Well posedness and approximation for nonhomogeneous fractional differential equations

Numerical Functional Analysis and Optimization ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/lnfa20 Well-Posedness and Approximation for Nonhomogeneous Fractional Differential Equations Ru Liu & Sergey Piskarev To cite this article: Ru Liu & Sergey Piskarev (2021): Well-Posedness and Approximation for Nonhomogeneous Fractional Differential Equations, Numerical Functional Analysis and Optimization, DOI: 10.1080/01630563.2021.1901117 To link to this article: https://doi.org/10.1080/01630563.2021.1901117 Published online: 22 Mar 2021 Submit your article to this journal Article views: View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=lnfa20 NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION https://doi.org/10.1080/01630563.2021.1901117 Well-Posedness and Approximation for Nonhomogeneous Fractional Differential Equations Ru Liua and Sergey Piskarevb a College of Computer Science, Chengdu University, Chengdu, Sichuan, P.R China; bScientific Research Computer Center, Lomonosov Moscow State University, Moscow, Russia ABSTRACT ARTICLE HISTORY In this paper, we consider the well-posedness and approximation for nonhomogeneous fractional differential equations in Banach spaces E Firstly, we get the necessary and sufficient condition for the well-posedness of nonhomogeneous fractional Cauchy problems in the spaces C0b ð½0, TŠ; EÞ: Secondly, by using implicit difference scheme and explicit difference scheme, we deal with the full discretization of the solutions of nonhomogeneous fractional differential equations in time variables, get the stability of the schemes and the order of convergence Received 30 September 2019 Revised March 2021 Accepted March 2021 KEYWORDS a-times resolvent family; discretization methods; explicit scheme; fractional Cauchy problem; implicit scheme; nonhomogeneous fractional equations MATHEMATICS SUBJECT CLASSIFICATION 45L05; 65M06; 65M12 Introduction A lot of works were devoted to the approximations of C0-semigroups, see [1–5] and the references therein While, other mathematicians considered the discrete approximation of integrated semigroups in their papers [6–8] We all know that Podlubny introduced fractional derivatives, fractional differential equations, some methods of their solutions and some of their applications in his book [9] Ashyralyev and Cakir considered the numerical solutions of fractional parabolic partial differential equations [10–15] In papers [16–19], we dealt with the discrete approximation of the homogeneous fractional differential equations and semilinear fractional differential equations in Banach spaces Especially in [18,19], we get the stability and the order of convergence of implicit difference scheme and explicit difference scheme for homogeneous fractional differential equations In this paper, we will consider the fulldiscrete approximation of the nonhomogeneous fractional differential equation in the space Cẵ0, T; Eị, which will be presented in section CONTACT Ru Liu srb40305079@163.com Sichuan, 610106, P.R China ß 2021 Taylor & Francis Group, LLC College of Computer Science, Chengdu University, Chengdu, R LIU AND S PISKAREV Let 0x, x E, ka1 ka IAị1 x ẳ NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION where b q ðÁÞ is denoted the Laplace transform of qðÁÞ: In the paper [16], we have proved that if the operator A generates an a-times resolvent family Sa ðÁ, AÞ which is satisfying jjSa ðt, AÞjj Mext , t ! 0, then the operator A is closed and densely defined Definition 1.1 [23] A family fSa ðt, AÞgt!0 & BðEÞ is called an a-times resolvent family generated by A if the following conditions are satisfied: (a) (b) (c) Sa ðt, AÞ is strongly continuous for t ! and Sa 0, Aị ẳ I; Sa t, AịDAị  DAị and ASa t, Aịx ẳ Sa ðt, AÞAx for DðAÞ, t ! 0; for x DðAÞ, Sa ðt, AÞx satisfies the resolvent equation ðt Sa t, Aịx ẳ x ỵ ga tsịSa s, AịAxds, t ! 0: all x2 Definition 1.2 An a-times resolvent family Sa ðÁ, AÞ is called analytic if Sa ðÁ, AÞ admits an analytic extension to a sector Rh0 nf0g for some h0 ð0, p=2Š, where Rh0 :¼ fk C : jargkjxg & qAị and 1 a k IAị x ẳ ekt Pa ðt, AÞxdt, Rek>x, x E: Remark 1.2 [23–25] If A generates an a-times resolvent family Sa ðt, AÞ for the case 10 that NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION jjga Sa t ỵ s, Aịga Sa t, AịjjE!E tỵs tỵs ẳ jj Pa s, AịdsjjE!E M1 sa1 ds t ẳ M1 ta1 s ¼ M1 ta s t s M2 : t M1 t a1 tỵs t ds t While, jjga Sa t ỵ s, Aịga Sa t, AịjjE!E M1 t ỵ sịa ỵ M1 ta M2 , then similar to the above process, one has (2.4) From the inequalities tỵs P0a s, AịdsjjE!E jjPa t ỵ s, AịPa t, AịjjE!E ẳ jj t tỵs ds M1 s M1 t a s s M1 ¼ M2 , 2Àa 2Àa s t t t t jjPa t ỵ s, AịPa t, AịjjE!E M1 t ỵ sịa ta ỵ M1 tỵs t M2 , t we get (2.5) w Remark 2.1 Actually, we can get (2.1) from the above lemma In fact, jjSa , AịxjjCb ẵ0, T ;Eị ẳ sup jjSa t, AịxjjE ỵ t T M1 jjxjjE ỵ sup tx1 , M M w jjkaÀ1 ðka IÀAÞÀ1 jjE!E jkj jkÀxj : Then, Sa ðt, AÞ is analytic Theorem 2.2 Let A be the generator of an analytic a-times resolvent family Then (1.1) is well posed in C0b ẵ0, T; Eị and the coercivity inequality (2.12) holds Proof If uðÁÞ is a solution to problem (1.1) in C0b ẵ0, T; Eị, then it is a solution iné Cẵ0, T; Eị, too Hence, we have the representation utị ẳ t Sa t, Aịx ỵ Pa ts, Aịf sịds :ẳ wtị ỵ vtị: We need to show that AwðtÞ, AvðtÞ, wðtÞ, vðtÞ belongs to C0b ẵ0, T; Eị and (2.12) holds Firstly, let us consider the estimate of jjAuịjjCb ẵ0, T;Eị : We know that wtị DAị and jjAwtịjjE ẳ jjSa ðt, AÞAxjjE M1 jjAxjjE , t T: Applying (2.2), we get that, for 00 and some 0