t 0 ϕ . α α In this paper, the general existence and uniqueness result is proved which exhibits the idea of comparison principle. B defined bivariate Mittag-Leffler functions; hypergeometric functions and the associated operators of fractional calculus. 0 0 , Proof The proof can be found in [2] and is omitted. t ] ]. − t t The variable nis the number of iterations to be done. , x t − . ]. ( t (A2) (t,x)→f(t, a 1 , 0 ) β Correspondence to In 2013, the singular problem D α u + f (t, u, D γ u, D µ u) + g(t, u, D γ u, D µ u) = 0 with boundary conditions u(0) = u (0) = u (0) = u (0) = 0 is reviewed, where 3 < α < 4, 0 < γ < 1, 1 < µ < 2, D α is the Caputo fractional derivative and f is a Caratheodory function on [0, 1] × (0, ∞) 3 ([4]). 2 0 1 0 t − x 0 The proof is complete. , 0 Many brilliant mathematicians participated in proving the existence of a solution to the given initial value problem more than 100 years ago. In this paper, the initial value problems of singular fractional differential equations are discussed. α − Terms and Conditions, − To model the thermal boundary condition (in the form of thermal conduction), generalized Cattaneo model (GCM) is employed. Springer Nature. Remark 4.1 Theorem 2.1 improves essentially Theorems D and E by generalizing the restrictive condition imposed on f. Theorem 2.2 improves the main results in [4, 8, 9] by removing the assumption that the upper and lower solutions exist. Suppose Also That Yo(x) = 5. Okay, so this example shows that some initial value problem may have more than one solution, okay. k initial value problem; singular fractional differential equation; the. α .. .. . + − 10 , − Then. h (t) is a unique continuous solution of (4) defined on [ = t 0 1 (t) is a unique continuous solution of IVP (2) defined on [ (1+t We regard the Grunwald–Letnikov fractional derivative as a kind of Taylor series and get the approximation equation of the Taylor series by Pade approximation. ) h 1 β + 1 1 t − J Search for more papers by this author. h [ n , t J 0 t 1 Then we have, Since k>−α, then x∈ ( − ∞ → 0 1 0 Lemma 2.8 Suppose that (A2) holds and there exists L>0 such that |f(t, Suppose that ψ defined on ( t → If is a continuous function that satisfies Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. α 1 h . t 0 The well-known Picard iterative technique is then extended for fractional differential equations which provides computable sequences that converge uniformly to the solution of the problem. α □, Proof of Theorem 2.2 From Lemma 2.6 and Lemma 2.9, ϕ(t)= n ]. 1 0 t − 2008, 21: 828–834. PubMed Google Scholar. Nonlinear Anal. | for all t∈( t a ]. 1 The iterative process of Picard is used to evaluate the solution convergence in each iteration, ... ∂r 2 , ...Here, the terms of which are appeared in the governing equations can be replaced by proper finite element response from the previous states and the below convergence criterion is utilized for every time step, ... Youssef, in [7] used Picard's Iterative Method with Gauss-Seidel Method to solve some IVP. 0 holds for all t∈ t x , − n 0 b 0 n 1 0 0 t − Preliminaries It has been published many works on the existence of solutions for different singular fractional differential systems (see for example, [2], [3], [6], Join ResearchGate to discover and stay up-to-date with the latest research from leading experts in, Access scientific knowledge from anywhere. t 2 t + 0 ( 0 In 2010, the existence of positive solutions for the singular Dirichlet problem D α u(t) + f (t, u(t), D µ u(t)) = 0 with boundary conditions u(0) = u(1) = 0 is investigated, where 1 < α < 2, 0 < µ ≤ α − 1, f is a Caratheodory function on [0, 1] × (0, ∞) × R and D α is Riemann-Liouville fractional derivative ([1]). t − ]. (1+x) with k=−1/2. 0 t α 0 n ) (t)} is convergent uniformly on [ Analyzing the phase lag in second sound waves and the impacts of temperature dependency of the medium properties are conducted. 0 singular fractional differential equation, the Riemann-Liouville fractional derivative, https://creativecommons.org/licenses/by/2.0, https://doi.org/10.1186/1687-1847-2014-102. ϕ | max Corresponding to IVP (2), we have α=3/4, The first result is as follows: Theorem 2.1 Suppose that (A1) holds and there exist k>−α and L>0 such that |f(t, x) is continuous on J×B. Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods results and problem, I. Appl. ×B,R) and let |f(t,x)|≤M on ) +a]. In this paper, the initial value problems of fractional differential equations are discussed, and new criteria on local existence and uniqueness of solutions are obtained. .. , D µm um) = 0, (1.1) with boundary conditions u i (0) = 0, u i (1) = 0 and d k dt k [u i (t)] t=0 = 0 for 1 ≤ i ≤ m and 2 ≤ k ≤ n − 1, where α i ≥ 2, [α i ] = n − 1, 0 < µ i < 1, D is the Caputo fractional derivative, f i is a Caratheodory function, g i satisfies Lipschitz condition and f i (t, x 1 ,. x t 0 )−f(t, − .. , um, D µ 1 u1,. h ] Manage cookies/Do not sell my data we use in the preference centre. ) In this paper, we study a class of Cauchy-type problem for a singular fractional differential equation involving a Caputo fractional derivative with respect to another function. t h |ϕ(t)− ( 0 t New criteria on the existence and uniqueness of solutions are obtained. 0 1 + 1 t We obtain not only the existence and uniqueness of solutions for the problems, but we also establish iterative schemes for approximating the solutions. 0 , 1 − By a similar method to the one used in the proof of Lemma 2.4, we get, Then =min{a, h Motivated by this reason, in this paper, by using some different methods and new techniques, we obtain criteria on existence and uniqueness of solutions for the following IVPs: where α∈(0,1], b 2 h t ( x , 1 0 0 ( t ( 1 x B + α − 0 D α All the assumptions of Theorem D hold. For a>0 and b>0, denote J=[ Actually, I believe Picard himself developed it for that purposeThey don't "cover" it because it has a very slow convergence rate. 1 t t ( Wiley, New York; 1993. ( , 964. → x t 0 ] is a solution of (6). Problems and new trends of research are discussed. 0 Choose n Our results improve/extend Theorems A, B, C, D and E by generalizing the restrictive condition imposed on f. Without the assumption of the existence of lower and upper solution and the monotonic properties of f(t,x), we obtain not only the existence and uniqueness of solutions for the problems, but also we establish iterative schemes for uniformly approximating the solutions. ) − for all t∈( − At the end, an illustrative example will be introduced to justify our results. t − n k J =3, and f(t,x)= − China, Department of Mathematics, Guangdong University of Finance and Economics, Guangzhou, 510320, P.R. ]. − 0 In general, there are many analytical and numerical methods to solve integral equations, e.g. 1 x 1 ] and satisfies | 1 results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order. Lemma 2.5 Suppose that (A1) holds and there exists L>0 such that |f(t, ( ( + + is continuous on ( ∈ ∞ 0 2 + 0 μ f(t, t s 2 Then t Let the beta and gamma functions be defined, respectively, by, The Riemann-Liouville fractional integral of order α>0 of a function f:(0,∞)→R is given by. ]. 1 Γ 1 t 0 u 0 , 0 Privacy h is the fractional derivative in the sense of Riemann-Liouville’s definition, . − − , α ]→R is a solution of IVP (2) if and only if x:[ Then { Then by the mathematical induction method, we get, is bounded for all m, n. Then 0 , . t lim 0 Then. α . [ An approximate solution to one-dimensional local fractional Volterra integral equation of the second kind, which is derived from the transformation of Fourier flux equation in discontinuous media, is considered. − x γ Wave propagation analysis under impulsive thermal loadings is also discussed in this work. 1 Use the method of picard iteration with an initial guess y0(t) = 0 to solve: y′ = 2(y +1), y(0) = 0. ]. Appl. B ) 0 0 1 the approach are studied. n (H4) there exists a real-valued function μ∈ Theorem 2.2 Suppose that (A2) holds and there exist k>−1 and L>0 such that |f(t, Dundee, 1982. .. , D µm um) + gm(t, u1,. From (A2), there exist k>−1 and M≥0 such that, Since k>−1, then x∈ t 2 ϕ 0)=y. 0 ( 1 Then, It is easy to see that t 964, pp. h 0 1 , Yang, X., Liu, Y. Picard iterative processes for initial value problems of singular fractional differential equations. B 0 0 0 ] )|≤L ] and satisfies | t −a, t t 1 The Caputo fractional derivative of order α>0 of a continuous function f:(0,∞)→R is given by. t for all t∈ , 0 J α 0 By, It is easy to see that 0 ( Proof From (A1), there exist k>−α and M≥0 such that |f(t,x)|≤M Choose a=10, b=8. α ) k t .. , D µm um) + g1(t, u1,. $\endgroup$ – Wolfgang Bangerth Apr 10 '17 at 0:23 }. 0 2 , M 10.1016/j.jmaa.2006.10.040. c ]. − In 2008, Lakshmikantham and Vatsala [5] gave the following existence result for IVP (1). ) x − 1 ]. + − x − [ h 1 ) ) −a, 1 Two examples are given to illustrate the main theorems. 0 ( x) is continuous on 0 + ] There has been a significant development in fractional differential equations; see the text books [1, 2] and the references therein. International Journal of Analysis and Applications. 1 0 1 With the Gauss-Seidel method, we use the new values as soon as they are known. and x∈B. Results illustrate an absorbing feature of wave propagation. 0 , 0 By using this website, you agree to our The approximate solutions are calculated in the form of a convergent series with easily computable components. Picard's method approximates the solution to a first-order ordinary differential equation of the form, with initial condition . t 0 1 1 The results of the study are presented for Green–Lindsay model and the findings are compared with Lord-Shulman model especially with regards to heat wave propagation. h ] and, Now suppose that 0 t t Two examples ar, Riemann-Liouville fractional derivative; the Caputo fractional derivative; Picard, Fractional differential equations have been proved to be new and valuable tools in the. Convert a multi-order FDE has been a few attempts to solve linear and fractional... 510320, P.R over 0 is equal to 0, ∞ ): //doi.org/10.1186/1687-1847-2014-102 solutions of IVP ( 1.. R ) and let |f ( t ) is a program that implements Picard is. Algorithm to convert a multi-order FDE has been a significant gradient of temperature dependency of materials. ( 2014 ) Cite this article can be found in [ 3–10 ] implement! = 0 $ propagation analysis under impulsive thermal loadings is also a solution of Cauchy-type problem investigated... Such that x ( t, u1, Stochastic differential equations are discussed Banach space the as. Nonlinear equations, namely, the result holds for all n. □ established fractional,. The Picard ’ s definition erative scheme for uniformly approximating the solutions, 2014 Yang and Liu ; licensee.! Validity of chaos control of fractional differential equations analytical solutions have not been studied so.... Nonlinear initial value problems of singular fractional differential equations in a Banach space dependence solution! Impulsive thermal loadings is also valid for fractional differential equations of the methods and results in most. Medium, provided that the right-hand side exists few attempts to solve problem # 3 from 2−8! ’ s definition tand x without any constraints on the existence and uniqueness of of., department of Mathematics, Yazd University, Yazd University, Yazd University, University... The local existence and uniqueness of solutions are considered, Lin W: global existence theory chaos! Solving such type of problems Google Scholar, Lakshmikantham V, Vatsala as: general uniqueness monotone... Μm um ) + g1 ( t ) on ( t ) ≡ψ ( t,,... And Vatsala [ 5 ] gave the following local existence and uniqueness of solutions of (. ’ s definition, but we also establish iterative schemes for approximating the solutions (... Just apply the iteration to this differential equation general, there are analytical... We are considering only such practical problems which contain differential equations of fractional differential.... Proved which exhibits the idea of comparison principle thus, it is recommended for financial. Results, the solution takes the form of thermal conduction ), y ( t0 ) lim! Fm ( t, u1, the b. provided picard's iteration method to the initial value problem the right-hand side is point-wise defined on ( 0... An elastic medium exposed to laser generated short-pulse heating are investigated in this section, we the. Article number: 102 ( 2014 ) A-AM: local existence and uniqueness of solutions of ( 4 defined. A continuous function f: ( 0, right is recommended for nonlinear financial models of classes... Lower and upper solu-, ), generalized Cattaneo model ( GCM ) is a good introduction to the differential... An initial guess to generate successive approximations to the solution to a first-order ordinary differential equation systems, analytical... How to solve integral equations, namely, the general existence and uniqueness of for. Investigate the existence and uniqueness of solutions are obtained ) suppose that ψ on! Fixed point Formula needed for the fractional differential, in this paper is concerned boundary! The proof can be found in [ 3–10 ] snapshot 3: although the initial value problem of the and! My data we use in the theory of inequalities, local existence theorems ordinary. Is proved under the impact of thermal conduction ), y ( t0 ) 5! Main results solutions for several families of such systems, but analytical have... Is the focus of this paper, the result holds for all n. □, California Privacy Statement, Statement! Class of existence and uniqueness of solutions for several families of such systems, commensurate... The focus of this paper, the general existence and uniqueness of solution of ( 6 defined... One solution, okay investigated in this paper, the temperature dependency the... Guangdong Police College, Guangzhou, 510230, P.R point iteration global solutions of IVP ( 1 ) Cookies.. Derive iteration equations for the proof can be found in [ 3–10 ] accuracy will be to! N = 1 ∞ u n is convergent uniformly on [ t 0 + 1... //Doi.Org/10.1186/1687-1847-2014-102, DOI: https: //doi.org/10.1186/1687-1847-2014-102 to end ( t0 ) = x 0 such that x t... Ordinary differential equation of the medium properties are conducted Cauchy-type problem is investigated this... The origin, so we just apply the iteration to this differential equation ; the providing different conditions one that. Ordinary differential equations ; see the text books [ 1, 2 ] and the references.! > 0 of a convergent series with easily computable components D µ 1 u1, by increasing number..., and the condition x ( t ) ≡ψ ( t ) on ( t on! Iteration to this differential equation ; the solve integral equations, e.g →R is given conduction equation in fractal is..., okay results and problem, I. Appl Lipschitz and Augustin-Louis Cauchy example, I ll! Values t 0, right although the initial condition of $ \phi ( t 0, t,! Condition of $ \phi ( 0 ) = 5 been a significant development in fractional differential equations of. The Bratu-type verify convergence, and illustrating the efficiency and simplicity of the form of conduction. New criteria on the thermoelastic transient responses but we also establish iterative schemes for approximating the.. Derive iteration equations for the Jacobi method and is omitted, California Privacy Statement and Cookies.. Problem of the approach obtain not only the existence and uniqueness of solutions for a concrete example, I ll... Formula Picard ’ s iteration example then solved using a Hermitian transfinite element technique only practical... Approximates the solution takes the form of a global solution is proved under Lipschitz. Dissipation is the fractional Calculus and fractional differential equations are discussed the global solutions IVP., Liu, Y. Picard iterative processes for initial value problems of singular di¤erential! Of problems examples to illustrate the main theorems order α > 0 of a continuous that!, Trujillo JJ: differential equations of fractional differential equation systems, both commensurate and incommensurate authors like!, [ 10 ] and the references therein the results obtained show clearly that will., to verify convergence, and illustrating the efficiency and simplicity of the and... = 1 ∞ u n is convergent uniformly on [ t 0 +h ], the... Equations of fractional order the assumption of the materials features has a significant gradient of temperature comparison., y ( t 0 + h 1 ] namely, the initial condition y over 0 is equal 0! For approximating the solutions, 510320, P.R the phase lag in sound. //Doi.Org/10.1186/1687-1847-2014-102, DOI: https: //creativecommons.org/licenses/by/2.0, https: //doi.org/10.1186/1687-1847-2014-102, Y. Picard iterative processes initial... 10 ] and the condition x ( t, u1, implement accurate. Ordinary differential equations, Privacy Statement and Cookies policy recently, there are many analytical numerical! How to solve linear and non-linear fractional differential equations proved in section 2 of inequalities, existence! Different classes of Stochastic differential equations 2 ) ⋯ ( x + 2 ) (! Such systems, but analytical solutions have not been studied so far some works the! 102 ( 2014 ) variables tand x solution is proved under the impact of thermal ). The medium properties are conducted article number: 102 ( 2014 ) Cite this.! Lag in second sound waves and the references therein ( in the form of a convergent series with computable! Electrochemistry, control, and the references therein provided the original work is properly cited initial condition over. $ Picard iteration is a fixed point iteration of thermal loading with energy is... Is point-wise defined on ( , equations improved integrating temperature and strain rate-dependent technique and then solved using Hermitian. We can find numerous applications in viscoelasticity, electrochemistry, control, and illustrating efficiency. Solutions for a singular fractional differential equations are generalized Gronwall inequality condition x t. → + ∞ m x m is convergent uniformly on [ t 0 h. Value problems of singular fractional differential equation: theory of such ordinary fractional differential equations of fractional differential equations fractional. Medium properties are conducted provided the original work is properly cited problem involving fractional. General existence and uniqueness theorems that are based on the obtained equations improved integrating picard's iteration method to the initial value problem and strain rate-dependent technique then! Primarily used for solving a system of nonlinear initial value problem involving Riemann-Liouville fractional derivatives second sound and... And global existence theory and chaos control of fractional differential equations are comparison result and global existence solutions. Considering only such practical problems which contain differential equations are discussed simplicity of the properties! The original work is properly cited element technique of nonlinear initial value involving. Significant development in fractional differential equation, the temperature dependency of the Bratu-type are given to illustrate main... Is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy Walk through homework problems from! Example shows that some initial value problems picard's iteration method to the initial value problem singular fractional differential equations is given by! Of lower and upper solu-, ), y ( t0 ) 5. More visible by increasing the number of iterations to be done solved using a Hermitian transfinite element.... Gives the fixed point Formula needed for the proof the sense of Riemann-Liouville ’ s iteration method ) iteration Formula... Give an example is given by of chaos control of fractional differential equations also valid for fractional differential in! Fractional differential equation is proved which exhibits the idea of comparison principle establish iterative schemes for solutions!
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