves convergence for the solution of the nonlinear Fredholm integral equations. In the present work, the upper limit for the error estimate is calculated using the φ-Caputo fractional differential operator. The φ-Haar-wavelet method for FDEs is shown to be convergent.Theorem 8.
Letynζ be continuous on interval α,β, and suppose ∃K>0, such that yφnζ≤K∀ζ∈α,β, where α,β∈ℝ+, yφnζ=1/φ′ζd/dζnyζ, and Dαρ,φymζ is the approximation of Dαρ,φyζ. Then, we have(23)Dαρ,φyζ−Dαρ,φymζE≤β−αKφ′βm−ρΓm−ρ+11km−ρ11−22ρ−m1/2.Proof.
Dαρ,φy can be approximated as follows:(24)Dαρ,φyζ=∑i=α∞cihiζ,where(25)ci=Dαρ,φyζ,hiζ=∫αβDαρ,φyζhiζdζ.
Suppose thatDαρ,φym is the following approximation of Dαρ,φy:(26)Dαρ,φymζ=∑i=0m−1cihiζ,where m=2κ+1, κ=1,2,3,⋯. Then,(27)Dαρ,φyζ−Dαρ,φymζ=∑i=m∞cihiζ=∑i=2κ+1∞cihiζ,which implies that(28)Dαρ,φyζ−Dαρ,φymζE2=∫αζDαρ,φyζ−Dαρ,φymζ2dζ=∑i=2κ+1∞∑i′=2κ+1∞cici′∫αζhiζhi′ζdζ.
By orthogonality of the sequencehmζ, we have ∫αβhmζhmζdζ=Im, where Im is the identity matrix of order m. Therefore,(29)Dαρ,φyζ−Dαρ,φymζE2=∑i′=2κ+1∞ci2.
From equation (25), we have(30)ci=∫αβDαρ,φyζhiζdζ=2j/2∫α+β−αk2−jα+β−αk+1/22−jDαρ,φyζdζ−∫α+β−αk+1/22−jα+β−αk+12−jDαρ,φyζdζ.
By the mean value theorem for integration, we have∃ζ1, ζ2∈α,β, such that(31)α+β−αk2−j<ζ1<α+β−αk+122−j,α+β−αk+122−j<ζ2<α+β−αk+12−j,ci=2j/2β−αα+k+122−j−α+k2−jDαρ,φyζ1−α+k+12−j−α+k+122−jDαρ,φyζ2=2j/2β−α2−j−1Dαρ,φyζ1−Dαρ,φyζ2.
Hence,(32)ci2=2−j−2β−α2Dαρ,φyζ1−Dαρ,φyζ22.
Employing the definition of theφ-Caputo fractional derivative, the fact that φ is increasing and the condition yφnζ≤K, we arrive at(33)Dαρ,φyζ1−Dαρ,φyζ2=1Γm−ρ∫αζ1φ′ζφζ1−φζm−ρ−1yφnζdζ−∫αζ2φ′ζφζ2−φζm−ρ−1yφnζdζ≤1Γm−ρ∫αζ1φ′ζφζ1−φζm−ρ−1yφnζdζ−∫αζ1φ′ζφζ2−φζm−ρ−1yφnζdζ+∫ζ1ζ2φ′ζφζ2−φζm−ρ−1yφnζdζ≤1Γm−ρ∫αζ1φ′ζφζ1−φζm−ρ−1−φζ2−φζm−ρ−1yφnζdζ+∫ζ1ζ2φ′ζφζ2−φζm−ρ−1yφnζdζ,where(34)m−ρ−1>0=KΓm−ρ+1φζ1−φαm−ρ−φζ2−φαm−ρ+2φζ2−φζ1m−ρ.
Sinceζ1>α, ζ2>α, and ζ2>ζ1 and φζ is an increasing function, so(35)φζ1−φαm−ρ−φζ2−φαm−ρ<0.
Therefore,(36)Dαρ,φyζ1−Dαρ,φyζ2≤2KΓm−ρ+1φζ2−φζ1m−ρ.
By mean value theorem,∃ϰ∈ζ,ζ2⊆α,β, such that φζ2−φζ1≤ζ2−ζ1φ′ϰ, we get(37)Dαρ,φyζ1−Dαρ,φyζ2≤2KΓm−ρ+1ζ2−ζ1φ′ϰm−ρ≤2KΓm−ρ+12jm−ρφ′βm−ρ,which gives(38)Dαρ,φyζ1−Dαρ,φyζ22≤4K2Γ2m−ρ+122jm−ρφ′β2m−ρ.
Putting equation (38) into equation (32), we get(39)ci2≤2−j−2β−α24K2Γ2m−ρ+122jm−ρφ′β2m−ρ.
Equations (29) and (39) give(40)Dαρ,φyζ−Dαρ,φymζE2=∑i=2κ+1∞ci2=∑j=κ+1∞∑i=2j2j+1−1ci2≤∑j=κ+1∞β−α2K2Γ2m−ρ+122jm−ρ+jφ′β2m−ρ2j+1−1−2j+1=β−α2K2φ′β2m−ρΓ2m−ρ+1∑j=κ+1∞122jm−ρ=β−α2K2φ′β2m−ρΓ2m−ρ+1122κ+1m−ρ11−22ρ−m,which implies that(41)Dαρ,φyζ−Dαρ,φymζE≤β−αKφ′βm−ρΓm−ρ+112κ+1m−ρ11−22ρ−m1/2.
Letk=2κ+1; (41) can also be written as(42)Dαρ,φyζ−Dαρ,φymζE≤β−αKφ′βm−ρΓm−ρ+11km−ρ11−22ρ−m1/2.
From the value ofK, we can get an upper bound for the error.
We first estimate the value ofK. Since ynζ is continuous and bounded on α,β, so yφnζ is also continuous and bounded on α,β and is given by(43)yφnζ≅∑i=0m−1cihiζ=CmTHmζ,where Cm=c0,c1,c2,⋯,cm−1T and Hmζ=h0ζ,h1ζ,h2ζ,⋯,hm−1ζT.
Integrating equation (43), we have(44)yφn−1ζ=∫αζyφnζdζ+yφn−1α=∫αζyφnζdζ≅CmTPm×m1,φHmζ.
Similarly,(45)yφn−2ζ=∫αζyφn−1ζdζ+yφn−2α=∫αζyφn−1ζdζ≅CmTPm×m2,φHmζ.
Proceeding in the same way, we get(46)yφζ≅CmTPm×mm,φHmζ.
By defining the pointsζj=j−1/2/m, j=0,1,2,⋯,m. Substituting ζj in equation (46), we have(47)yφζj≅CmTPm×mm,φHmζj.
The matrix form of equation (47) is as follows:(48)yφ≅CmTPm×mm,φHmζj,where yφ=yφζ1,yφζ2,yφζ3,⋯,yφζm.
By using equation (48), we can investigate CmT. From equation (43), we may know the value of Dm,φζ for each ζ∈α,β.
Supposeti∈α,β, for i=1,2,3,⋯,l, ti=ti=i−1/l/l, and we calculateyφnti for i=1,2,3,⋯,l; then, ε+maxyφnti may be measured as the approximation for K.
Obviously, this approximation would have additional precision ifl increases and ε is selected as β.☐☐Theorem 9.
LetDαρ,φym, achieved by applying the φ-Haar-wavelet, be the estimation of Dαρ,φy; then, the actual upper-bound of error is given as follows:(49)yζ−ymζE≤KNΓρ+1Γm−ρ+11km−ρ11−22ρ−m1/2,where N=maxβ−αφβm−ρφζ−φ0ρ.Theorem9 can be proven simply via Theorem 8. From equation (49), we can understand that yζ−ymζE tends to 0 as m tends to ∞, which shows that the φ-Haar-wavelet technique converges.Example 10.
To demonstrate optimality of the upper bound in equation (49), we consider the following φ-FDE:(50)D0ρ,φyζ+yζ=φζ2ρ+Γ2ρ+1Γρ+1φζρ,0<ρ≤1,ζ∈0,1,with initial condition y0=0, having the exact solution yζ=φζ2ρ.Table1 shows the optimal values of the upper bound of error obtained for various options J and ρ=0.25.Table 1
Optimal value of the upper bound of error at differentJ and α=0.25.
Jyexact−yapproxxEOptimality of the upper bound of error43.5102×10−40.071452.8937×10−50.021666.8632×10−60.054273.2381×10−60.0139


## 4. Numerical Solutions ofφ-FDEs

In this section, we provide the solution to some problems in linear and nonlinearφ-FDEs by employing the φ-Haar-wavelet operational matrix technique.


### 4.1. Linear Case

Here, we consider two examples of linearφ-FDEs for the numerical solution by the proposed method.Example 11.
Consider the composite oscillation equation of a fractional order with theφ-Caputo fractional derivative:(51)D0ρ,φyζ+yζ=φζ2+2Γ3−ρφζ2−ρ,0<ρ≤1,ζ∈0,1,with the initial condition y0=0. The exact solution of equation (51) is given by yζ=φζ2. For numerical solutions, we approximate D0ρ,φyζ as(52)D0ρ,φyζ=CmTHmζ.
Integrating equation (52) with the φ-Caputo integral operator, we have(53)yζ=I0ρ,φCmTHmζ+c1=CmTPm×mρ,φHmζ+c1.Substituting the initial conditions in equation (51), we get(54)yζ=CmTPm×mρ,φHmζ+φ02.
Substituting equations (52) and (54) for equation (51), we have(55)CmTHmζ+Pm×mρ,φHmζ=fζ,where fζ=φζ2+2/Γ3−ρφζ2−ρ−φ02.
Equation (55) can be expressed in the matrix form as follows:(56)CmTHmζ+APm×mρ,φHmζ=F,where F=fζ. The value of CmT can be obtained from equation (56). By using CmT in equation (54), we can obtain the numerical solutions. Table 2 represents approximate solutions obtained for various choices of ρ and J. The exact and numerical solutions of equation (51) and the maximum absolute error are plotted in Figures 2(a) and 2(b), respectively, for J=5, ρ=0.6, and φζ=sinζ.Example 12.
In this example, consider the FDE involving theφ-Caputo derivative:(57)D0ρ,φyζ+yζ=φζ4−12φζ3−3Γ4−ρφζ3−ρ+24Γ5−ρφζ4−ρ,where 0<ρ≤1, ζ∈0,1, and the initial condition(58)y0=0.
The exact solution of the problem (57) is given as follows: yζ=φζ4−1/2φζ3. To get numerical solutions, the φ-Haar-wavelet method is employed as follows. Let(59)D0ρ,φyζ=CmTHmζ.
Integrating equation (59) with the φ-Caputo integral operator, we have(60)yζ=I0ρ,φCmTHmζ+c1.
Substituting initial conditions in equation (60), we get c1=y0. Equation (60) becomes(61)yζ=CmTPρ,φHmζ+y0.
Substituting equations (59) and (60) for equation (57), we get(62)CmTHmζ+aζPρ−κ,φHmζ+bζPρ,φHmζ=Fζ,where Fζ=φζ4−1/2φζ3−3/Γ4−ρφζ3−ρ+24/Γ5−ρφζ4−ρ. Approximate solutions are obtained by solving equations (61) and (62). The exact solution, approximate solutions, and the maximum absolute error are plotted in Figure 3 for J=6 and ρ=0.6. Also, the maximum absolute errors obtained for various choices of ρ and J are given in Table 3. We noticed that the maximum absolute error decreases with an increase in J.Table 2
Maximum absolute error for various choices ofρ and J.
Jρ=0.50ρ=0.70ρ=0.90ρ=10.53.2914×10−42.4211×10−42.3518×10−42.4036×10−40.61.1220×10−46.9659×10−55.9464×10−56.0560×10−50.73.8646×10−52.0316×10−51.5089×10−51.5199×10−50.81.3413×10−55.9901×10−63.8355×10−63.8072×10−6Figure 2
ForJ=5, ρ=0.6, and φζ=sinζ: (a) approximate and exact solutions; (b) maximum absolute error.
(a)(b)Figure 3
(a) Approximate and exact solutions of equation (57) for ρ=0.6 and φζ=ζ2−ζ. (b) Maximum absolute error. (c) Approximate solutions of equation (57) for φζ=ζ2 and various choices of ρ.
(a)(b)(c)Table 3
Maximum absolute error for various choices ofρ and J.
Jρ=0.60ρ=0.70ρ=0.80ρ=0.900.54.7700×10−54.4509×10−53.833×10−53.4673×10−50.61.4816×10−51.2922×10−51.0871×10−59.0534×10−60.74.6410×10−63.7700×10−62.9758×10−62.3511×10−60.81.4674×10−61.1071×10−68.1696×10−76.1074×10−7


### 4.2. Nonlinear Case

Example 13.
Consider the fractional-order Riccati differential equation with theφ-Caputo fractional derivative:(63)D0ρ,φyζ=−y2ζ+1,0<ρ≤1,ζ∈0,1,subject to the initial condition y0=0. For ρ=1, the exact solution of equation (63) is given by yζ=eφ2ζ−1/eφ2ζ+1. For numerical solutions, we first utilize the quasilinearization techniques to make the nonlinear terms of equation (63) linear and then solve the linearized problem with the φ-Haar-wavelet method. The linearized form of (63) is(64)D0ρ,φyr+1ζ+2yrζyr+1ζ=yr2ζ+1,ζ>0,0<ρ≤1,with the initial condition yr+10=0.
Now, we apply theφ-Haar-wavelet method to equation (64). Let(65)D0ρ,φyr+1ζ=CmTHmζ.
Operating theφ-Caputo integral on equation (65), we get(66)yr+1ζ=Iρ,φCmTHmζ+c1=CmTPm×mρ,φHmζ+c1.Putting the initial conditions in equation (66) gives(67)yr+1ζ=CmTPm×mρ,φHmζ.
Substituting equations (65) and (66) for equation (64), we have(68)CmTHmζ+2yrζPm×mρ,φHmζ=1+yr2ζ.
The matrix form of equation (68) is given by(69)CmTHmζ+2yrPm×mρ,φHmζ=F,where F=1+yr2. By solving the algebraic system given by equation (69) for CmT and substituting this value into equation (67), we will have the required numerical solution. In Table 4, the maximum absolute error is given for φζ=ζ3. Also, approximate solutions for different choices of the function φ are plotted in Figure 4.Example 14.
Finally, consider the Riccati differential equation of fractional order having theφ-Caputo fractional derivative:(70)D0ρ,φyζ=2yζ−yζ2+1,where 0<ρ≤1 and ζ∈0,1.
Then, we subject this to the initial condition:(71)y0=0.
Whenρ=1, yζ=1+2tanh2φζ+1/2log2−1/2+1 is the actual solution of problem (70). For numerical solutions, we first utilize the quasilinearization technique to linearize the nonlinear terms in equation (70) and then solve the linearized FDE by the φ-Haar-wavelet method.
Equation (70) in the linearized form is given by(72)D0ρ,φyr+1−2−2yrζyr+1ζ=yr2ζ+1,ζ>0and0<ρ≤1,with the initial condition yr+10=0.
Consider(73)D0ρ,φyr+1=CmTHmζ.Taking the φ-Caputo integral of (73),(74)yr+1=I0ρ,φCmTHmζ+c1.Substituting the initial condition in equation (74), we have c1=0.
Usingc1=0 in equation (74), we get(75)yr+1=CmTPm×mρ,φHm.Substituting equations (73) and (75) for equation (70), we get(76)CmTHmζ−2−2yrζPm×mρ,φHmζ=Fζ.
Required approximate solutions can be obtained by using the value ofCmT from equation (76) in equation (75). Table 5 shows that the maximum absolute error decreases by increasing the values of J. Also, the approximate solutions are displayed in Figure 5 for various values of ρ.Example 15.
For comparison with another method, we consider the following problem:(77)D03/2,φyζ+2Γ3/2yζ=2Γ3/21+φζ3/2,ζ∈0,1,with the initial condition y0=0. The exact solution of equation (77) is given by yζ=φζ2. This problem is studied in [31] by using the operational matrix of the φ-shifted Legendre polynomials.Table 4
Maximum absolute error forφζ=ζ3 and various choices of J and ρ.
Jρ=0.60ρ=0.70ρ=0.80ρ=0.90ρ=10.52.9805×10−42.6189×10−49.0556×10−58.4937×10−54.0843×10−50.69.3858×10−58.2375×10−57.5629×10−54.5815×10−53.0209×10−50.78.0570×10−57.5617×10−56.8581×10−54.2393×10−62.5527×10−60.84.6741×10−53.3469×10−55.6175×10−63.0351×10−66.3818×10−70.93.2109×10−52.3126×10−57.8318×10−62.6165×10−71.5954×10−7Figure 4
Approximate solutions forρ=1, J=5, and different choices of φζ.Table 5
Maximum absolute error for various choices ofρ and J.
ρJ=0.5J=0.6J=0.7J=0.80.603.9081×10−42.1491×10−41.6723×10−43.0569×10−50.703.6472×10−41.7153×10−43.2854×10−53.7074×10−50.803.2019×10−41.3570×10−42.8061×10−55.8283×10−60.902.6195×10−49.0689×10−51.8584×10−54.1523×10−60.12.2700×10−45.8851×10−51.4983×10−53.7800×10−6Figure 5
Exact solution forρ=1 and numerical solutions for various choices of ρ.Forφζ=ζ2/2+ζ/2, a comparison of the results obtained in [31] and by the proposed method is given in Table 6.Table 6
Comparison of results obtained in [31] and by our method for φζ=ζ2/2+ζ/2.
ζy-Exacty-Approximate by [31]Error by [31]Error by our method0.00.00000.0006060×10−460×10−40.10.00310.003770×10−460×10−40.20.01440.015180×10−470×10−40.30.03800.038890×10−480×10−40.40.07830.079310×10−410×10−40.50.14160.142711×10−412×10−30.60.23130.232411×10−311×10−30.70.35420.355312×10−411×10−40.80.52740.528410×10−310×10−30.90.74110.742110×10−410×10−41.01.00001.000770×10−470×10−4


## 4.1. Linear Case

Here, we consider two examples of linearφ-FDEs for the numerical solution by the proposed method.Example 11.
Consider the composite oscillation equation of a fractional order with theφ-Caputo fractional derivative:(51)D0ρ,φyζ+yζ=φζ2+2Γ3−ρφζ2−ρ,0<ρ≤1,ζ∈0,1,with the initial condition y0=0. The exact solution of equation (51) is given by yζ=φζ2. For numerical solutions, we approximate D0ρ,φyζ as(52)D0ρ,φyζ=CmTHmζ.
Integrating equation (52) with the φ-Caputo integral operator, we have(53)yζ=I0ρ,φCmTHmζ+c1=CmTPm×mρ,φHmζ+c1.Substituting the initial conditions in equation (51), we get(54)yζ=CmTPm×mρ,φHmζ+φ02.
Substituting equations (52) and (54) for equation (51), we have(55)CmTHmζ+Pm×mρ,φHmζ=fζ,where fζ=φζ2+2/Γ3−ρφζ2−ρ−φ02.
Equation (55) can be expressed in the matrix form as follows:(56)CmTHmζ+APm×mρ,φHmζ=F,where F=fζ. The value of CmT can be obtained from equation (56). By using CmT in equation (54), we can obtain the numerical solutions. Table 2 represents approximate solutions obtained for various choices of ρ and J. The exact and numerical solutions of equation (51) and the maximum absolute error are plotted in Figures 2(a) and 2(b), respectively, for J=5, ρ=0.6, and φζ=sinζ.Example 12.
In this example, consider the FDE involving theφ-Caputo derivative:(57)D0ρ,φyζ+yζ=φζ4−12φζ3−3Γ4−ρφζ3−ρ+24Γ5−ρφζ4−ρ,where 0<ρ≤1, ζ∈0,1, and the initial condition(58)y0=0.
The exact solution of the problem (57) is given as follows: yζ=φζ4−1/2φζ3. To get numerical solutions, the φ-Haar-wavelet method is employed as follows. Let(59)D0ρ,φyζ=CmTHmζ.
Integrating equation (59) with the φ-Caputo integral operator, we have(60)yζ=I0ρ,φCmTHmζ+c1.
Substituting initial conditions in equation (60), we get c1=y0. Equation (60) becomes(61)yζ=CmTPρ,φHmζ+y0.
Substituting equations (59) and (60) for equation (57), we get(62)CmTHmζ+aζPρ−κ,φHmζ+bζPρ,φHmζ=Fζ,where Fζ=φζ4−1/2φζ3−3/Γ4−ρφζ3−ρ+24/Γ5−ρφζ4−ρ. Approximate solutions are obtained by solving equations (61) and (62). The exact solution, approximate solutions, and the maximum absolute error are plotted in Figure 3 for J=6 and ρ=0.6. Also, the maximum absolute errors obtained for various choices of ρ and J are given in Table 3. We noticed that the maximum absolute error decreases with an increase in J.Table 2
Maximum absolute error for various choices ofρ and J.
Jρ=0.50ρ=0.70ρ=0.90ρ=10.53.2914×10−42.4211×10−42.3518×10−42.4036×10−40.61.1220×10−46.9659×10−55.9464×10−56.0560×10−50.73.8646×10−52.0316×10−51.5089×10−51.5199×10−50.81.3413×10−55.9901×10−63.8355×10−63.8072×10−6Figure 2
ForJ=5, ρ=0.6, and φζ=sinζ: (a) approximate and exact solutions; (b) maximum absolute error.
(a)(b)Figure 3
(a) Approximate and exact solutions of equation (57) for ρ=0.6 and φζ=ζ2−ζ. (b) Maximum absolute error. (c) Approximate solutions of equation (57) for φζ=ζ2 and various choices of ρ.
(a)(b)(c)Table 3
Maximum absolute error for various choices ofρ and J.
Jρ=0.60ρ=0.70ρ=0.80ρ=0.900.54.7700×10−54.4509×10−53.833×10−53.4673×10−50.61.4816×10−51.2922×10−51.0871×10−59.0534×10−60.74.6410×10−63.7700×10−62.9758×10−62.3511×10−60.81.4674×10−61.1071×10−68.1696×10−76.1074×10−7


## 4.2. Nonlinear Case

Example 13.
Consider the fractional-order Riccati differential equation with theφ-Caputo fractional derivative:(63)D0ρ,φyζ=−y2ζ+1,0<ρ≤1,ζ∈0,1,subject to the initial condition y0=0. For ρ=1, the exact solution of equation (63) is given by yζ=eφ2ζ−1/eφ2ζ+1. For numerical solutions, we first utilize the quasilinearization techniques to make the nonlinear terms of equation (63) linear and then solve the linearized problem with the φ-Haar-wavelet method. The linearized form of (63) is(64)D0ρ,φyr+1ζ+2yrζyr+1ζ=yr2ζ+1,ζ>0,0<ρ≤1,with the initial condition yr+10=0.
Now, we apply theφ-Haar-wavelet method to equation (64). Let(65)D0ρ,φyr+1ζ=CmTHmζ.
Operating theφ-Caputo integral on equation (65), we get(66)yr+1ζ=Iρ,φCmTHmζ+c1=CmTPm×mρ,φHmζ+c1.Putting the initial conditions in equation (66) gives(67)yr+1ζ=CmTPm×mρ,φHmζ.
Substituting equations (65) and (66) for equation (64), we have(68)CmTHmζ+2yrζPm×mρ,φHmζ=1+yr2ζ.
The matrix form of equation (68) is given by(69)CmTHmζ+2yrPm×mρ,φHmζ=F,where F=1+yr2. By solving the algebraic system given by equation (69) for CmT and substituting this value into equation (67), we will have the required numerical solution. In Table 4, the maximum absolute error is given for φζ=ζ3. Also, approximate solutions for different choices of the function φ are plotted in Figure 4.Example 14.
Finally, consider the Riccati differential equation of fractional order having theφ-Caputo fractional derivative:(70)D0ρ,φyζ=2yζ−yζ2+1,where 0<ρ≤1 and ζ∈0,1.
Then, we subject this to the initial condition:(71)y0=0.
Whenρ=1, yζ=1+2tanh2φζ+1/2log2−1/2+1 is the actual solution of problem (70). For numerical solutions, we first utilize the quasilinearization technique to linearize the nonlinear terms in equation (70) and then solve the linearized FDE by the φ-Haar-wavelet method.
Equation (70) in the linearized form is given by(72)D0ρ,φyr+1−2−2yrζyr+1ζ=yr2ζ+1,ζ>0and0<ρ≤1,with the initial condition yr+10=0.
Consider(73)D0ρ,φyr+1=CmTHmζ.Taking the φ-Caputo integral of (73),(74)yr+1=I0ρ,φCmTHmζ+c1.Substituting the initial condition in equation (74), we have c1=0.
Usingc1=0 in equation (74), we get(75)yr+1=CmTPm×mρ,φHm.Substituting equations (73) and (75) for equation (70), we get(76)CmTHmζ−2−2yrζPm×mρ,φHmζ=Fζ.
Required approximate solutions can be obtained by using the value ofCmT from equation (76) in equation (75). Table 5 shows that the maximum absolute error decreases by increasing the values of J. Also, the approximate solutions are displayed in Figure 5 for various values of ρ.Example 15.
For comparison with another method, we consider the following problem:(77)D03/2,φyζ+2Γ3/2yζ=2Γ3/21+φζ3/2,ζ∈0,1,with the initial condition y0=0. The exact solution of equation (77) is given by yζ=φζ2. This problem is studied in [31] by using the operational matrix of the φ-shifted Legendre polynomials.Table 4
Maximum absolute error forφζ=ζ3 and various choices of J and ρ.
Jρ=0.60ρ=0.70ρ=0.80ρ=0.90ρ=10.52.9805×10−42.6189×10−49.0556×10−58.4937×10−54.0843×10−50.69.3858×10−58.2375×10−57.5629×10−54.5815×10−53.0209×10−50.78.0570×10−57.5617×10−56.8581×10−54.2393×10−62.5527×10−60.84.6741×10−53.3469×10−55.6175×10−63.0351×10−66.3818×10−70.93.2109×10−52.3126×10−57.8318×10−62.6165×10−71.5954×10−7Figure 4
Approximate solutions forρ=1, J=5, and different choices of φζ.Table 5
Maximum absolute error for various choices ofρ and J.
ρJ=0.5J=0.6J=0.7J=0.80.603.9081×10−42.1491×10−41.6723×10−43.0569×10−50.703.6472×10−41.7153×10−43.2854×10−53.7074×10−50.803.2019×10−41.3570×10−42.8061×10−55.8283×10−60.902.6195×10−49.0689×10−51.8584×10−54.1523×10−60.12.2700×10−45.8851×10−51.4983×10−53.7800×10−6Figure 5
Exact solution forρ=1 and numerical solutions for various choices of ρ.Forφζ=ζ2/2+ζ/2, a comparison of the results obtained in [31] and by the proposed method is given in Table 6.Table 6
Comparison of results obtained in [31] and by our method for φζ=ζ2/2+ζ/2.
ζy-Exacty-Approximate by [31]Error by [31]Error by our method0.00.00000.0006060×10−460×10−40.10.00310.003770×10−460×10−40.20.01440.015180×10−470×10−40.30.03800.038890×10−480×10−40.40.07830.079310×10−410×10−40.50.14160.142711×10−412×10−30.60.23130.232411×10−311×10−30.70.35420.355312×10−411×10−40.80.52740.528410×10−310×10−30.90.74110.742110×10−410×10−41.01.00001.000770×10−470×10−4


## 5. Conclusion

In this article, theφ-FDEs are solved numerically by introducing the φ-Haar-wavelet operational matrix of integration of fractional order. This operational matrix has been used to solve both linear and nonlinear problems with success. In comparison to the other methods, this approach is simple and more convergent. The developed method is used to solve a number of linear and nonlinear problems, demonstrating its efficiency and accuracy. Furthermore, the method’s error analysis is thoroughly examined. As a future work, the proposed method may be applied to different wavelets as well as other operators.



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