Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders

Elec onic Resea ch A chi e h p://www.aimsp ess.com/jou nal/e a ERA, 32(2): 733–761. DOI: 10.3934/e a.2024035 Recei ed: 26 Sep embe 2023 Re ised: 03 Decembe 2023 Accep ed: 15 Decembe 2023 Published: 10 Janua y 2024 Resea ch a icle Exis ence, uniqueness and nume ical solu ion o s ochas ic ac ional di e en ial equa ions wi h in ege and non-in ege o de s Seda IGRET ARAZ1,2,*, Mehme Aki CETIN3and Abdon ATANGANA2,4,5 1Sii Uni e si y, Depa men o Ma hema ics Educa ion, Sii , Tu key 2Ins i u e o G oundwa e S udies, Facul y o Na u al and Ag icul u al Sciences, Uni e si y o he F ee S a e, Sou h A ica 3ALTSO Voca ional School, Alanya Alaaddin Keykuba Uni e si y, An alya, Tu key 4Depa men o Medical Resea ch, China Medical Uni e si y Hospi al, China Medical Uni e si y, Taichung, Taiwan 5IT4Inno a ions, VSB–Technical Uni e si y o Os a a, Os a a-Po uba 70800, Czech Republic *Co espondence: Email: [email p o ec ed]. Abs ac : The pa ame ized app oach is ex ended in his s udy o ind solu ions o di e en ial equa ions wi h ac al, ac ional, ac al- ac ional, and piecewise de i a i es wi h he inclusion o a s ochas ic componen . The exis ence and uniqueness o he solu ion o he s ochas ic A angana-Baleanu ac- ional di e en ial equa ion a e es ablished using Ca a heodo y’s exis ence heo em. Fo he solu ion o di e en ial equa ions using piecewise di e en ial ope a o s, which ake in o accoun combining de e minis ic and s ochas ic p ocesses u ilizing ce ain signi ican ma hema ical ools such as ac al and ac al- ac ional de i a i es, he applicabili y o he pa ame ized echnique is being examined. We discuss he c osso e beha io s o he model ob ained by including hese ope a o s and we p esen some illus a i e examples o some p oblems wi h piecewise di e en ial ope a o s. Keywo ds: Ca a heodo y’s condi ions; ac al- ac ional di e en ia ion; piecewise calculus; pa ame ized me hod 1. In oduc ion F ac ional analysis is a heo y ha s a ed wi h Leibniz asking i he e is a de i a i e o o de 1 / 2 o a unc ion. This heo y in e es ed many esea che s when di e en ypes o ac ional de i a i es we e in oduced. One well-known de ini ion is he Riemann-Liou ille ac ional de i a i e whe e he powe -law ke nel is inco po a ed. Capu o [1] in oduced a de i a i e wi h a modi ica ion on he 734 Riemann-Liou ille ac ional de i a i e [2] because i was use ul in heo y bu no app op ia e o sol ing eal-li e p oblems. These ope a o s, which a e used o model powe law p ocesses, ha e beha io ha is bo h nonlocal and singula . E en hough some p ocesses a e unique, ano he o m o ma h is needed o desc ibe p ocesses ha beha e di e en ly. Capu o and Fab izio [3] ha e c ea ed a ma hema ical concep called a de i a i e wi h ading memo y, which uses an exponen ial pa e n. This de i a i e deals wi h p ocesses ha beha e p edic ably and wi hin a small a ea. Howe e , we needed a de i a i e ha is p edic able bu ac s o e a la ge a ea. The A angana-Baleanu ac ional de i a i e [4] is a ma hema ical ool ha mee s his equi emen , and i u ilizes he Mi ag-Le le unc ion. The ac al de i a i e o Hausdo de i a i e [5] is a di e en kind o de i a i e used o measu ing ac als in ac al geome y. F ac al de i a i es we e made o s udy how hings sp ead in a s ange way when no mal ways o s udying do no conside he ac al shape o he hing ha hings a e sp eading h ough. A ac al measu e changes i s size in ela ion o aised o he powe o β . This ype o de i a i e is only used in a speci ic a ea, unlike he ac ional de i a i e, which is used in a simila way. La e , A angana in oduced ac al- ac ional de i a i es [6] by combining he concep s o ac al and ac ional de i a i es. Al hough he e is no doub ha ac ional di e en ial ope a o s a e use ul in modeling ele an p ocesses [7–11], hese ope a o s canno be used o model c osso e p ocesses such as om s ochas ic o powe -law o om ading memo y o s ochas ic [12–15]. Concluding ha a new class o di e en ial ope a o s was needed o his, A angana and A az in oduced piecewise di e en ial ope a o s [16], which can be c ea ed by including a ious di e en ial ope a o s o model such p ocesses. These ope a o s, which can be used o desc ibe many p ocesses, om modeling he di e en a es (o e en s opping) o an indi idual’s hea bea o e a pe iod o ime, o modeling he sp ead o a i us, i s cumula i ely and hen daily, ha e become ocus o a en ion o esea che s. In o de o be e unde s and and analyze he p ocesses discussed, i is necessa y o sol e he equa ions ha ep esen hese p ocesses. Because i is di icul o sol e hese equa ions using analy ic me hods when he ope a o s men ioned abo e and he nonlinea i y o he associa ed equa ions a e in ol ed, we ha e o use nume ical me hods o ob ain solu ions o such equa ions. The pa ame ized me hod, which deals wi h he app oxima ion o a unc ion wi h cons an s depending on a pa ame e , is one o he well-known nume ical me hods. While he pa ame e ized me hod is p esen ed in he li e a u e [8–10] o classical di e en ial equa ions, A angana and A az [17] ex ended his me hod o sol e ac ional and ac al- ac ional di e en ial equa ions. The pa ame ized me hod was compa ed wi h exis ing me hods in he li e a u e in [17] and i was shown ha he me hod is mo e e ec i e han o he me hods, especially when he pa ame e is close o 1. Howe e , in [17], he applica ion o he ele an me hod o s ochas ic di e en ial equa ions wi h ac ional, ac al- ac ional and piecewise de i a i es [16] is no aken in o accoun . The e o e, in his s udy, we p esen he de i a ion o his me hod o s ochas ic di e en ial equa ions wi h ac ional, ac al- ac ional and piecewise de i a i es. We employ he pa ame ized me hod o sol e di e en ypes o equa ions ob ained by inco po a ing hese ma hema ical ools in o di e en ial equa ions. Be o e p esen ing he associa ed me hod, i s he de ini ions o he abo e-men ioned ac ional, ac al ac ional and piecewise de i a i es will be p esen ed. In he ollowing sec ion, wi h he help o Ca a h ´ eodo y condi ions [15,16], he exis ence and uniqueness o he solu ion o A angana-Baleanu s ochas ic di e en ial equa ions [18] will be in es iga ed. In he emaining sec ions, in addi ion o he de i a ion o he pa ame ized me hod wi h hese de i a i es, some illus a i e examples will be included. Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 735 2. P elimina ies In his sec ion, he de ini ions o ac ional de i a i es wi h powe law beha io , ading memo y and exhibi ing powe law beha io a e ading memo y, ac al- ac ional de i a i es and piecewise de i a i es, which can be ep esen ed in di e en ways by including ac ional and ac al- ac ional de i a i es, will be discussed. The Capu o-Fab izio ac ional de i a i e [3] o he unc ion ( )∈H1(0,T)is de ined by CF 0Dα ( )=1 1−αZ 0 0(τ)exp −α 1−α( −τ)dτ, (2.1) whe e 0 <α<1 and H1(0,T)desc ibes he Hilbe space. The associa ed in eg al is gi en as CF 0Jα ( )=(1−α) ( )+αZ 0 (τ)dτ. (2.2) The Capu o ac ional de i a i e [1] o he unc ion ( )∈H1(0,T)is de ined by C 0Dα ( )=1 Γ(1−α)Z 0 0(τ) ( −τ)−αdτ, (2.3) whe e 0 < α ≤ 1 and he Riemann-Liou ille ac ional de i a i e o he unc ion ( )∈C(0,T) is de ined by RL 0Dα ( )=1 Γ(1−α) d d Z 0 (τ) ( −τ)−αdτ. (2.4) The in eg al wi h powe -law ke nel [2] is gi en by RL 0Jα ( )=1 Γ(α)Z 0 (τ) ( −τ)α−1dτ. (2.5) The ollowing o mulas desc ibe he A angana-Baleanu ac ional de i a i e [4], which has he c osso e beha io om s e ched exponen ial o powe -law, ABC 0Dα ( )=1 1−αZ 0 0(τ)Eα−α 1−α( −τ)αdτ, (2.6) and ABR 0Dα ( )=1 1−α d d Z 0 (τ)Eα−α 1−α( −τ)αdτ. (2.7) The abo e ope a o s a e called A angana-Baleanu ac ional de i a i e in he Capu o sense and A angana- Baleanu ac ional de i a i e in he Riemann-Liou ille sense [4], espec i ely. The associa ed in eg al is gi en by AB 0Jα ( )=(1−α) ( )+α Γ(α)Z 0 (τ) ( −τ)α−1dτ. (2.8) The concep o ac al- ac ional di e en ia ion and in eg a ion has appea ed p e iously wi h he idea o combining he ac al and ac ional de i a i es. The ac al- ac ional de i a i e [6] wi h powe -law ke nel is de ined by Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 736 FFP 0Dα,β ( )=1 Γ(1−α) d d βZ 0 (τ) ( −τ)−αdτ, (2.9) whe e he de ini ion o ac al de i a i e [5] is d d β ( )=lim → 1 ( )− ( 1) β− β 1 .(2.10) The associa ed ac al- ac ional in eg al [6] wi h powe -law ke nel is gi en by FFP 0Jα ( )=1 Γ(α)Z 0 βτβ−1 (τ) ( −τ)α−1dτ. (2.11) The ac al- ac ional de i a i e wi h Mi ag-Le le ke nel [6] is de ined by FFM 0Dα,β ( )=1 1−α d d βZ 0 (τ)Eα−α 1−α( −τ)αdτ(2.12) and he associa ed ac al- ac ional in eg al is gi en by FFM 0Jα ( )=(1−α)β β−1 ( )+α Γ(α)Z 0 βτβ−1 (τ) ( −τ)α−1dτ. (2.13) The ac al- ac ional de i a i e wi h exponen ial decay ke nel [6] is de ined by FFE 0Dα,β ( )=1 1−α d d βZ 0 (τ)exp −α 1−α( −τ)dτ(2.14) and he associa ed ac al- ac ional in eg al is gi en by FFE 0Jα ( )=(1−α)β β−1 ( )+αZ 0 βτβ−1 (τ)dτ. (2.15) We now p esen he de ini ions o he piecewise de i a i e and in eg al ope a o s, which made signi ican con ibu ion o li e a u e [16]. The piecewise de i a i e wi h classical and ac ional de i a i e wi h powe -law ke nel such ha i can be aken as [16] PRL 0Dα y( )=(y0( )i 0 ≤ ≤ 0 RL 0Dα y( )i 0≤ ≤T(2.16) whe e PRL 0Dα ep esen s he classical de i a i e wi hin 0 ≤ ≤ 0 and he Riemann-Liou ille ac ional de i a i e wi hin 0≤ ≤T. The piecewise wi h Capu o de i a i e is gi en as [16] PC 0Dα y( )=(y0( )i 0 ≤ ≤ 0 C 0Dα y( )i 0≤ ≤T(2.17) whe e he unc ion y( ) is con inuous bu no necessa ily di e en iable in [ 0,T]. He e, PRL 0Dα ep esen s he classical de i a i e on 0 ≤ ≤ 0 and he Capu o ac ional de i a i e [1] on 0≤ ≤T . The associa ed piecewise in eg al o yis gi en as [16] Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 737 PPLI y( )=      R 0 0y(τ)dτi 0 ≤ ≤ 0 1 Γ(α)R 0y(τ) ( −τ)α−1dτi 0≤ ≤T(2.18) whe e PPL 0Iα ep esen s he classical in eg al on 0 ≤ ≤ 0 and he in eg al wi h powe -law ke nel on 0≤ ≤T. The piecewise de i a i e wi h classical de i a i e and exponen ial decay ke nel is gi en as [16] PCF 0Dα y( )=(y0( )i 0 ≤ ≤ 0 CF 0Dα y( )i 0≤ ≤T(2.19) whe e PCF 0Dα is he classical de i a i e on 0 ≤ ≤ 0 and he Capu o-Fab izio ac ional de i a i e [3] on 0≤ ≤T. He e, i is assumed ha he unc ion y( ) is di e en iable. A piecewise in eg al is gi en as [16] PCF I y( )=      R 0 0y(τ)dτi 0 ≤ ≤ 0 1−α M(α)y( )+α M(α)R 0y(τ)dτi 0≤ ≤T.(2.20) The piecewise de i a i e wi h classical de i a i e and Mi ag-Le le ke nel is de ined by [16] PAB 0Dα y( )=(y0( )i 0 ≤ ≤ 0 ABC 0Dα y( )i 0≤ ≤T(2.21) whe e PAB 0Dα ep esen s he classical de i a i e on 0 ≤ ≤ 0 and he A angana-Baleanu ac ional de i a i e [4] on 0≤ ≤T.The associa ed piecewise in eg al is gi en as [16] PABI y( )=      R 0 0y(τ)dτi 0 ≤ ≤ 0 (1−α)y( )+α Γ(α)R 0y(τ) ( −τ)α−1dτi 0≤ ≤T.(2.22) Lemma 1. (The gene aliza ion o he G onwall inequali y) Assume ha b≥ 0 , α > 0 , and x( ) is a nonnega i e unc ion locally in eg able on 0 ≤ <T, and assume ha y( ) is nonnega i e and locally in eg able on 0≤ <T wi h y( )≤x( )+bZ 0 y(τ) ( −τ)α−1dτ. (2.23) Then, y( )≤x( )+Z 0 ∞ X n=1 (bΓ(α))n Γ(nα)y(τ) ( −τ)nα−1x(τ)dτ. (2.24) De ini ion 1. (S i ling o mula) The S i ling o mula o he Gamma unc ion is o mula ed by Γ(x)∼√2πe−xxx−1 2.(2.25) Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 738 3. Ca a heodo y’s heo y o exis ence and uniqueness o a gene al Cauchy p oblem wi h s ochas ic A angana-Baleanu ac ional de i a i e In his sec ion, we p o e he exis ence and uniqueness o he solu ion o he A angana-Baleanu s ochas ic di e en ial equa ion [18] by employing Ca a h ´ eodo y’s exis ence heo y [19,20], which is a mo e gene al e sion o Peano’s exis ence heo em. I is wo h no ing ha he exis ence and uniqeness o he solu ion o s ochas ic di e en ial equa ions wi h he Capu o ac ional de i a i e is p esen ed in [21]. He e, we will examine he exis ence and uniqueness o he s ochas ic di e en ial equa ion wi h A angana-Baleanu ac ional de i a i e. The di e en ial equa ion unde in es iga ion is ep esen ed in he o m: AB 0Dα y( )= 1( ,y)d +σy( )dB ( ), ≥0 (3.1) y( 0)=y0 unde he condi ions E1) Fo all y,¯y∈H, he e is a cons an k>0 such ha | 1( ,y)− 1( ,¯y)|2,| 2( ,y)− 2( ,¯y)|2≤k|y−¯y|2, ≥0.(3.2) E2) Fo all y∈H, he e is a cons an ¯ k>0 such ha | 1( ,y)|2,| 2( ,y)|2≤¯ k1+|y|2, ≥0 (3.3) whe e H is a Banach space. No e ha condi ions E1 and E2 a e known as he Lipschi z condi ion and he g ow h condi ion, espec i ely. Theo em 1. Fo each y0∈L2(Ω,H) , Eq (26) has a unique mild solu ion y∈C[0,T],L2(Ω,H) = S such ha sup 0≤ ≤T E|y|2<∞. P oo . Fo he p oo , we will use he con ac ion mapping p inciple. Be o e p oceeding wi h he p oo , we de ine he no m kηk2 γ=sup 0≤ ≤T E|η( )|2(3.4) whe e Edeno es he expec a ion. Fo any ∈[0,T] and y∈S , we de ine he mapping subjec o Ω = C[0,T],L2(Ω,H)→ C[0,T],L2(Ω,H) (Λy) ( )=y0+(1−α) 1( ,y)+(1−α)σy( )B0( )(3.5) +α Γ(α)Z 0 1(s,y) ( −s)α−1ds +ασ Γ(α)Z 0 y(s) ( −s)α−1dB (s). Thus, we w i e E|(Λy) ( )−(Λ¯y) ( )|2=E (1−α) ( 1( ,y)− 1( ,¯y)) +(1−α)σ(y( )−¯y( )) B0( ) +α Γ(α)R 0( 1(s,y)− 1(s,¯y)) ( −s)α−1ds +ασ Γ(α)R 0(y(s)−¯y(s)) ( −s)α−1dB (s)  2 .(3.6) Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 739 Taking 2 α− 1 > 0, by he Cauchy-Schwa z inequali y, I o’s isome y o mula and he Lipschi z condi ion [22], we ha e E|(Λy) ( )−(Λ¯y) ( )|2≤4(1−α)2kσ1+|B0|2E|y−¯y|2(3.7) +(T+1)4α2k Γ2(α)Z 0 E|y−¯y|2( −s)2α−2ds ≤4(1−α)2σk1+|B0|2ky−¯ykγ +(T+1)4α2k Γ2(α) 2α−1 (2α−1)ky−¯ykγ ≤4σ(1−α)2k 1+sup ∈[0,T]|B0|2!ky−¯ykγ +(T+1)4α2k Γ2(α) 2α−1 (2α−1)ky−¯ykγ ≤4σ(1−α)2k1+kB0k∞ky−¯ykγ +(T+1)4σα2k Γ2(α) 2α−1 (2α−1)ky−¯ykγ ≤˜ kky−¯ykγ, whe e ˜ k=4σ(1−α)2k1+kB0k∞+(T+1)4σα2k Γ2(α) 2α−1 (2α−1).(3.8) Using he gene alized G onwall inequali y [23], we w i e EΛ2y( )−Λ2¯y( )≤4σ(1−α)2k1+kB0k∞E|Λy−Λ¯y|2(3.9) +4σα2k(T+1) Γ2(α)Z 0 ( −s)2α−2E|Λy−Λ¯y|2ds ≤4σ(1−α)2k1+kB0k∞"4(1−α)2k(1+kB0k∞) +(T+1)4α2k Γ2(α) 2α−1 (2α−1)#(3.10) +4σα2k(T+1) Γ2(α)Z 0 ( −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 he induc ion o mula o n, we can hen w i e E|(Λny) ( )−(Λn¯y) ( )|≤  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−14σα2k(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) Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 740 ≤Lky−¯ykγ whe e 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−14σα2k(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 p o e he heo em holds, we will show ha L< 1 o su icien la ge n . Le us conside he ollowing se ies o posi i e e ms ∞ 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−14σα2k(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 he d’Alembe disc iminan me hod 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 equi alen o 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 he S i ling o mula [21], we ha e he ollowing o las e m lim n→∞ 4σα2k(T+1) Γ2(α)Γ(2α−1)T(2α−1)e(2α−1) √n+1 √nn n+1n(2α−1)1 ((n+1)(2α−1))(2α−1)=0 (3.17) and knowing ha α < 1,we can ha e 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−14σα2k(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 gua an ees ha L< 1 holds. This p o es ha Λ y( ) is a con ac ion mapping, which comple es he p oo . Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 741 4. Pa ame ized me hod o a gene al Cauchy p oblem wi h s ochas ic ac ional de i a i es In his sec ion, we de elop he pa ame ized app oach o nume ically sol ing di e en ial equa ions wi h ac ional de i a i es ha inco po a e s ochas ic componen s. Be o e p esen ing he ex ension o he me hod o he solu ions o di e en di e en ial equa ions, we shall ecall he o mula ion o he pa ame ized app oach [17,24–26]. The app oach is o mula ed by he ollowing: ϕ1( ,y)≈" 1−1 2ξ!ϕ1 k,yk+1 2ξϕ1 k+1,˜yk+1#.(4.1) 4.1. Pa ame ized me hod o a gene al Cauchy p oblem wi h s ochas ic componen To de i e he associa ed me hod, in his subsec ion, we conside a gene al Cauchy p oblem wi h s ochas ic componen gi en by dy ( )=ϕ1( ,y)d +σy( )dB ( ).(4.2) We con e he abo e in o an in eg al equa ion, by applying on bo h sides he classical in eg al y( )=y(0)+Z 0 ϕ1(τ, y)dτ+Z 0 σy(τ)dB (τ).(4.3) A = k+1,we w i e y( k+1)=y(0)+Z k+1 0 ϕ1(τ, y)dτ+Z k+1 0 σy(τ)dB (τ)(4.4) and a = k y( k)=y(0)+Z k 0 ϕ1(τ, y)dτ+Z k 0 σy(τ)dB (τ).(4.5) Subs ac ing hese wo equali ies gi es y( k+1)=y( k)+Z k+1 k ϕ1(τ, y)dτ+Z k+1 k σy(τ)dB (τ).(4.6) The unc ion ϕ1(τ, y) can be app oxima ed by using he pa ame ized app oach [17,24 – 26] p esen ed ea lie . A e simpli ica ion, we ha e he p edic o -co ec o o mula [27] yk+1=yk+h" 1−1 2ξ!ϕ1 k,yk+1 2ξϕ1 k+1,˜yk+1#(4.7) +σy(ck) (B( k+1)−B( k)) , whe e ck∈[ k, k+1]and he p edic o e m ˜yk+1=yk+hϕ1 k,yk.(4.8) Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 748 5.3. Pa ame ized me hod o a gene al Cauchy p oblem wi h s ochas ic ac al- ac ional de i a i e wi h powe -law ke nel In his sec ion, we ob ain he nume ical solu ion o a gene al Cauchy p oblem wi h s ochas ic ac al- ac ional de i a i e [18] wi h powe -law ke nel by using he pa ame ized me hod [17]. The associa ed p oblem unde conside a ion is ep esen ed by (FFP 0Dα y( )=ϕ1( ,y)+σy( )dB ( ),i >0, y(0)=y0,i =0.(5.11) Applying he ac al- ac ional de i a i e [6] wi h powe -law ke nel, we ha e y( )=β Γ(α)Z 0 τβ−1ϕ1(τ, y) ( −τ)α−1dτ+β Γ(α)Z 0 σy(τ)τβ−1( −τ)α−1dB (τ).(5.12) A = k+1,we ha e y( k+1)=β Γ(α) k X n=0Z n+1 n τβ−1ϕ1(τ, y) ( k+1−τ)α−1dτ(5.13) +β Γ(α) k X n=0Z n+1 n τβ−1σy(τ) ( k+1−τ)α−1B0(τ)dτ. Replacing he unc ion ϕ1(τ, y)by i s pa ame ized app oxima ion, we ha e yk+1=β Γ(α) k X n=0" 1−1 2ξ!ϕ1( n,yn)+1 2ξϕ1 n+1,˜yn+1#(5.14) ×Z n+1 n τβ−1( k+1−τ)α−1dτ +β hΓ(α) k X n=0 σy(cn) (B( n+1)−B( n)) ×Z n+1 n τβ−1( k+1−τ)α−1dτ. The in eg al on he igh hand side o he abo e equa ion is calcula ed by using he change o a iables τ= k+1uand dτ= k+1du as ollows: Z n+1 n τβ−1( k+1−τ)α−1dτ= α+β−1 k+1Z n+1 n uβ−1(1−u)α−1du = α+β−1 k+1 B n+1 k+1 , β, α!−B n k+1 , β, α!!, whe e he unc ion B(·,·,·) is he incomple e Be a unc ion. By calcula ion o hese in eg als, he ollowing nume ical scheme is ob ained: yk+1=β Γ(α) k X n=0" 1−1 2ξ!ϕ1( n,yn)+1 2ξϕ1 n+1,˜yn+1#(5.15) Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 749 × α+β−1 k+1 B n+1 k+1 , β, α!−B n k+1 , β, α!! +β hΓ(α) k X n=0 σy(cn) (B( n+1)−B( n)) × α+β−1 k+1 B n+1 k+1 , β, α!−B n k+1 , β, α!!. We know ha he e m ˜yn+1is p edic ed by he ollowing: ˜yn+1=y0+β Γ(α) k X n=0 ϕ1( n,yn) α+β−1 k+1 B n+1 k+1 , β, α!−B n k+1 , β, α!!.(5.16) 5.4. Pa ame ized me hod o a gene al Cauchy p oblem wi h s ochas ic ac al- ac ional de i a i e wi h Mi ag-Le le ke nel To examine he solu ion o a gene al Cauchy p oblem wi h s ochas ic ac al- ac ional de i a i e wi h Mi ag-Le le ke nel [18], we conside he ollowing p oblem: (FFM 0Dα y( )=ϕ1( ,y)+σy( )dB ( ),i >0, y(0)=y0,i =0(5.17) A e aking he associa ed in eg al, he abo e can be a anged as ollows: y( )=(1−α)ϕ1( ,y)+(1−α)σy( )dB ( )(5.18) +αβ Γ(α)Z 0 τβ−1ϕ1(τ, y) ( −τ)α−1dτ +αβ Γ(α)Z 0 σy(τ)τβ−1( −τ)α−1B0(τ)dτ. A = k+1,we ha e y( )=(1−α)ϕ1 k+1,yk+1+(1−α)σy( k+1)dB ( k+1)(5.19) +αβ Γ(α)Z k+1 0 τβ−1ϕ1(τ, y) ( k+1−τ)α−1dτ +αβ Γ(α)Z k+1 0 σy(τ)τβ−1( k+1−τ)α−1dB (τ). Using he ϕ1(τ, y)app oxima ions, we ha e yk+1=(1−α)ϕ1 k+1,yk+1+(1−α)σy(ck+1) (B( k+1)−B( k)) (5.20) +αβ Γ(α) k X n=0" 1−1 2ξ!ϕ1( n,yn)+1 2ξϕ1 n+1,˜yn+1# ×Z n+1 n τβ−1( k+1−τ)α−1dτ Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 750 +αβ hΓ(α) k X n=0 σy(cn) (B( n+1)−B( n)) ×Z n+1 n τβ−1( k+1−τ)α−1dτ. Using he calcula ions o hese in eg als and a anging he abo e, we ha e yk+1=(1−α)ϕ1 k+1,yk+1+(1−α)σy(ck+1) B( k+1)−B( k) h!(5.21) +αβ Γ(α) k X n=0" 1−1 2ξ!ϕ1( n,yn)+1 2ξϕ1 n+1,˜yn+1# × α+β−1 k+1 B n+1 k+1 , β, α!−B n k+1 , β, α!! +αβ hΓ(α) k X n=0 σy(cn) (B( n+1)−B( n)) × α+β−1 k+1 B n+1 k+1 , β, α!−B n k+1 , β, α!!, whe e he p edic o o mula is s a ed as: ˜yk+1=y0+(1−α)β β−1 k+1ϕ1 k+1,yk+1+αβ Γ(α) k X n=0 ϕ1( n,yn)(5.22) × α+β−1 k+1 B n+1 k+1 , β, α!−B n k+1 , β, α!!. 6. Pa ame ized me hod o a gene al Cauchy p oblem wi h piecewise de i a i e In his sec ion, we de i e he pa ame ized me hod [17] o some e sions o nonlinea di e en ial equa ions wi h piecewise di e en ia ion. We shall s a wi h he e sion o nonlinea di e en ial equa ions wi h piecewise de i a i e [16], in which classical p ocesses can be used in he i s ime in e al, p ocesses wi h powe -law a e ading memo y in he second ime in e al, and s ochas ic p ocesses can be used in he hi d ime in e al. The associa ed model is ep esen ed by he ollowing:                            dy d =ϕ( ,y), i 0 ≤ ≤ 1 y(0)=y0, ABC 1Dα y=ϕ( ,y),i 1≤ ≤ 2 y( 1)=y1, dy ( )=ϕ( ,y)d +σydB ( ), i 2≤ ≤T y( 2)=y2 .(6.1) The unc ion ϕ( ,y) can be app oxima ed by employing he pa ame ized o mula ion [17], hus in e- g a ing wi hin [ n, n+1], we ha e he ollowing co ec o o mula wi h p edic o e m: Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 751 yk+1=         y0+hk1 j1=0h1−1 2ξϕ1 j1,yj1+1 2ξϕ1 j1+1,˜yj1+1i, i 0 ≤ ≤ 1 ,(6.2)                                      y1+(1−α)ϕ1 k2+1,˜yk2+1 +(1−α)σyck2+1B k2+1−B k2 +hα Γ(α) k2 j2=k1+1h1−1 2ξϕ1 j2,yj2+1 2ξϕ1 j2+1,˜yj2+1i ×(k2−j2+1)α−(k2−j2)α +hα−1 Γ(α) k2 j2=k1+1σycj2B j2+1−B j2 ×(k2−j2+1)α−(k2−j2)α, i 1≤ ≤ 2,            y2+hk j3=k2+1h1−1 2ξϕ1 j3,yj3+1 2ξϕ1 j3+1,˜yj3+1i +σy(ck) (B( k+1)−B( k)) , i 2≤ ≤T. The p edic o componen s o each in e al a e calcula ed as                    n˜yk1+1=y0+hk1 j1=0ϕ1 j1,yj1,i 0 ≤ ≤ 1,        ˜yk2+1=y1+(1−α)ϕ1 k2,yk2+hα Γ(α) k2 j2=k1+1ϕ1 j2,yj2 ×(k2−j2+1)α−(k2−j2)α,i 1≤ ≤ 2, n˜yk3+1=y2+hk j3=k2+1ϕ1 j3,yj3,i 2≤ ≤T. (6.3) Now, we p oceed wi h an ano he e sion o nonlinea di e en ial equa ions wi h piecewise de i a i es [16]. In he i s ime in e al, ading memo y p ocesses can be u ilized, while s ochas ic p ocesses can be used in he second ime in e al. Fo he hi d ime in e al, p ocesses ha deal wi h powe -law beha io s ha ing ac al p ope ies can be employed. The model ha explains he p ocess p esen ed he e is shown as ollows:                            CF 0Dα y=ϕ( ,y), i 0 ≤ ≤ 1 y(0)=y0, dy ( )=ϕ( ,y)d +σydB ( ),i 1≤ ≤ 2 y( 1)=y1, FFP 2Dα y=ϕ( ,y), i 2≤ ≤T y( 2)=y2. (6.4) Using he a o emen ioned concep o nume ical scheme, he nume ical scheme o he Cauchy p oblem in he amewo k o piecewise de i a i e [16] is achie ed as yk+1=                      y0+(1−α)ϕ1 k1+1,yk1+1 +αhk1 j1=0h1−1 2ξϕ1 j1,yj1+1 2ξϕ1 j1+1,˜yj1+1i, i 0 ≤ ≤ 1 (6.5)            y1+hk2 j2=k1+1h1−1 2ξϕ1 j2,yj2+1 2ξϕ1 j2+1,˜yj2+1i +σyck2B k2+1−B k2, i 1≤ ≤ 2 Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 752                            β Γ(α) k j3=k2+1h1−1 2ξϕ1 j3,yj3+1 2ξϕ1 j3+1,˜yj3+1i × α+β−1 k+1B j3+1 k+1, β, α−B j3 k+1, β, α +β hΓ(α) k j3=k2+1σycj3B j3+1−B j3 × α+β−1 k+1B j3+1 k+1, β, α−B j3 k+1, β, α, i 2≤ ≤T. The p edic o componen s o each in e al a e de e mined as                        n˜yk1+1=y0+hk1 j1=0ϕ1 j1,yj1,i 0 ≤ ≤ 1, n˜yk2+1=y1+hk2 j2=k1+1ϕ1 j2,yj2,i 1≤ ≤ 2,          ˜yk1+1=(1−α)β β−1 k1ϕ1 k1,yk1+αβ Γ(α) k j3=k2+1ϕ1 j3,yj3 × α+β−1 k1+1B j3+1 k1+1, β, α−B j3 k1+1, β, α,i 2≤ ≤T. (6.6) 7. Illus a i e examples In his sec ion, we will in es iga e he applicabili y o he pa ame ized me hod o di e en ial equa ions wi h piecewise de i a i es wi h he help o some illus a i e examples. This will be pe o med wi h he combina ion o de e minis ic and s ochas ic p ocesses whe e he concep s o classical, s ochas ic, ac ional, and ac al- ac ional a e added. We will s a wi h a simple piecewise Cauchy p oblem in which he i s pa is wi h classical de e minis ic, he second pa is wi h A angana-Baleanu de i a i e and las pa is wi h he classical s ochas ic. Ano he simple scena io will be p esen ed wi h classical de e minis ic, Capu o ac ional de i a i e and he classical s ochas ic. Finally, we will conside an anxie y model [28] employing he di e en e sions o he piecewise de i a i e. Example 1. We conside a gene al Cauchy p oblem wi h piecewise de i a i e                            dy d =− , i 0 ≤ ≤ 1 y(0)=0, ABC 1Dα y=− ,i 1≤ ≤ 2 y( 1)=y1, dy ( )=− d +σydB ( ), i 2≤ ≤T y( 2)=y2. (7.1) The nume ical solu ion o abo e p oblem is ep esen ed by yk+1=         y0+hk1 j1=0h−1−1 2ξ j1−1 2ξ j1+1i, i 0 ≤ ≤ 1 ,(7.2)                                y1−(1−α) k2+1+(1−α)σyck2+1B k2+1−B k2 +hα Γ(α) k2 j2=k1+1h−1−1 2ξ j2−1 2ξ j2+1i ×(k2−j2+1)α−(k2−j2)α +hα−1 Γ(α) k2 j2=k1+1σycj2B j2+1−B j2 ×(k2−j2+1)α−(k2−j2)α, i 1≤ ≤ 2, Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 753            y2+hk j3=k2+1h−1−1 2ξ j3−1 2ξ j3+1i +σy(ck) (B( k+1)−B( k)) , i 2≤ ≤T. The p edic o e ms a e as ollows:                    n˜yk1+1=y0+hk1 j1=0− j1,i 0 ≤ ≤ 1,        ˜yk2+1=y1−(1−α) k2+1−hα Γ(α) k2 j2=k1+1 j2 ×(k2−j2+1)α−(k2−j2)α,i 1≤ ≤ 2, n˜yk3+1=y2−hk j3=k2+1 j3,i 2≤ ≤ . (7.3) No ing ha he s ochas ic cons an σis aken as 0.1, he ollowing ini ial condi ions a e as ollows: y(0)=0,(7.4)                      y( 1)=−45 i α=0.9 y( 1)=−47.9 i α=0.8 y( 1)=−48.49 i α=0.6 y( 1)=−48.8 i α=0.4 y( 1)=−48.2 i α=0.2 ,                      y( 2)=−145 i α=0.9 y( 2)=−123.6 i α=0.8 y( 2)=−85.2 i α=0.6 y( 2)=−76 i α=0.4 y( 2)=−60.2 i α=0.2 . In Figu e 1, he nume ical simula ion o he conside ed p oblem wi h piecewise de i a i e is pe o med by conside ing di e en alues o ac ional o de s. Figu e 1. The g aphical isualiza ion o he piecewise Cauchy p oblem. Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 754 Example 2. We conside a gene al Cauchy p oblem wi h piecewise de i a i e                            dy d =sin , i 0 ≤ ≤ 1 y(0)=1, ABC 1Dα y=sin ,i 1≤ ≤ 2 y( 1)=y1, dy ( )=sin d +σydB ( ), i 2≤ ≤T y( 2)=y2. (7.5) The nume ical solu ion o abo e p oblem is ep esen ed by yk+1=         y0+hk1 j1=0h1−1 2ξsin  j1+1 2ξsin  j1+1i, i 0 ≤ ≤ 1 ,(7.6)                                y1+(1−α)sin  k2+1+(1−α)σyck2+1B k2+1−B k2 +hα Γ(α) k2 j2=k1+1h1−1 2ξsin  j2+1 2ξsin  j2+1i ×(k2−j2+1)α−(k2−j2)α +hα−1 Γ(α) k2 j2=k1+1σycj2B j2+1−B j2 ×(k2−j2+1)α−(k2−j2)α, i 1≤ ≤ 2,            y2+hk j3=k2+1h1−1 2ξsin  j3+1 2ξsin  j3+1i +σy(ck) (B( k+1)−B( k)) , i 2≤ ≤T. The p edic o componen s o each in e al a e calcula ed as                    n˜yk1+1=y0+hk1 j1=0sin  j1,i 0 ≤ ≤ 1,        ˜yk2+1=y1+(1−α)sin  k2+1+hα Γ(α) k2 j2=k1+1sin  j2 ×(k2−j2+1)α−(k2−j2)α,i 1≤ ≤ 2, n˜yk3+1=y2+hk j3=k2+1sin  j3,i 2≤ ≤T. (7.7) No ing ha he s ochas ic cons an σis aken as 0.1, he ini ial condi ions a e as ollows: y(0)=1,(7.8)                      y( 1)=1.6 i α=0.9 y( 1)=2 i α=0.8 y( 1)=2.5 i α=0.6 y( 1)=3.2 i α=0.4 y( 1)=3.4 i α=0.2 ,                      y( 2)=2 i α=0.9 y( 2)=2.48 i α=0.8 y( 2)=3.36 i α=0.6 y( 2)=4.7 i α=0.4 y( 2)=4.27 i α=0.2 . Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 755 In Figu e 2, he nume ical simula ion o he conside ed p oblem wi h piecewise de i a i e is pe o med by conside ing di e en alues o ac ional o de s. Figu e 2. The g aphical isualiza ion o he piecewise Cauchy p oblem. Example 3. (Ma hema ical modeling o anxie y o ma hema ics) Ins uc ional and social psychologi- cal en i onmen a e some o he a i ude a ibu e o s uden s and possible ac o s a ec ing he s uden s’ disliking o liking o ma hema ics and ma hema ics anxie y is closely ela ed o a b oad spec um o cogni i e, psychological, and beha io al p oblems [28,29]. We nex conside a ma hema ical model as- socia ed wi h he anxie y o ma hema ics [28]. The ma hema ical model unde in es iga ion is p esen ed by he ollowing: dS d =(1−ε)χ+ωR+ρ(1−η)P−θ N(A+ϕQ)+κS(7.9) dP d =εχ −(κ+(1−η)ρ)P dE d =θ N(A+ϕQ)S−(κ+υ)E dA d =(1−%)υE−(κ+δ+ς)A dQ d =δA−κQ dR d =ςA+%υE−(κ+ω)R and he ini ial condi ions a e aken as S(0)≥0,P(0)≥0,E(0)≥0,A(0)≥0,Q(0)≥0,R(0)≥0.(7.10) He e, S : anxie y owa ds ma hema ics suscep ible s uden s; P : anxie y owa ds ma hema ics p o ec ed s uden s; E : anxie y owa ds ma hema ics exposed s uden s; A : s uden s who ha e anxie y owa ds Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 756 ma hema ics; Q : s uden s who ha e pe manen anxie y owa ds ma hema ics; R : s uden s eco e ed om anxie y owa ds ma hema ics. Replacing he classical de i a i e by he piecewise di e en ial ope a o s and simpli ying he model wi h piecewise de i a i e, we ge he ollowing modi ied model o anxie y:                            dU d =ψ( ,U), i 0 ≤ ≤ 1 U(0)=U0, C 1Dα U=ψ( ,U),i 1≤ ≤ 2 U( i)=U1, dU ( )=ψ( ,U)d +σiUdBi( ), i 2≤ ≤T U( 2)=U2, (7.11) whe e U=  S P E A Q R  , ψ ( ,U)=  (1−ε)χ+ωR+ρ(1−η)P−θ N(A+ϕQ)+κS 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 he sugges ed me hod o each in e al, he nume ical solu ion can be ob ained as Uk+1=         U0+hk1 j1=0h1−1 2ξψ j1,Uj1+1 2ξψ j1+1,e Uj1+1i, i 0 ≤ ≤ 1 ,(7.13)                          U1+hα Γ(α+1) k2 j2=k1+1h1−1 2ξψ j2,Uj2+1 2ξψ j2+1,e Uj2+1i ×(k2−j2+1)α−(k2−j2)α +hα−1 Γ(α+1) k2 j2=k1+1σiUcj2Bi j2+1−Bi j2 ×(k2−j2+1)α−(k2−j2)α, i 1≤ ≤ 2,            U2+hk j3=k2+1h1−1 2ξψ j3,Uj3+1 2ξψ j3+1,e Uj3+1i +σy(ck) (B( k+1)−B( k)) , i 2≤ ≤T. The p edic o componen s o each in e al a e calcula ed as                        ne Uk1+1=U0+hk1 j1=0ψ j1,Uj1,i 0 ≤ ≤ 1,           e Uk2+1=U1+hα Γ(α+1) k2 j2=k1+1ψ j2,Uj2"(k2−j2+1)α −(k2−j2)α#, i 1≤ ≤ 2, ne Uk3+1=U2+hk j3=k2+1ψ j3,Uj3,i 2≤ ≤T. (7.14) In Figu e 3, we simula e he nume ical solu ion o he anxie y model wi h piecewise de i a i e o α=0.9. Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761. 757 Figu e 3. The g aphical isualiza ion o he anxie y model wi h piecewise se ing. We p esen ano he case o ou model since we know ha he model can be modi ied wi h di e en de i a i es in each in e als. Fo ano he case o ou model, i can be w i en as ollows:                            CF 0Dα U=ψ( ,U), i 0 ≤ ≤ 1 U(0)=U0, dU ( )=ψ( ,U)d +σUdB ( ),i 1≤ ≤ 2 U( 1)=U1, FFP 2Dα,β U=ψ( ,U), i 2≤ ≤T U( 2)=U2. (7.15) Fo such a model, we ob ain Uk+1=                      U0+(1−α)ψ k1+1,Uk1+1 +αhk1 j1=0h1−1 2ξψ j1,Uj1+1 2ξψ j1+1,e Uj1+1i, i 0 ≤ ≤ 1 (7.16) Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.