An Efficient Approach for Solving Differential Equations in the Frame of a New Fractional Derivative Operator

Citation: Attia, N.; Akgül, A.; Seba, D.; Nour, A.; la Sen, M.D.; Bayram, M. An Efficient Approach for Solving Differential Equations in the Frame of a New Fractional Derivative Operator. Symmetry 2023,15, 144. https://doi.org/10.3390/sym15010144 Academic Editors: Dongfang Li and Calogero Vetro Received: 17 November 2022 Revised: 10 December 2022 Accepted: 27 December 2022 Published: 3 January 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article An Efficient Approach for Solving Differential Equations in the Frame of a New Fractional Derivative Operator Nourhane Attia 1,* , Ali Akgül 2,3,4 , Djamila Seba 5, Abdelkader Nour 5, Manuel De la Sen 6 and Mustafa Bayram 7 1 Ecole Nationale Supérieure des Sciences de la Mer et de l’Aménagement du Littoral, Campus Universitaire de Dely Ibrahim, Bois des Cars, B.P. 19, Alger 16320, Algeria 2 Department of Computer Science and Mathematics, Lebanese American University, Beirut 1102 2801, Lebanon 3Art and Science Faculty, Department of Mathematics, Siirt University, Siirt 56100, Turkey 4Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, Nicosia 99138, Turkey 5Dynamic of Engines and Vibroacoustic Laboratory, Faculty of Engineer’s Sciences, University M’hamed Bougara of Boumerdes, Boumerdes 35000, Algeria 6Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, 48940 Leioa, Bizkaia, Spain 7Department of Computer Engineering, Biruni University, Topkapı, Istanbul 34010, Turkey *Correspondence: [email protected] Abstract: Recently, a new fractional derivative operator has been introduced so that it presents the combination of the Riemann–Liouville integral and Caputo derivative. This paper aims to enhance the reproducing kernel Hilbert space method (RKHSM, for short) for solving certain fractional differential equations involving this new derivative. This is the first time that the application of the RKHSM is employed for solving some differential equations with the new operator. We illustrate the convergence analysis of the applicability and reliability of the suggested approaches. The results confirm that the RKHSM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed method. Keywords: fractional differential equations; proportional-Caputo hybrid operator; constant proportionalCaputo operator; reproducing kernel Hilbert space method MSC: 46E22; 34A08 1. Introduction The subject of the derivative concept has become considerably important and popular due to its many applications in the broad disciplines of chemistry, biology, engineering, applied physics, and many others. Fractional calculus has an important role in modeling various fascinating complex phenomena in the form of ordinary or partial differential equations: to mention a few, time-fractional Schrödinger equations [ 1 , 2 ], the time-fractional Benjamin–Bona-Mahony equation [ 2 ], time-fractional Burgers equation [ 3 ], time-fractional Korteweg–de Vries equation [ 4 ], and time-fractional Kuramoto–Sivashinsky equation [ 5 ]. The symmetric and anti-symmetric solutions of the fractional Schrödinger equation have been studied in [ 6 ]. In [ 7 ], the authors extended the Lie symmetry analysis to the time fractional generalized KdV equations. However, using the classical concept of derivative does not fill the gap that exists in different fields. This latter need modernized the classical concept of the derivative to the rich concept of fractional derivative [ 8 , 9 ]. For the most part, the non-local behaviors of many processes can be formulated as fractional differential equations (FDEs, for short) [ 10 ]. Here, the exact solutions of the FDEs are too complicated to be found. From this point, some well-known numerical techniques have been needed to find an approximate approach to these equations [11–13]. Symmetry 2023,15, 144. https://doi.org/10.3390/sym15010144 https://www.mdpi.com/journal/symmetry Symmetry 2023,15, 144 2 of 25 Many researchers worked on developing the fractional calculus concept and investigating new ways to define fractional derivatives which range from Riemann–Liouville to new hybrid proportional-Caputo fractional derivatives. The hybrid proportional-Caputo is the novel suggested fractional operator [ 14 ]. In their paper, they used an elementary FDE to discover that their new operator is deeply connected with the bivariate Mittag–Leffler function, which arises naturally from the modeling of certain systems of the real world. In addition, their new hybrid fractional operator may be useful as a tool to study the anomalous behavior of dynamical systems in diverse real data, such as chaotic systems, visco-elasticity, electro-chemistry, and physics. In this research, motivated by the work of Baleanu et al. [ 14 ], we apply the RKHSM to fractional differential equations with respect to the new hybrid fractional operator. The RKHSM is a widely used numerical method for solving non-linear systems. This method was proposed in 1908 [ 15 ] and is an effective numerical method for complex nonlinear problems without discretization. Many researchers applied it to solve several types of equations [ 16 – 21 ]. Its principal advantages are the feature that it is easy to be applied, especially because it is meshfree, and its capability to deal with diverse complex differential equations. The highlights of the manuscript can be summarized as follows: (i) an efficient numerical technique is employed for solving some differential equations with the new operator; (ii) the effect of the new fractional derivative is shown in the obtained outcomes; (iii) the superior performance of the used method is confirmed via comparing the numerical solutions with the true ones. In Section 2, we will discuss some basic tools to apply the RKHSM. After doing the preparations we need, we will describe how to apply the RKHSM in Section 3. In Section 4, some applications are presented. Finally, the conclusion is given. 2. Mathematical Concepts 2.1. The New Fractional Derivative Operator Definition 1. The Caputo derivative of order of }(τ)is described by [9] C 0Dγ τ}(τ) =RL 0I1−γ τ}0(τ) = 1 Γ(1−γ)Zτ 0 }0(η)(τ−η)−γdη, (1) where RL 0Iγ τis the Riemann–Liouville integral which is given by the definition below. Definition 2. Let } be an integrable function. The Riemann–Liouville integral of order γ> 0of } is given by [9] RL 0Iγ τ}(τ) = 1 Γ(γ)Zτ 0 }(η)(τ−η)γ−1dη. (2) The more essential properties related to the above operators can be obtained from [ 22 ]. Definition 3. Let 0 ≤γ≤ 1and K0 , K1∈C([0, 1]×R,R+) .The proportional derivative operator of order γof a differentiable function }is given by [23] PDγ}(τ) = K1(γ,τ)}(τ) + K0(γ,τ)}0(τ), (3) where the functions K0and K1satisfy the following conditions: lim γ→0+ K0(γ,τ) = 0; lim γ→1− K0(γ,τ) = 1; K0(γ,τ)6=0, 0 <γ≤1; ∀τ∈R, (4) lim γ→0+ K1(γ,τ) = 1; lim γ→1− K1(γ,τ) = 0; K1(γ,τ)6=0, 0 ≤γ<1; ∀τ∈R. (5) Remark 1. For the special cases of γ ,we can obtain from (3) – (5) two different cases. The first one, if γ= 0, then (3) reduces to the function itself, i.e., PD0}(τ) = }(τ) .In the case γ= 1, (3) reduces to the standard differentiation operator, i.e., PD1}(τ) = d dτ}(τ) = }0(τ). Symmetry 2023,15, 144 3 of 25 Remark 2. The proportional (conformable) derivative goes back to Khalil et al. [ 24 ]. In [ 25 ], some properties of the conformable derivative were investigated. In the same year, a modified proportional derivative was explored in [ 26 ] with more properties. Definition 3represents a precise definition of the proportional derivative. For more details related to the proportional operator see, for instance, [27]. It is interesting to note that the proportional derivative operator (3) of order γcan be expressed as a special case where K0 and K1 depend only on γ , they are constant functions with respect to τ. So as a consequence, this particular case can be defined as follows. Definition 4. Let 0 ≤γ≤ 1and K0 , K1∈C([0, 1],R+) .The constant proportional (CP, for short) derivative operator of order γof a differentiable function }is given by [14] CPDγ}(τ) = K1(γ)}(τ) + K0(γ)}0(τ), (6) where the functions K0and K1satisfy the following conditions: lim γ→0+ K0(γ) = 0; lim γ→1− K0(γ) = 1; K0(γ)6=0, 0 <γ≤1; (7) lim γ→0+ K1(γ) = 1; lim γ→1− K1(γ) = 0; K1(γ)6=0, 0 ≤γ<1. (8) Recently, Baleanu et al. [ 14 ] introduced the concept of a new hybrid fractional operator which can be defined in two possible ways. One is to combine both proportional and Caputo definitions. The other is to combine both constant proportional and Caputo definitions. The concept of these operators is formalized as follows. Definition 5. Let the function } be differentiable and let } with its derivative }0 be locally L1 functions on R+[14]. 1. The proportional-Caputo (PC, for short) hybrid operator of order γ,for }is given by PC 0Dγ τ}(τ) =RL 0I1−γ τhPDγ}(τ)i =     1 Γ(1−γ)Rτ 0(K1(γ,η)}(η) + K0(γ,η)}0(η))(τ−η)−γdη, 0 <γ<1, Rτ 0}(η)dη,γ=0, }0(τ),γ=1. (9) 2. The constant proportional-Caputo (CPC, for short) hybrid operator of order γ ,for } is given by CPC0Dγ τ}(τ) =RL 0I1−γ τhCPDγ}(τ)i =     1 Γ(1−γ)Rτ 0(K1(γ)}(η) + K0(γ)}0(η))(τ−η)−γdη, 0 <γ<1, Rτ 0}(η)dη,γ=0, }0(τ),γ=1. (10) Proposition 1. The hybrid CPC and PC fractional operators are non-local and singular [14]. Theorem 1. The Laplace transform of the hybrid CPC operator CPC 0Dγ τis represented by [14] LhCPC 0Dγ τ}(τ)i=K1(γ) s+K0(γ)sγˆ }(s)−K0(γ)sγ−1}(0), (11) where the function }(τ) is differentiable. } , with its derivative }0 , are locally L1 functions on R+ , and ˆ }(s)exists. Symmetry 2023,15, 144 4 of 25 Among the main properties of the PC and CPC fractional operators, we mention the following ones, namely, the inversion relations: •PC 0Iγ τPC 0Dγ τ}(τ) = }(τ)−exp−Rτ 0 K1(γ,s) K0(γ,s)ds}(0), •CPC 0Iγ τCPC 0Dγ τ}(τ) = }(τ)−exp−K1(γ) K0(γ)τ}(0), •PC 0Dγ τPC 0Iγ τ}(τ) = CPC 0Dγ τCPC 0Iγ τ}(τ) = }(τ)−τ−γ Γ(1−γ)lim τ→0 RL 0Iγ τ}(τ), where PC 0Iγ τ and CPC 0Iγ τ denote the inverse of the operators PC 0Dγ τ and CPC 0Dγ τ , respectively. They were constructed in [14] as follows. Proposition 2. The inverse operators to the CPC and PC fractional derivatives are defined, respectively, as [14] PC 0Iγ τ}(τ) = Zτ 0exp−Zτ η K1(γ,s) K0(γ,s)dsRL 0D1−γ η}(η) K0(γ,η)dη, (12) CPC 0Iγ τ}(τ) = 1 K0(γ)Zτ 0exp−K1(γ) K0(γ)(τ−η)RL 0D1−γ η}(η)dη. (13) These satisfy the inversion relations which are mentioned just above. The RL 0Dγ τ denotes the Riemann–Liouville derivative which is defined as follows: RL 0Dγ τ}(τ) = 1 Γ(1−γ) d dτZτ 0 }(η)(τ−η)−γdη. (14) For more details on CPC and PC hybrid fractional operators see [14]. Remark 3. Of special interest, Baleanu et al. [14] realized that the specific case K0(γ,τ) = γτ1−γ,K1(γ,τ) = (1−γ)τγ. (15) will not be useful in applications because of the lack of dimensional agreement in (3) while this is important for physical consistency. Remark 4. We shall pay special attention in this paper to two specific cases when 1. For any σ∈(0, +∞),we take K0(γ) = γσ1−γ,K1(γ) = (1−γ)σγ(16) 2. For any σ∈(0, +∞),we take K0(γ) = γσ1+γ,K1(γ) = (1−γ)σγ(17) 2.2. The Reproducing Kernel Theory The following are some fundamental definitions and theorems required from the reproducing kernel theory. Definition 6. We say that a function K:S×S→C is a reproducing kernel of the space H provided [28]: 1. K(·,t)∈H,∀t∈S. 2. hf,K(·,t)i=f(t),∀f∈H and ∀t∈S. where H is a Hilbert space over S and S 6=∅. Remark 5. Note that the assertion (2)is called the reproducing property (RP). Remark 6. An RKHS “H” is a Hilbert space endowed with an RK “K”. Symmetry 2023,15, 144 5 of 25 Definition 7 ([ 28 ]) . The function space W2 2[ 0, T] consists of all functions f for which f and f0 are absolutely continuous functions on [0, T],f00 ∈L2[0, T],and f(0) = 0. Definition 8 ([28]).If f,g∈W2 2[0, T],then the inner product and norm are hf,giW2 2= 1 ∑ ı=0 f(ı)(0)g(ı)(0) + ZT 0f(m)(t)g(m)(t)dt, and kfkW2 2=qhf,fiW2 2. Theorem 2. We obtain the RK function Sη(t)of W2 2[0, T]as: Sη(t) = tη+1/2 ηt2−1/6 t3,t≤η, −1/6 η3+1/2 tη2+tη,t>η.(18) Proof. We must prove f,SηW2 2=f(η). We have f,SηW2 2=f(0)Sη(0) + f0(0)S0 η(0) + f0(0)S0 η(0) + ZT 0f00(t)S00 η(t)dt. Applying integration by parts, we obtain: f,SηW2 2=f(0)Sη(0) + f0(0)S0 η(0) + f0(T)S00 η(T)−f0(0)S00 η(0)−ZT 0f0(t)S(3) η(t)dt. Since f(t)∈W2 2[0, T], we have f(0) = 0. (19) Then f,SηW2 2=f0(0)S0 η(0) + f0(T)S00 η(T)−f0(0)S00 η(0)−ZT 0f0(t)S(3) η(t)dt. We need to compute S0 η(0),S00 η(0), and S00 η(T): S0 η(0) = η, S00 η(0) = η, S00 η(T) = 0. By using the above equations, we obtain: f,SηW2 2=−ZT 0f0(t)S(3) η(t)dt. (20) We have S(3) η(t) = −1 , t≤η, 0 , t>η. Symmetry 2023,15, 144 6 of 25 Thus f,SηW2 2=−Zη 0f0(t)S(3) η(t)dt−ZT ηf0(t)S(3) η(t)dt =Zη 0f0(t)dt =f(η)−f(0), and, since f(t)∈W2 2[0, T], we deduce f,SηW2 2=f(η). Definition 9 ([ 28 ]) . The function space W1 2[ 0, T] consists of all functions f for which f is absolutely continuous function on [0, T]and f0∈L2[0, T]. Definition 10 ([28]).If f,g∈W1 2[0, T],then the inner product and norm are hf,giW1 2=f(0)g(0) + ZT 0f0(t)g0(t)dt, and kfkW1 2=qhf,fiW1 2. Theorem 3. We obtain the RK function Rη(t)of W1 2[0, T]as: Rη(t) = 1+t,t≤η, η+1 , t>η.(21) Proof. We must prove f,RηW1 2=f(η). We have f,RηW1 2=f(0)Rη(0) + ZT 0f0(x)R0 η(x)dx. We need to compute Rη(0): Rη(0) = 1, By using the above equation, we obtain: f,RηW1 2=f(0) + ZT 0f0(t)R0 η(t)dt. We have R0 η(t) = 1 , t≤η, 0 , t>η. Thus f,RηW1 2=f(0) + Zη 0f0(t)R0 η(t)dt+ZT ηf0(t)R0 η(t)dt =f(0) + Zη 0f0(t)dt =f(0) + f(η)−f(0), Symmetry 2023,15, 144 7 of 25 so, we deduce f,RηW1 2=f(η). 3. The RKHS Approach Considering the following fractional initial value problem: CPC 0Dγ tf(t) = φ(t,f(t)),t∈[0, T], f(0) = µ.(22) where CPC 0Dγ tf(t)is given in (10). One skillful way to investigate the considered problem by using RKHSM is to homogenize the initial condition f(0) = µ. To do so, the transformation has the form: g(t) = f(t)−µ. Then, (22) becomes CPC 0Dγ tg(t) = Λ(t,g(t)),t∈[0, T], g(0) = 0. (23) where Λ(t,g(t)) = φ(t,g(t) + µ)−µt1−γK1(γ)/Γ(2−γ). The first step is to define a linear operator O:W2 2[0, T]→W1 2[0, T]defined by Og(t) = CPC 0Dγ tg(t). (24) Theorem 4. The operator O:W2 2[0, T]→W1 2[0, T]is bounded and linear. Proof. For checking the linearity, let g(t),m(t)∈W2 2[0, T]. Then, O(g+m)(t) = CPC 0Dγ t(g+m)(t), =1 Γ(1−γ)Zt 0K1(γ)(g+m)(τ) + K0(γ)(g+m)0(τ)(t−τ)−γdτ, =CPC 0Dγ tg(t) + CPC 0Dγ tm(t), =Og(t) + Om(t). Additionally, let g(t)∈W2 2[0, T]and ξ∈R. Then O(ξg)(t) = CPC 0Dγ t(ξg)(t), =1 Γ(1−γ)Zt 0K1(γ)(ξg)(τ) + K0(γ)(ξg)0(τ)(t−τ)−γdτ, =ξCPC 0Dγ tg(t), =ξOg(t). We can now prove that Ois bounded. This shows that kOgkW1 2≤ΞkgkW2 2, with Ξ>0. From Definition 10, we have kOg(t)k2 W1 2=hOg(t),Og(t)iW1 2=[Og(0)]2+ZT 0Og0(t)2dt. Symmetry 2023,15, 144 8 of 25 By virtue of the RP, we have g(t) = hg(),St()iW2 2. In addition, Og(t) = hg(),OSt()iW2 2, Og0(t) = hg(),∂t(OSt())iW2 2. Using the Schwarz inequality and the continuity of St()to obtain |Og(t)|=hg(),OSt()iW2 2≤kgkW2 2kOSt()kW2 2≤Ξ1kgkW2 2. (25) In the same way, Og0(t)≤Ξ2kgkW2 2. Hence kOg(t)k2 W2 2≤Ξ2 1kgk2 W2 2+ZT 0 Ξ2 2kgk2 W2 2dt, =Ξ2 1+TΞ2 2kgk2 W2 2. (26) From (26) , we conclude that Hence kOg(t)k2 W2 2≤Ξ2 1kgk2 W2 2+ZT 0 Ξ2 2kgk2 W2 2dt, =Ξ2 1+TΞ2 2kgk2 W2 2.(26) From (26), we conclude that kOg(τ)kW1 2≤ΞkgkW2 2,where Ξ=Ξ2 1+TΞ2 2. By applying (24), we can rewrite (23) as follow    Og(t) = Λ(t, g(t)), t ∈[0, T], g(0) = 0.(27) where Λ(t, g(t)) = φ(t, g(t) + µ)−(µt1−γK1(γ)) /Γ(2 −γ). Prior to construct the numerical solution of (27), we should first construct the orthogonal function system of W2 2[0, T].For this, let us introduce the useful functions: κı(t) = Rtı(t)and ψı(t) = O∗κı(t), where • Rtı(t)represents the RKF associated with W1 2[0, T]. •O∗is the formal adjoint of O. • {tı}∞ ı=1 is a dense countable set in [0, T]. The Gram-Schmidt’s process gives the following orthonormal system {¯ ψı}∞ ı=1: ¯ ψı(t) = ı X k=1 $ıkψk(t), $ıı >0, ı = 1,2,.... (28) Here {ψı}∞ ı=1 denotes a function system in W2 2[0, T]where its expression can be determined by the following way: $ı =       1 kψ1kfor ı== 1, 1 eıfor ı=6= 1, −1 eıPı−1 k=Cık$k for ı > , (29) where eı=qkψık2−Pı−1 k=1 C2 ık, Cık =ψı,¯ ψkW2 2 . 12 . By applying (24), we can rewrite (23) as follows Og(t) = Λ(t,g(t)),t∈[0, T], g(0) = 0. (27) where Λ(t,g(t)) = φ(t,g(t) + µ)−µt1−γK1(γ)/Γ(2−γ). Prior to constructing the numerical solution of (27) , we should first construct the orthogonal function system of W2 2[0, T]. For this, let us introduce the useful functions: κı(t) = Rtı(t)and ψı(t) = O∗κı(t), where •Rtı(t)represents the RKF associated with W1 2[0, T]. •O∗is the formal adjoint of O. •{tı}∞ ı=1is a dense countable set in [0, T]. The Gram–Schmidt process gives the following orthonormal system {¯ ψı}∞ ı=1: ¯ ψı(t) = ı ∑ k=1 vıkψk(t),vıı >0, ı=1, 2, . . . . (28) Here, {ψı}∞ ı=1 denotes a functioning system in W2 2[ 0, T] where its expression can be determined by the following way: vı=     1 kψ1kfor ı==1, 1 eıfor ı=6=1, −1 eı∑ı−1 k=Cıkvkfor ı>, (29) where eı=qkψık2−∑ı−1 k=1C2 ık,Cık =hψı,¯ ψkiW2 2. Symmetry 2023,15, 144 9 of 25 Remark 7. Not that the formula for ψı(t)is given by ψı(t) = O∗κı(t) = hO∗κı(η),St(η)iW2 2=hκı(η),OSt(η)iW1 2=hRtı(η),OSt(η)iW1 2=OηSt(η)|η=tı. The operator Oηmeans that Ois applied to ηvariables. Theorem 5. Assume {tı}∞ ı=1 is dense on [ 0, T] ,then {ψı}∞ ı=1 is the complete system of the space W2 2[0, T]. Proof. We see easily that ψı(t)∈W2 2[0, T]. Therefore, for each fixed g(t)∈W2 2[0, T], hg(t),ψı(t)iW2 2=0, ı=1, 2, . . . , Since hg(t),ψı(t)iW2 2=hg(t),O∗κı(t)iW2 2=hOg(t),κı(t)iW1 2=Og(tı) = 0. Due to the density of {tı}∞ ı=1in [0, T], one can get Og(t) = 0. Additionally, the existence of O−1implies. g(t) = 0. Lemma 1. Assume g(t)∈W2 2[0, T],then   g(ı)(t)  C≤zkg(t)kW2 2,ı=0, 1. zis non-negative and kg(t)kC=max t∈[0,T]|g(t)|. Proof. ∀t∈[0, T]we have g(ı)(t) = Dg(),∂(ı) tSt()EW2 2 ,ı=0, 1, and   ∂(ı) tSt  W2 2 ≤zı,ı=0, 1. As a result, g(ı)(t)=Dg(),∂(ı) tSt()EW2 2 ≤  ∂(ı) tSt  W2 2 kgkW2 2≤zıkgkW2 2,ı=0, 1. (30) Here z=max ı=0,1{zı}. Theorem 6. Assume {tı}∞ ı=1 is dense in [ 0, T] and there exists a unique solution g(t) for (27) in W2 2[0, T],then the solution’s representation of (27)is given by g(t) = ∞ ∑ ı=1 ı ∑ k=1 vıkΛ(tk,g(tk)) ¯ ψı(t), (31) Symmetry 2023,15, 144 16 of 25 Figure 3. Comparison of numerical solutions of the RKHSM by the ES for the FS of Example 1. Symmetry 2023,15, 144 17 of 25 (a) Exact and the RKHSM’s solutions. (b) Absolute error of the RKHSM. Figure 4. Comparison of numerical solutions of the RKHSM by the ES for the SS of Example 1. Symmetry 2023,15, 144 18 of 25 (a) Exact and the RKHSM’s solutions. (b) Absolute error of the RKHSM. Figure 5. Comparison of numerical solutions of the RKHSM by the ES for the SS of Example 1. Symmetry 2023,15, 144 19 of 25 Figure 6. Comparison of numerical solutions of the RKHSM by the ES for the SS of Example 1. Example 2. Considering the following simple problem: CPC 0Dγ tf(t) = t, 0 <t≤1, f(0) = 0. (35) The exact solution to the above problem takes the following form f(t) = exp−tK1(γ) K0(γ)tγ−Γ(1+γ)+Γ1+γ,−tK1(γ) K0(γ)−tK1(γ) K0(γ)−γ Γ(1+γ)K1(γ). 1. First situation (FS, for short): K0(γ) = γσ1−γ,K1(γ) = (1−γ)σγ,σ=1 2. 2. Second situation (SS, for short): K0(γ) = γσ1+γ,K1(γ) = (1−γ)σγ,σ=1 2. This RKHSM is tested with the grid points tı=ı n , ı= 1, 2, . . . , n .The ES, AS, AE, and RE of (35) are shown in Tables 5–8when γ∈ { 0.25, 0.5, 0.75, 0.9 } and t∈[ 0, 1 ] for both the FS and SS. Additionally, the comparisons between the ES and RKHSM’s solution for both cases are depicted in Figure 7for the FS and Figure 8for the SS, when γ= 0.25, 0.5, 0.75, 0.9. From the results, we can see that the numerical solutions are found to be in excellent agreement with the exact solutions. Symmetry 2023,15, 144 20 of 25 Table 5. First and second situations: results for Example 2when γ=0.25. First Situation Second Situation tES AS AE RE ES AS AE RE 0.1 0.278390226 0.278390225 8.0 ×10−10 2.873664108 ×10−90.366578617 0.366578616 1.7 ×10−94.637477257 ×10−9 0.2 0.559438322 0.559438324 2.5 ×10−94.468767876 ×10−90.694736216 0.694736218 1.3 ×10−91.871213807 ×10−9 0.3 0.794667664 0.794667667 3.2 ×10−94.026840585 ×10−90.941462921 0.941462918 2.9 ×10−93.080312495 ×10−9 0.4 0.986000956 0.986000987 3.1 ×10−83.184581091 ×10−81.125538939 1.125538968 2.9 ×10−82.576543467 ×10−8 0.5 1.141251776 1.141251787 1.1 ×10−89.638539217 ×10−91.265695498 1.265695503 5.0 ×10−93.950397238 ×10−9 0.6 1.268231856 1.268231894 3.8 ×10−82.996297548 ×10−81.375697794 1.375697830 3.6 ×10−82.616853800 ×10−8 0.7 1.373458393 1.373458428 3.5 ×10−82.548311633 ×10−81.464920275 1.464920304 2.9 ×10−81.979629915 ×10−8 0.8 1.462030835 1.462030879 4.4 ×10−83.009512450 ×10−81.539590658 1.539590698 4.0 ×10−82.598093187 ×10−8 0.9 1.537826398 1.537826474 7.6 ×10−84.942040278 ×10−81.603828534 1.603828600 6.6 ×10−84.115153123 ×10−8 1 1.603753900 1.603753849 5.1 ×10−83.180039032 ×10−80.001239376 1.660377912 3.0 ×10−81.806817547 ×10−8 Table 6. First and second situations: results for Example 2when γ=0.5. First Situation Second Situation tES AS AE RE ES AS AE RE 0.1 0.064667410 0.064667411 9.9 ×10−10 1.530910242 ×10−80.124390490 0.124390493 2.9 ×10−92.331367938 ×10−8 0.2 0.175914718 0.175914719 1.5 ×10−98.526859025 ×10−90.326098542 0.326098549 6.5 ×10−91.993262514 ×10−8 0.3 0.311030443 0.311030447 4.4 ×10−91.414652521 ×10−80.556730844 0.556730856 1.2 ×10−82.227288130 ×10−8 0.4 0.461172981 0.461172979 1.7 ×10−93.686252383 ×10−90.798597356 0.798597370 1.4 ×10−81.803161493 ×10−8 0.5 0.621108506 0.621108511 4.7 ×10−97.567115817 ×10−91.042442942 1.265695503 1.9 ×10−81.822641756 ×10−8 0.6 0.787336310 0.787336312 2.1 ×10−92.667221077 ×10−91.283029637 1.375697830 3.6 ×10−82.805858881 ×10−8 0.7 0.957370780 0.957370788 8.2 ×10−98.565124582 ×10−91.517362373 1.464920304 4.2 ×10−82.767961161 ×10−8 0.8 1.129387212 1.129387215 3.0 ×10−92.656307746 ×10−91.743794486 1.539590698 4.6 ×10−82.637925603 ×10−8 0.9 1.302020083 1.302020093 1.0 ×10−87.680373084 ×10−91.961520086 1.603828600 8.3 ×10−84.231412368 ×10−8 1 1.474236919 1.474236919 0 0 2.170265034 2.170264989 4.5 ×10−82.073479473 ×10−8 Symmetry 2023,15, 144 21 of 25 Table 7. First and second situations: results for Example 2when γ=0.75. First Situation Second Situation tES AS AE RE ES AS AE RE 0.1 0.01738193388 0.01738193250 1.38 ×10−97.939277698 ×10−80.048404880 0.048404879 1.1 ×10−92.355134427 ×10−8 0.2 0.05796964429 0.05796964023 4.06 ×10−97.003665539 ×10−80.158979414 0.158979408 5.5 ×10−93.459567419 ×10−8 0.3 0.1168624847 0.1168624762 8.5 ×10−97.273506140 ×10−80.315694535 0.315694524 1.1 ×10−83.579409442 ×10−8 0.4 0.1917111802 0.1917111584 2.2 ×10−81.137127213 ×10−70.510261243 0.510261205 3.8 ×10−87.427567849 ×10−8 0.5 0.2809210058 0.2809209865 1.9 ×10−86.870258757 ×10−80.736856811 0.736856784 2.7 ×10−83.596356798 ×10−8 0.6 0.3832748550 0.3832748188 3.6 ×10−89.444919104 ×10−80.990974626 0.990974570 5.6 ×10−85.600547032 ×10−8 0.7 0.4977838273 0.4977837878 4.0 ×10−87.935171421 ×10−81.268951107 1.268951049 5.8 ×10−84.570704078 ×10−8 0.8 0.6236125669 0.6236125196 4.7 ×10−87.584837527 ×10−81.567719773 1.567719703 7.0 ×10−84.465083697 ×10−8 0.9 0.7600365805 0.7600365012 7.9 ×10−81.043370833 ×10−71.884664677 1.884664559 1.2 ×10−76.261060731 ×10−8 1 0.9064155258 0.9064155820 5.6 ×10−86.200246840 ×10−82.217524640 2.217524727 8.7 ×10−83.923293497 ×10−8 Table 8. First and second situations: results for Example 2when γ=0.9. First Situation Second Situation tES AS AE RE ES AS AE RE 0.1 0.0081861790 0.0081861771 1.9 ×10−92.313655737 ×10−70.028351089 0.028351084 5.2 ×10−91.844726329 ×10−7 0.2 0.0304849118 0.0304849070 4.8 ×10−91.567988792 ×10−70.105006807 0.105006793 1.4 ×10−81.314200522 ×10−7 0.3 0.0657211749 0.0657211649 1.0 ×10−81.526144354 ×10−70.225161367 0.225161338 2.9 ×10−81.279082660 ×10−7 0.4 0.1132757197 0.1132756975 2.2 ×10−81.959819815 ×10−70.386004901 0.386004835 6.7 ×10−81.725366693 ×10−7 0.5 0.1727089417 0.1727089198 2.2 ×10−81.268029309 ×10−70.585395620 0.585395557 6.4 ×10−81.088152999 ×10−7 0.6 0.2436739618 0.2436739235 3.8 ×10−81.571772368 ×10−70.821549439 0.821549325 1.1 ×10−71.380318635 ×10−7 0.7 0.3258809599 0.3258809169 4.3 ×10−81.319500225 ×10−71.092910710 1.092910583 1.3 ×10−71.162034545 ×10−7 0.8 0.4190787537 0.4190787040 5.0 ×10−81.185934614 ×10−71.398085171 1.398085022 1.5 ×10−71.065743369 ×10−7 0.9 0.5230442467 0.5230441624 8.4 ×10−81.611718330 ×10−71.735800499 1.735800248 2.5 ×10−71.446018711 ×10−7 1 0.6375755634 0.6375756254 6.2 ×10−89.724337562 ×10−82.104880901 2.104881081 1.8 ×10−78.551552723 ×10−8 Symmetry 2023,15, 144 22 of 25 (a) Exact solutions for different values of γ. (b) Approximate solutions for different values of γ. Figure 7. Comparison of numerical solutions of the RKHSM by the ES for the FS of Example 2. Symmetry 2023,15, 144 23 of 25 (a) Exact solutions for different values of γ. (b) Approximate solutions for different values of γ. Figure 8. Comparison of numerical solutions of the RKHSM by the ES for the SS of Example 2. 5. Conclusions In this paper, an efficient method has been applied successfully for solving FDEs. The approximate solution gn(t) and its derivative both converge uniformly. The accuracy and applicability of the proposed method are validated by computing the numerical solutions at many grid points. The boundedness of the linear operator is demonstrated. The results show that the proposed method is a powerful tool to approximate many other non-linear problems which are described by the new hybrid fractional derivative operator. This research opens the way for the use of the RKHSM to study the mentioned problem for various new fractional derivatives. As part of our purpose, we plan to apply the RKHSM Symmetry 2023,15, 144 24 of 25 to multidimensional fractional partial differential equations that are described with CPC derivative, which will be new in the literature. Author Contributions: Conceptualization, D.S.; methodology, N.A.; software, A.A.; validation, A.A. and D.S.; formal analysis, N.A.; investigation, A.A.; resources, M.D.l.S.; data curation, A.A.; writing—original draft preparation, N.A.; writing—review and editing, N.A.; visualization, A.A.; supervision, M.B.; project administration, A.A., M.D.l.S., M.B., D.S. and A.N.; funding acquisition, M.D.l.S. and M.B. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Data are included within this research. Acknowledgments: The authors are grateful to the Basque Government for its support through Grant IT1555-22 and to MCIN/AEI 269.10.13039/501100011033 for Grant PID2021-1235430B-C21/C22. Conflicts of Interest: The authors declare no conflict of interest. References 1. Kapoor, M.; Shah, N.A.; Weera, W. Analytical solution of time-fractional Schrödinger equations via Shehu Adomian Decomposition Method. AIMS Math. 2022,7, 19562–19596. [CrossRef] 2. Ozkan, E.M. New exact solutions of some important nonlinear fractional partial differential Equations with beta derivative. Fractal Fract. 2022,6, 173. [CrossRef] 3. Zhang, Y.; Feng, M. A local projection stabilization virtual element method for the time-fractional Burgers equation with high Reynolds numbers. Appl. Math. Comput. 2023,436, 127509. [CrossRef] 4. Iqbal, J.; Shabbir, K.; Guran, L. Stability analysis and computational interpretation of an effective semi analytical scheme for fractional order non-linear partial differential equations. 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