Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders
Abstract
The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory's existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators.
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Electronic Research Archive http://www.aimspress.com/journal/era ERA, 32(2): 733–761. DOI: 10.3934/era.2024035 Received: 26 September 2023 Revised: 03 December 2023 Accepted: 15 December 2023 Published: 10 January 2024 Research article Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders Seda IGRET ARAZ1,2,*, Mehmet Akif CETIN3and Abdon ATANGANA2,4,5 1Siirt University, Department of Mathematics Education, Siirt, Turkey 2Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa 3ALTSO Vocational School, Alanya Alaaddin Keykubat University, Antalya, Turkey 4Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan 5IT4Innovations, VSB–Technical University of Ostrava, Ostrava-Poruba 70800, Czech Republic *Correspondence: Email: [email protected]. Abstract: The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory’s existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators. Keywords: Caratheodory’s conditions; fractal-fractional differentiation; piecewise calculus; parametrized method 1. Introduction Fractional analysis is a theory that started with Leibniz asking if there is a derivative of order 1 / 2 of a function. This theory interested many researchers when different types of fractional derivatives were introduced. One well-known definition is the Riemann-Liouville fractional derivative where the power-law kernel is incorporated. Caputo [1] introduced a derivative with a modification on the
734 Riemann-Liouville fractional derivative [2] because it was useful in theory but not appropriate for solving real-life problems. These operators, which are used to model power law processes, have behavior that is both nonlocal and singular. Even though some processes are unique, another form of math is needed to describe processes that behave differently. Caputo and Fabrizio [3] have created a mathematical concept called a derivative with fading memory, which uses an exponential pattern. This derivative deals with processes that behave predictably and within a small area. However, we needed a derivative that is predictable but acts over a larger area. The Atangana-Baleanu fractional derivative [4] is a mathematical tool that meets this requirement, and it utilizes the Mittag-Leffler function. The fractal derivative or Hausdorffderivative [5] is a different kind of derivative used for measuring fractals in fractal geometry. Fractal derivatives were made to study how things spread in a strange way when normal ways of studying do not consider the fractal shape of the thing that things are spreading through. A fractal measure t changes its size in relation to t raised to the power of β . This type of derivative is only used in a specific area, unlike the fractional derivative, which is used in a similar way. Later, Atangana introduced fractal-fractional derivatives [6] by combining the concepts of fractal and fractional derivatives. Although there is no doubt that fractional differential operators are useful in modeling relevant processes [7–11], these operators cannot be used to model crossover processes such as from stochastic to power-law or from fading memory to stochastic [12–15]. Concluding that a new class of differential operators was needed for this, Atangana and Araz introduced piecewise differential operators [16], which can be created by including various differential operators to model such processes. These operators, which can be used to describe many processes, from modeling the different rates (or even stopping) of an individual’s heartbeat over a period of time, to modeling the spread of a virus, first cumulatively and then daily, have become focus of attention for researchers. In order to better understand and analyze the processes discussed, it is necessary to solve the equations that represent these processes. Because it is difficult to solve these equations using analytic methods when the operators mentioned above and the nonlinearity of the associated equations are involved, we have to use numerical methods to obtain solutions to such equations. The parametrized method, which deals with the approximation of a function with constants depending on a parameter, is one of the well-known numerical methods. While the parameterized method is presented in the literature [8–10] for classical differential equations, Atangana and Araz [17] extended this method to solve fractional and fractal-fractional differential equations. The parametrized method was compared with existing methods in the literature in [17] and it was shown that the method is more effective than other methods, especially when the parameter is close to 1. However, in [17], the application of the relevant method to stochastic differential equations with fractional, fractal-fractional and piecewise derivatives [16] is not taken into account. Therefore, in this study, we present the derivation of this method for stochastic differential equations with fractional, fractal-fractional and piecewise derivatives. We employ the parametrized method to solve different types of equations obtained by incorporating these mathematical tools into differential equations. Before presenting the associated method, first the definitions of the above-mentioned fractional, fractal fractional and piecewise derivatives will be presented. In the following section, with the help of Carath ´ eodory conditions [15,16], the existence and uniqueness of the solution of Atangana-Baleanu stochastic differential equations [18] will be investigated. In the remaining sections, in addition to the derivation of the parametrized method with these derivatives, some illustrative examples will be included. Electronic Research Archive Volume 32, Issue 2, 733–761.
735 2. Preliminaries In this section, the definitions of fractional derivatives with power law behavior, fading memory and exhibiting power law behavior after fading memory, fractal-fractional derivatives and piecewise derivatives, which can be represented in different ways by including fractional and fractal-fractional derivatives, will be discussed. The Caputo-Fabrizio fractional derivative [3] of the function f(t)∈H1(0,T)is defined by CF 0Dα tf(t)=1 1−αZt 0 f0(τ)exp −α 1−α(t−τ)dτ, (2.1) where 0 <α<1 and H1(0,T)describes the Hilbert space. The associated integral is given as CF 0Jα tf(t)=(1−α)f(t)+αZt 0 f(τ)dτ. (2.2) The Caputo fractional derivative [1] of the function f(t)∈H1(0,T)is defined by C 0Dα tf(t)=1 Γ(1−α)Zt 0 f0(τ) (t−τ)−αdτ, (2.3) where 0 < α ≤ 1 and the Riemann-Liouville fractional derivative of the function f(t)∈C(0,T) is defined by RL 0Dα tf(t)=1 Γ(1−α) d dt Zt 0 f(τ) (t−τ)−αdτ. (2.4) The integral with power-law kernel [2] is given by RL 0Jα tf(t)=1 Γ(α)Zt 0 f(τ) (t−τ)α−1dτ. (2.5) The following formulas describe the Atangana-Baleanu fractional derivative [4], which has the crossover behavior from stretched exponential to power-law, ABC 0Dα tf(t)=1 1−αZt 0 f0(τ)Eα−α 1−α(t−τ)αdτ, (2.6) and ABR 0Dα tf(t)=1 1−α d dt Zt 0 f(τ)Eα−α 1−α(t−τ)αdτ. (2.7) The above operators are called Atangana-Baleanu fractional derivative in the Caputo sense and AtanganaBaleanu fractional derivative in the Riemann-Liouville sense [4], respectively. The associated integral is given by AB 0Jα tf(t)=(1−α)f(t)+α Γ(α)Zt 0 f(τ) (t−τ)α−1dτ. (2.8) The concept of fractal-fractional differentiation and integration has appeared previously with the idea of combining the fractal and fractional derivatives. The fractal-fractional derivative [6] with power-law kernel is defined by Electronic Research Archive Volume 32, Issue 2, 733–761.
736 FFP 0Dα,β tf(t)=1 Γ(1−α) d dtβZt 0 f(τ) (t−τ)−αdτ, (2.9) where the definition of fractal derivative [5] is d dtβf(t)=lim t→t1 f(t)−f(t1) tβ−tβ 1 .(2.10) The associated fractal-fractional integral [6] with power-law kernel is given by FFP 0Jα tf(t)=1 Γ(α)Zt 0 βτβ−1f(τ) (t−τ)α−1dτ. (2.11) The fractal-fractional derivative with Mittag-Leffler kernel [6] is defined by FFM 0Dα,β tf(t)=1 1−α d dtβZt 0 f(τ)Eα−α 1−α(t−τ)αdτ(2.12) and the associated fractal-fractional integral is given by FFM 0Jα tf(t)=(1−α)βtβ−1f(t)+α Γ(α)Zt 0 βτβ−1f(τ) (t−τ)α−1dτ. (2.13) The fractal-fractional derivative with exponential decay kernel [6] is defined by FFE 0Dα,β tf(t)=1 1−α d dtβZt 0 f(τ)exp −α 1−α(t−τ)dτ(2.14) and the associated fractal-fractional integral is given by FFE 0Jα tf(t)=(1−α)βtβ−1f(t)+αZt 0 βτβ−1f(τ)dτ. (2.15) We now present the definitions of the piecewise derivative and integral operators, which made significant contribution to literature [16]. The piecewise derivative with classical and fractional derivative with power-law kernel such that it can be taken as [16] PRL 0Dα ty(t)=(y0(t)if 0 ≤t≤t0 RL t0Dα ty(t)if t0≤t≤T(2.16) where PRL 0Dα t represents the classical derivative within 0 ≤t≤t0 and the Riemann-Liouville fractional derivative within t0≤t≤T. The piecewise with Caputo derivative is given as [16] PC 0Dα ty(t)=(y0(t)if 0 ≤t≤t0 C t0Dα ty(t)if t0≤t≤T(2.17) where the function y(t) is continuous but not necessarily differentiable in [t0,T]. Here, PRL 0Dα t represents the classical derivative on 0 ≤t≤t0 and the Caputo fractional derivative [1] on t0≤t≤T . The associated piecewise integral of yis given as [16] Electronic Research Archive Volume 32, Issue 2, 733–761.
737 PPLIty(t)= Rt0 0y(τ)dτif 0 ≤t≤t0 1 Γ(α)Rt t0y(τ) (t−τ)α−1dτif t0≤t≤T(2.18) where PPL 0Iα t represents the classical integral on 0 ≤t≤t0 and the integral with power-law kernel on t0≤t≤T. The piecewise derivative with classical derivative and exponential decay kernel is given as [16] PCF 0Dα ty(t)=(y0(t)if 0 ≤t≤t0 CF t0Dα ty(t)if t0≤t≤T(2.19) where PCF 0Dα t is the classical derivative on 0 ≤t≤t0 and the Caputo-Fabrizio fractional derivative [3] on t0≤t≤T. Here, it is assumed that the function y(t) is differentiable. A piecewise integral is given as [16] PCF Ity(t)= Rt0 0y(τ)dτif 0 ≤t≤t0 1−α M(α)y(t)+α M(α)Rt t0y(τ)dτif t0≤t≤T.(2.20) The piecewise derivative with classical derivative and Mittag-Leffler kernel is defined by [16] PAB 0Dα ty(t)=(y0(t)if 0 ≤t≤t0 ABC t0Dα ty(t)if t0≤t≤T(2.21) where PAB 0Dα t represents the classical derivative on 0 ≤t≤t0 and the Atangana-Baleanu fractional derivative [4] on t0≤t≤T.The associated piecewise integral is given as [16] PABIty(t)= Rt0 0y(τ)dτif 0 ≤t≤t0 (1−α)y(t)+α Γ(α)Rt t0y(τ) (t−τ)α−1dτif t0≤t≤T.(2.22) Lemma 1. (The generalization of the Gronwall inequality) Assume that b≥ 0 , α > 0 , and x(t) is a nonnegative function locally integrable on 0 ≤t<T, and assume that y(t) is nonnegative and locally integrable on 0≤t<T with y(t)≤x(t)+bZt 0 y(τ) (t−τ)α−1dτ. (2.23) Then, y(t)≤x(t)+Zt 0 ∞ X n=1 (bΓ(α))n Γ(nα)y(τ) (t−τ)nα−1x(τ)dτ. (2.24) Definition 1. (Stirling formula) The Stirling formula for the Gamma function is formulated by Γ(x)∼√2πe−xxx−1 2.(2.25) Electronic Research Archive Volume 32, Issue 2, 733–761.
738 3. Caratheodory’s theory for existence and uniqueness for a general Cauchy problem with stochastic Atangana-Baleanu fractional derivative In this section, we prove the existence and uniqueness of the solution for the Atangana-Baleanu stochastic differential equation [18] by employing Carath ´ eodory’s existence theory [19,20], which is a more general version of Peano’s existence theorem. It is worth noting that the existence and uniqeness of the solution for stochastic differential equations with the Caputo fractional derivative is presented in [21]. Here, we will examine the existence and uniqueness of the stochastic differential equation with Atangana-Baleanu fractional derivative. The differential equation under investigation is represented in the form: AB 0Dα ty(t)=f1(t,y)dt +σy(t)dB (t),t≥0 (3.1) y(t0)=y0 under the conditions E1) For all y,¯y∈H, there is a constant k>0 such that |f1(t,y)−f1(t,¯y)|2,|f2(t,y)−f2(t,¯y)|2≤k|y−¯y|2,t≥0.(3.2) E2) For all y∈H,there is a constant ¯ k>0 such that |f1(t,y)|2,|f2(t,y)|2≤¯ k1+|y|2,t≥0 (3.3) where H is a Banach space. Note that conditions E1 and E2 are known as the Lipschitz condition and the growth condition, respectively. Theorem 1. For each y0∈L2(Ω,H) , Eq (26) has a unique mild solution y∈C[0,T],L2(Ω,H) = S such that sup 0≤t≤T E|y|2<∞. Proof. For the proof, we will use the contraction mapping principle. Before proceeding with the proof, we define the norm kηk2 γ=sup 0≤t≤T E|η(t)|2(3.4) where Edenotes the expectation. For any t∈[0,T] and y∈S , we define the mapping subject to Ω = C[0,T],L2(Ω,H)→ C[0,T],L2(Ω,H) (Λy) (t)=y0+(1−α)f1(t,y)+(1−α)σy(t)B0(t)(3.5) +α Γ(α)Zt 0 f1(s,y) (t−s)α−1ds +ασ Γ(α)Zt 0 y(s) (t−s)α−1dB (s). Thus, we write E|(Λy) (t)−(Λ¯y) (t)|2=E (1−α) ( f1(t,y)−f1(t,¯y)) +(1−α)σ(y(t)−¯y(t)) B0(t) +α Γ(α)Rt 0(f1(s,y)−f1(s,¯y)) (t−s)α−1ds +ασ Γ(α)Rt 0(y(s)−¯y(s)) (t−s)α−1dB (s) 2 .(3.6) Electronic Research Archive Volume 32, Issue 2, 733–761.
739 Taking 2 α− 1 > 0, by the Cauchy-Schwartz inequality, Ito’s isometry formula and the Lipschitz condition [22], we have E|(Λy) (t)−(Λ¯y) (t)|2≤4(1−α)2kσ1+|B0|2E|y−¯y|2(3.7) +(T+1)4α2k Γ2(α)Zt 0 E|y−¯y|2(t−s)2α−2ds ≤4(1−α)2σk1+|B0|2ky−¯ykγ +(T+1)4α2k Γ2(α) t2α−1 (2α−1)ky−¯ykγ ≤4σ(1−α)2k 1+sup t∈[0,T]|B0|2!ky−¯ykγ +(T+1)4α2k Γ2(α) t2α−1 (2α−1)ky−¯ykγ ≤4σ(1−α)2k1+kB0k∞ky−¯ykγ +(T+1)4σα2k Γ2(α) t2α−1 (2α−1)ky−¯ykγ ≤˜ kky−¯ykγ, where ˜ k=4σ(1−α)2k1+kB0k∞+(T+1)4σα2k Γ2(α) t2α−1 (2α−1).(3.8) Using the generalized Gronwall inequality [23], we write EΛ2y(t)−Λ2¯y(t)≤4σ(1−α)2k1+kB0k∞E|Λy−Λ¯y|2(3.9) +4σα2k(T+1) Γ2(α)Zt 0 (t−s)2α−2E|Λy−Λ¯y|2ds ≤4σ(1−α)2k1+kB0k∞"4(1−α)2k(1+kB0k∞) +(T+1)4α2k Γ2(α) t2α−1 (2α−1)#(3.10) +4σα2k(T+1) Γ2(α)Zt 0 (t−s)2α−2"4(1−α)2k(1+kB0k∞) +(T+1)4α2k Γ2(α) s2α−1 (2α−1)#ds(3.11) ≤ 4σ(1−α)2k(1+kB0k∞)2 +(T+1)4σα2k Γ2(α) T2α−1 (2α−1)4(1−α)2(1+kB0k∞) +4σ(1−α)2k(1+kB0k∞)4α2k(T+1) Γ2(α)T2α−1 (2α−1) +4σα2k(T+1) Γ2(α)2Γ2(2α−1) Γ(4α−2) T4α−2 (2α−1) ky−¯ykγ. By the induction formula for n, we can then write E|(Λny) (t)−(Λn¯y) (t)|≤ 4σ(1−α)2(1+kB0k∞)n +4σ(1−α)2(1+kB0k∞) (T+1)4α2k Γ2(α) T2α−1 (2α−1)n−1 +4σ(1−α)2(1+kB0k∞)n−14σα2k(T+1) Γ2(α)Tn(2α−1) Γ(n(2α−1)) +4σα2k(T+1) Γ2(α)nTn(2α−1) (2α−1) Γn(2α−1) Γ(n(2α−1)) ky−¯ykγ(3.12) Electronic Research Archive Volume 32, Issue 2, 733–761.
740 ≤Lky−¯ykγ where L= 4σ(1−α)2(1+kB0k∞)n +4σ(1−α)2(1+kB0k∞) (T+1)4α2k Γ2(α) T2α−1 (2α−1)n−1 +4σ(1−α)2(1+kB0k∞)n−14σα2k(T+1) Γ2(α)Tn(2α−1) Γ(n(2α−1)) +4σα2k(T+1) Γ2(α)nTn(2α−1) (2α−1) Γn(2α−1) Γ(n(2α−1)) .(3.13) To prove the theorem holds, we will show that L< 1 for sufficient large n . Let us consider the following series of positive terms ∞ n=1 4σ(1−α)2(1+kB0k∞)n +4σ(1−α)2(1+kB0k∞) (T+1)4α2k Γ2(α) T2α−1 (2α−1)n−1 +4σ(1−α)2(1+kB0k∞)n−14σα2k(T+1) Γ2(α)Tn(2α−1) Γ(n(2α−1)) +4σα2k(T+1) Γ2(α)nTn(2α−1) (2α−1) Γn(2α−1) Γ(n(2α−1)) .(3.14) Using the d’Alembert discriminant method lim n→∞ 4α2k(T+1) Γ2(α)n+1T(n+1)(2α−1) (2α−1) Γn+1(2α−1) Γ((n+1)(2α−1)) 4σα2k(T+1) Γ2(α)nTn(2α−1) (2α−1) Γn(2α−1) Γ(n(2α−1)) <1 (3.15) which is equivalent to lim n→∞ 4σα2k(T+1) Γ2(α)T(2α−1)Γ(2α−1)Γ(n(2α−1)) Γ((n+1) (2α−1)) <1.(3.16) Using the Stirling formula [21], we have the following for last term lim n→∞ 4σα2k(T+1) Γ2(α)Γ(2α−1)T(2α−1)e(2α−1) √n+1 √nn n+1n(2α−1)1 ((n+1)(2α−1))(2α−1)=0 (3.17) and knowing that α < 1,we can have lim n→∞ 4σ(1−α)2(1+kB0k∞)n +4σ(1−α)2(1+kB0k∞) (T+1)4α2k Γ2(α) T2α−1 (2α−1)n−1 +4σ(1−α)2(1+kB0k∞)n−14σα2k(T+1) Γ2(α)Tn(2α−1) Γ(n(2α−1)) +4σα2k(T+1) Γ2(α)nTn(2α−1) (2α−1) Γn(2α−1) Γ(n(2α−1)) =0.(3.18) This guarantees that L< 1 holds. This proves that Λ y(t) is a contraction mapping, which completes the proof. Electronic Research Archive Volume 32, Issue 2, 733–761.
741 4. Parametrized method for a general Cauchy problem with stochastic fractional derivatives In this section, we develop the parametrized approach to numerically solving differential equations with fractional derivatives that incorporate stochastic components. Before presenting the extension of the method to the solutions of different differential equations, we shall recall the formulation of the parametrized approach [17,24–26]. The approach is formulated by the following: ϕ1(t,y)≈" 1−1 2ξ!ϕ1tk,yk+1 2ξϕ1tk+1,˜yk+1#.(4.1) 4.1. Parametrized method for a general Cauchy problem with stochastic component To derive the associated method, in this subsection, we consider a general Cauchy problem with stochastic component given by dy (t)=ϕ1(t,y)dt +σy(t)dB (t).(4.2) We convert the above into an integral equation, by applying on both sides the classical integral y(t)=y(0)+Zt 0 ϕ1(τ, y)dτ+Zt 0 σy(τ)dB (τ).(4.3) At t=tk+1,we write y(tk+1)=y(0)+Ztk+1 0 ϕ1(τ, y)dτ+Ztk+1 0 σy(τ)dB (τ)(4.4) and at t=tk y(tk)=y(0)+Ztk 0 ϕ1(τ, y)dτ+Ztk 0 σy(τ)dB (τ).(4.5) Substracting these two equalities gives y(tk+1)=y(tk)+Ztk+1 tk ϕ1(τ, y)dτ+Ztk+1 tk σy(τ)dB (τ).(4.6) The function ϕ1(τ, y) can be approximated by using the parametrized approach [17,24 – 26] presented earlier. After simplification, we have the predictor-corrector formula [27] yk+1=yk+h" 1−1 2ξ!ϕ1tk,yk+1 2ξϕ1tk+1,˜yk+1#(4.7) +σy(ck) (B(tk+1)−B(tk)) , where ck∈[tk,tk+1]and the predictor term ˜yk+1=yk+hϕ1tk,yk.(4.8) Electronic Research Archive Volume 32, Issue 2, 733–761.
748 5.3. Parametrized method for a general Cauchy problem with stochastic fractal-fractional derivative with power-law kernel In this section, we obtain the numerical solution of a general Cauchy problem with stochastic fractal-fractional derivative [18] with power-law kernel by using the parametrized method [17]. The associated problem under consideration is represented by (FFP 0Dα ty(t)=ϕ1(t,y)+σy(t)dB (t),if t>0, y(0)=y0,if t=0.(5.11) Applying the fractal-fractional derivative [6] with power-law kernel, we have y(t)=β Γ(α)Zt 0 τβ−1ϕ1(τ, y) (t−τ)α−1dτ+β Γ(α)Zt 0 σy(τ)τβ−1(t−τ)α−1dB (τ).(5.12) At t=tk+1,we have y(tk+1)=β Γ(α) k X n=0Ztn+1 tn τβ−1ϕ1(τ, y) (tk+1−τ)α−1dτ(5.13) +β Γ(α) k X n=0Ztn+1 tn τβ−1σy(τ) (tk+1−τ)α−1B0(τ)dτ. Replacing the function ϕ1(τ, y)by its parametrized approximation, we have yk+1=β Γ(α) k X n=0" 1−1 2ξ!ϕ1(tn,yn)+1 2ξϕ1tn+1,˜yn+1#(5.14) ×Ztn+1 tn τβ−1(tk+1−τ)α−1dτ +β hΓ(α) k X n=0 σy(cn) (B(tn+1)−B(tn)) ×Ztn+1 tn τβ−1(tk+1−τ)α−1dτ. The integral on the right hand side of the above equation is calculated by using the change of variables τ=tk+1uand dτ=tk+1du as follows: Ztn+1 tn τβ−1(tk+1−τ)α−1dτ=tα+β−1 k+1Ztn+1 tn uβ−1(1−u)α−1du =tα+β−1 k+1 B tn+1 tk+1 , β, α!−B tn tk+1 , β, α!!, where the function B(·,·,·) is the incomplete Beta function. By calculation of these integrals, the following numerical scheme is obtained: yk+1=β Γ(α) k X n=0" 1−1 2ξ!ϕ1(tn,yn)+1 2ξϕ1tn+1,˜yn+1#(5.15) Electronic Research Archive Volume 32, Issue 2, 733–761.
749 ×tα+β−1 k+1 B tn+1 tk+1 , β, α!−B tn tk+1 , β, α!! +β hΓ(α) k X n=0 σy(cn) (B(tn+1)−B(tn)) ×tα+β−1 k+1 B tn+1 tk+1 , β, α!−B tn tk+1 , β, α!!. We know that the term ˜yn+1is predicted by the following: ˜yn+1=y0+β Γ(α) k X n=0 ϕ1(tn,yn)tα+β−1 k+1 B tn+1 tk+1 , β, α!−B tn tk+1 , β, α!!.(5.16) 5.4. Parametrized method for a general Cauchy problem with stochastic fractal-fractional derivative with Mittag-Leffler kernel To examine the solution of a general Cauchy problem with stochastic fractal-fractional derivative with Mittag-Leffler kernel [18], we consider the following problem: (FFM 0Dα ty(t)=ϕ1(t,y)+σy(t)dB (t),if t>0, y(0)=y0,if t=0(5.17) After taking the associated integral, the above can be arranged as follows: y(t)=(1−α)ϕ1(t,y)+(1−α)σy(t)dB (t)(5.18) +αβ Γ(α)Zt 0 τβ−1ϕ1(τ, y) (t−τ)α−1dτ +αβ Γ(α)Zt 0 σy(τ)τβ−1(t−τ)α−1B0(τ)dτ. At t=tk+1,we have y(t)=(1−α)ϕ1tk+1,yk+1+(1−α)σy(tk+1)dB (tk+1)(5.19) +αβ Γ(α)Ztk+1 0 τβ−1ϕ1(τ, y) (tk+1−τ)α−1dτ +αβ Γ(α)Ztk+1 0 σy(τ)τβ−1(tk+1−τ)α−1dB (τ). Using the ϕ1(τ, y)approximations, we have yk+1=(1−α)ϕ1tk+1,yk+1+(1−α)σy(ck+1) (B(tk+1)−B(tk)) (5.20) +αβ Γ(α) k X n=0" 1−1 2ξ!ϕ1(tn,yn)+1 2ξϕ1tn+1,˜yn+1# ×Ztn+1 tn τβ−1(tk+1−τ)α−1dτ Electronic Research Archive Volume 32, Issue 2, 733–761.
750 +αβ hΓ(α) k X n=0 σy(cn) (B(tn+1)−B(tn)) ×Ztn+1 tn τβ−1(tk+1−τ)α−1dτ. Using the calculations for these integrals and arranging the above, we have yk+1=(1−α)ϕ1tk+1,yk+1+(1−α)σy(ck+1) B(tk+1)−B(tk) h!(5.21) +αβ Γ(α) k X n=0" 1−1 2ξ!ϕ1(tn,yn)+1 2ξϕ1tn+1,˜yn+1# ×tα+β−1 k+1 B tn+1 tk+1 , β, α!−B tn tk+1 , β, α!! +αβ hΓ(α) k X n=0 σy(cn) (B(tn+1)−B(tn)) ×tα+β−1 k+1 B tn+1 tk+1 , β, α!−B tn tk+1 , β, α!!, where the predictor formula is stated as: ˜yk+1=y0+(1−α)βtβ−1 k+1ϕ1tk+1,yk+1+αβ Γ(α) k X n=0 ϕ1(tn,yn)(5.22) ×tα+β−1 k+1 B tn+1 tk+1 , β, α!−B tn tk+1 , β, α!!. 6. Parametrized method for a general Cauchy problem with piecewise derivative In this section, we derive the parametrized method [17] for some versions of nonlinear differential equations with piecewise differentiation. We shall start with the version of nonlinear differential equations with piecewise derivative [16], in which classical processes can be used in the first time interval, processes with power-law after fading memory in the second time interval, and stochastic processes can be used in the third time interval. The associated model is represented by the following: dy dt =ϕ(t,y), if 0 ≤t≤t1 y(0)=y0, ABC t1Dα ty=ϕ(t,y),if t1≤t≤t2 y(t1)=y1, dy (t)=ϕ(t,y)dt +σydB (t), if t2≤t≤T y(t2)=y2 .(6.1) The function ϕ(t,y) can be approximated by employing the parametrized formulation [17], thus integrating within [tn,tn+1], we have the following corrector formula with predictor term: Electronic Research Archive Volume 32, Issue 2, 733–761.
751 yk+1= y0+hk1 j1=0h1−1 2ξϕ1tj1,yj1+1 2ξϕ1tj1+1,˜yj1+1i, if 0 ≤t≤t1 ,(6.2) y1+(1−α)ϕ1tk2+1,˜yk2+1 +(1−α)σyck2+1Btk2+1−Btk2 +hα Γ(α) k2 j2=k1+1h1−1 2ξϕ1tj2,yj2+1 2ξϕ1tj2+1,˜yj2+1i ×(k2−j2+1)α−(k2−j2)α +hα−1 Γ(α) k2 j2=k1+1σycj2Btj2+1−Btj2 ×(k2−j2+1)α−(k2−j2)α, if t1≤t≤t2, y2+hk j3=k2+1h1−1 2ξϕ1tj3,yj3+1 2ξϕ1tj3+1,˜yj3+1i +σy(ck) (B(tk+1)−B(tk)) , if t2≤t≤T. The predictor components for each interval are calculated as n˜yk1+1=y0+hk1 j1=0ϕ1tj1,yj1,if 0 ≤t≤t1, ˜yk2+1=y1+(1−α)ϕ1tk2,yk2+hα Γ(α) k2 j2=k1+1ϕ1tj2,yj2 ×(k2−j2+1)α−(k2−j2)α,if t1≤t≤t2, n˜yk3+1=y2+hk j3=k2+1ϕ1tj3,yj3,if t2≤t≤T. (6.3) Now, we proceed with an another version of nonlinear differential equations with piecewise derivatives [16]. In the first time interval, fading memory processes can be utilized, while stochastic processes can be used in the second time interval. For the third time interval, processes that deal with power-law behaviors having fractal properties can be employed. The model that explains the process presented here is shown as follows: CF 0Dα ty=ϕ(t,y), if 0 ≤t≤t1 y(0)=y0, dy (t)=ϕ(t,y)dt +σydB (t),if t1≤t≤t2 y(t1)=y1, FFP t2Dα ty=ϕ(t,y), if t2≤t≤T y(t2)=y2. (6.4) Using the aforementioned concept of numerical scheme, the numerical scheme for the Cauchy problem in the framework of piecewise derivative [16] is achieved as yk+1= y0+(1−α)ϕ1tk1+1,yk1+1 +αhk1 j1=0h1−1 2ξϕ1tj1,yj1+1 2ξϕ1tj1+1,˜yj1+1i, if 0 ≤t≤t1 (6.5) y1+hk2 j2=k1+1h1−1 2ξϕ1tj2,yj2+1 2ξϕ1tj2+1,˜yj2+1i +σyck2Btk2+1−Btk2, if t1≤t≤t2 Electronic Research Archive Volume 32, Issue 2, 733–761.
752 β Γ(α) k j3=k2+1h1−1 2ξϕ1tj3,yj3+1 2ξϕ1tj3+1,˜yj3+1i ×tα+β−1 k+1Btj3+1 tk+1, β, α−Btj3 tk+1, β, α +β hΓ(α) k j3=k2+1σycj3Btj3+1−Btj3 ×tα+β−1 k+1Btj3+1 tk+1, β, α−Btj3 tk+1, β, α, if t2≤t≤T. The predictor components for each interval are determined as n˜yk1+1=y0+hk1 j1=0ϕ1tj1,yj1,if 0 ≤t≤t1, n˜yk2+1=y1+hk2 j2=k1+1ϕ1tj2,yj2,if t1≤t≤t2, ˜yk1+1=(1−α)βtβ−1 k1ϕ1tk1,yk1+αβ Γ(α) k j3=k2+1ϕ1tj3,yj3 ×tα+β−1 k1+1Btj3+1 tk1+1, β, α−Btj3 tk1+1, β, α,if t2≤t≤T. (6.6) 7. Illustrative examples In this section, we will investigate the applicability of the parametrized method to differential equations with piecewise derivatives with the help of some illustrative examples. This will be performed with the combination of deterministic and stochastic processes where the concepts of classical, stochastic, fractional, and fractal-fractional are added. We will start with a simple piecewise Cauchy problem in which the first part is with classical deterministic, the second part is with Atangana-Baleanu derivative and last part is with the classical stochastic. Another simple scenario will be presented with classical deterministic, Caputo fractional derivative and the classical stochastic. Finally, we will consider an anxiety model [28] employing the different versions of the piecewise derivative. Example 1. We consider a general Cauchy problem with piecewise derivative dy dt =−t, if 0 ≤t≤t1 y(0)=0, ABC t1Dα ty=−t,if t1≤t≤t2 y(t1)=y1, dy (t)=−tdt +σydB (t), if t2≤t≤T y(t2)=y2. (7.1) The numerical solution of above problem is represented by yk+1= y0+hk1 j1=0h−1−1 2ξtj1−1 2ξtj1+1i, if 0 ≤t≤t1 ,(7.2) y1−(1−α)tk2+1+(1−α)σyck2+1Btk2+1−Btk2 +hα Γ(α) k2 j2=k1+1h−1−1 2ξtj2−1 2ξtj2+1i ×(k2−j2+1)α−(k2−j2)α +hα−1 Γ(α) k2 j2=k1+1σycj2Btj2+1−Btj2 ×(k2−j2+1)α−(k2−j2)α, if t1≤t≤t2, Electronic Research Archive Volume 32, Issue 2, 733–761.
753 y2+hk j3=k2+1h−1−1 2ξtj3−1 2ξtj3+1i +σy(ck) (B(tk+1)−B(tk)) , if t2≤t≤T. The predictor terms are as follows: n˜yk1+1=y0+hk1 j1=0−tj1,if 0 ≤t≤t1, ˜yk2+1=y1−(1−α)tk2+1−hα Γ(α) k2 j2=k1+1tj2 ×(k2−j2+1)α−(k2−j2)α,if t1≤t≤t2, n˜yk3+1=y2−hk j3=k2+1tj3,if t2≤t≤t. (7.3) Noting that the stochastic constant σis taken as 0.1, the following initial conditions are as follows: y(0)=0,(7.4) y(t1)=−45 if α=0.9 y(t1)=−47.9 if α=0.8 y(t1)=−48.49 if α=0.6 y(t1)=−48.8 if α=0.4 y(t1)=−48.2 if α=0.2 , y(t2)=−145 if α=0.9 y(t2)=−123.6 if α=0.8 y(t2)=−85.2 if α=0.6 y(t2)=−76 if α=0.4 y(t2)=−60.2 if α=0.2 . In Figure 1, the numerical simulation for the considered problem with piecewise derivative is performed by considering different values of fractional orders. Figure 1. The graphical visualization for the piecewise Cauchy problem. Electronic Research Archive Volume 32, Issue 2, 733–761.
754 Example 2. We consider a general Cauchy problem with piecewise derivative dy dt =sin t, if 0 ≤t≤t1 y(0)=1, ABC t1Dα ty=sin t,if t1≤t≤t2 y(t1)=y1, dy (t)=sin tdt +σydB (t), if t2≤t≤T y(t2)=y2. (7.5) The numerical solution of above problem is represented by yk+1= y0+hk1 j1=0h1−1 2ξsin tj1+1 2ξsin tj1+1i, if 0 ≤t≤t1 ,(7.6) y1+(1−α)sin tk2+1+(1−α)σyck2+1Btk2+1−Btk2 +hα Γ(α) k2 j2=k1+1h1−1 2ξsin tj2+1 2ξsin tj2+1i ×(k2−j2+1)α−(k2−j2)α +hα−1 Γ(α) k2 j2=k1+1σycj2Btj2+1−Btj2 ×(k2−j2+1)α−(k2−j2)α, if t1≤t≤t2, y2+hk j3=k2+1h1−1 2ξsin tj3+1 2ξsin tj3+1i +σy(ck) (B(tk+1)−B(tk)) , if t2≤t≤T. The predictor components for each interval are calculated as n˜yk1+1=y0+hk1 j1=0sin tj1,if 0 ≤t≤t1, ˜yk2+1=y1+(1−α)sin tk2+1+hα Γ(α) k2 j2=k1+1sin tj2 ×(k2−j2+1)α−(k2−j2)α,if t1≤t≤t2, n˜yk3+1=y2+hk j3=k2+1sin tj3,if t2≤t≤T. (7.7) Noting that the stochastic constant σis taken as 0.1, the initial conditions are as follows: y(0)=1,(7.8) y(t1)=1.6 if α=0.9 y(t1)=2 if α=0.8 y(t1)=2.5 if α=0.6 y(t1)=3.2 if α=0.4 y(t1)=3.4 if α=0.2 , y(t2)=2 if α=0.9 y(t2)=2.48 if α=0.8 y(t2)=3.36 if α=0.6 y(t2)=4.7 if α=0.4 y(t2)=4.27 if α=0.2 . Electronic Research Archive Volume 32, Issue 2, 733–761.
755 In Figure 2, the numerical simulation for the considered problem with piecewise derivative is performed by considering different values of fractional orders. Figure 2. The graphical visualization for the piecewise Cauchy problem. Example 3. (Mathematical modeling of anxiety of mathematics) Instructional and social psychological environment are some of the attitude attribute of students and possible factors affecting the students’ disliking or liking of mathematics and mathematics anxiety is closely related to a broad spectrum of cognitive, psychological, and behavioral problems [28,29]. We next consider a mathematical model associated with the anxiety of mathematics [28]. The mathematical model under investigation is presented by the following: dS dt =(1−ε)χ+ωR+ρ(1−η)P−θ N(A+ϕQ)+κS(7.9) dP dt =εχ −(κ+(1−η)ρ)P dE dt =θ N(A+ϕQ)S−(κ+υ)E dA dt =(1−%)υE−(κ+δ+ς)A dQ dt =δA−κQ dR dt =ςA+%υE−(κ+ω)R and the initial conditions are taken as S(0)≥0,P(0)≥0,E(0)≥0,A(0)≥0,Q(0)≥0,R(0)≥0.(7.10) Here, S : anxiety towards mathematics susceptible students; P : anxiety towards mathematics protected students; E : anxiety towards mathematics exposed students; A : students who have anxiety towards Electronic Research Archive Volume 32, Issue 2, 733–761.
756 mathematics; Q : students who have permanent anxiety towards mathematics; R : students recovered from anxiety towards mathematics. Replacing the classical derivative by the piecewise differential operators and simplifying the model with piecewise derivative, we get the following modified model of anxiety: dU dt =ψ(t,U), if 0 ≤t≤t1 U(0)=U0, C t1Dα tU=ψ(t,U),if t1≤t≤t2 U(ti)=U1, dU (t)=ψ(t,U)dt +σiUdBi(t), if t2≤t≤T U(t2)=U2, (7.11) where U= S P E A Q R , ψ (t,U)= (1−ε)χ+ωR+ρ(1−η)P−θ N(A+ϕQ)+κS εχ −(κ+(1−η)ρ)P θ N(A+ϕQ)S−(κ+υ)E (1−%)υE−(κ+δ+ς)A δA−κQ ςA+%υE−(κ+ω)R .(7.12) Using the suggested method for each interval, the numerical solution can be obtained as Uk+1= U0+hk1 j1=0h1−1 2ξψtj1,Uj1+1 2ξψtj1+1,e Uj1+1i, if 0 ≤t≤t1 ,(7.13) U1+hα Γ(α+1) k2 j2=k1+1h1−1 2ξψtj2,Uj2+1 2ξψtj2+1,e Uj2+1i ×(k2−j2+1)α−(k2−j2)α +hα−1 Γ(α+1) k2 j2=k1+1σiUcj2Bitj2+1−Bitj2 ×(k2−j2+1)α−(k2−j2)α, if t1≤t≤t2, U2+hk j3=k2+1h1−1 2ξψtj3,Uj3+1 2ξψtj3+1,e Uj3+1i +σy(ck) (B(tk+1)−B(tk)) , if t2≤t≤T. The predictor components for each interval are calculated as ne Uk1+1=U0+hk1 j1=0ψtj1,Uj1,if 0 ≤t≤t1, e Uk2+1=U1+hα Γ(α+1) k2 j2=k1+1ψtj2,Uj2"(k2−j2+1)α −(k2−j2)α#, if t1≤t≤t2, ne Uk3+1=U2+hk j3=k2+1ψtj3,Uj3,if t2≤t≤T. (7.14) In Figure 3, we simulate the numerical solution of the anxiety model with piecewise derivative for α=0.9. Electronic Research Archive Volume 32, Issue 2, 733–761.
757 Figure 3. The graphical visualization for the anxiety model with piecewise setting. We present another case for our model since we know that the model can be modified with different derivatives in each intervals. For another case of our model, it can be written as follows: CF 0Dα tU=ψ(t,U), if 0 ≤t≤t1 U(0)=U0, dU (t)=ψ(t,U)dt +σUdB (t),if t1≤t≤t2 U(t1)=U1, FFP t2Dα,β tU=ψ(t,U), if t2≤t≤T U(t2)=U2. (7.15) For such a model, we obtain Uk+1= U0+(1−α)ψtk1+1,Uk1+1 +αhk1 j1=0h1−1 2ξψtj1,Uj1+1 2ξψtj1+1,e Uj1+1i, if 0 ≤t≤t1 (7.16) Electronic Research Archive Volume 32, Issue 2, 733–761.