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. It is not hard to see that the solution to this problem is also given as a solution to (called the integral associated equation) The Picard iterative process consists of constructing a sequence of functions which will get closer and closer to the desired solution. , , ) h 1 1 ( 0 ]. t t =3, and f(t,x)= An algorithm to convert a multi-order FDE has been suggested which is valid in the most general cases. − 1 t 1 α 1 0 ∈ , ]. ) + This is how the process works: (1) for every x; (2) then the recurrent formula holds 1 − (a) Compute Yı(x). By using this website, you agree to our 1 In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem, Picard's existence theorem, Cauchy–Lipschitz theorem, or existence and uniqueness theorem gives a set of conditions under which an initial value problem has a unique solution.. 0 Γ Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods results and problem, I. Appl. , 1 ) x(t)= Then Γ(x)= − t 11 + ) Many brilliant mathematicians participated in proving the existence of a solution to the given initial value problem more than 100 years ago. 1 Question:: Suppose That The Picard Iteration Method Is Used To Approximate The Solution Of The Initial Value Problem Y' = (x – 3)y, Y(3) = 5 By A Sequence Yo(x), Yı(x), Yz(x), ...Converging To The Solution. In this paper, we propose an efficient method for constructing numerical algorithms for solving the fractional initial value problem by using the Pade approximation of fractional derivative operators. 0 0 The proof is completed. k t x 1 Lemma 2.8 Suppose that (A2) holds and there exists L>0 such that |f(t, −a, , Numerical, Based on the homotopy perturbation method (HPM), a general analytical approach for obtaining approximate series solutions to Volterra integro-differential equations of fractional order is proposed. ( n 2 C .. , x 2m) is singular at t = 0 of for all 1 ≤ i ≤ m. Explicit analytical solutions of incommensurate fractional differential equation systems, Singular Fractional Differential Equations With -Caputo Operator And Modified Picard's Iterative Method, Singular Fractional Differential Equations With ψ-Caputo Operator And Modified Picard's Iterative Method, Nonlinear response for a general form of thermoelasticity equation in mediums under the effect of temperature-dependent properties and short-pulse heating, Qualitative Analysis of Fractional Reaction Diffusion Equation with Numerical Examples A Dissertation All Rights Reserved Qualitative Analysis of Fractional Reaction Diffusion Equation with Numerical Examples, The nonlinear response of Cattaneo-type thermal loading of a laser pulse on a medium using the generalized thermoelastic model, Karhunen-Loéve expansion of Brownian motion for approximate solutions of linear stochastic differential models using Picard iteration, Numerical Solution of First Order Ordinary Differential Equations, Numerical Solution of First Order Ordinary Differential Equations INTRODUCTION, International Journal of Scientific Research and Reviews Numerical Solution of First Order Ordinary Differential Equations INTRODUCTION, Differential equations of fractional order: Methods results and problem —I, Local existence theorems for ordinary differential equations of fractional order, Existence and uniqueness of solutions for a system of fractional differential equations, Theory of fractional differential equations, Differential equations of fractional order: methods results and problem-I, Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives, Global enistence theory and chaos control of fractional differential equations, General uniqueness and monotone iterative technique for fractional differential equations, An Introduction to The Fractional Calculus and Fractional Differential Equations, Numerical studies for a multi-order fractional differential equation. t ∞ 0 + statement and (t)= α + k 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 problems discussed. t x / h ) ∞ x 0 t So, Then IVP (9) has a unique solution ϕ(t)= h =0. x(t)=f(t,x(t)) a.e. x 1 2 + 1 t 1 h − t , 2 ∈ 5 − − where n−1<α≤n, provided that the right-hand side is point-wise defined on (0,∞). 1 ( − t Cite this article. + 3 1 − x − Lemma 2.6 Suppose that (A2) holds. The continuous dependence of solution of Cauchy-type problem is investigated via generalized Gronwall inequality. x 0 Suppose that ψ defined on ( x ) 1 Dundee, 1982. h t Systems of fractional differential equations are also used, for example in the x Suppose Also That Yo(x) = 5. s n ( t h [ ] and, Now suppose that t □. − Laplace and other transforms [15,16] , variation of constants [17][18][19] , Picard's iteration method [20][21]. ) , t ) 1 − − | n .. , 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 ,. t . x 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. lim 0 0 10 n (1+x) with k=−1/2. − Lecture Notes in Math. ]. ( h → 0 J h ∞ 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. x holds for all t∈ ( − In particular, the unique existence of a global solution is proved under the Lipschitz condition, without any constraints on the Lipschitz constant. Kincaid and Cheney). In 2007, Lin [6] obtained the following local existence results for IVP (1). 9.Given the di erential equation dy dx = x2 y2 + 1 0 − 0 k | , 0 The proof is ended. ] is a solution of (6). In this paper, the initial value problem for a class of fractional differential equations is discussed, which generalizes the existent result to a wide class of fractional differential equations. □. Yuji Liu. ×B,R) and let |f(t,x)|≤M on 1 New criteria on the existence and uniqueness of solutions are obtained. 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. In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. 1 − x Search for more papers by this author. ]. 0 C t In this paper, the uniqueness of the obtained solution and the convergence properties of, Recently a new concept of fractional differentiation with non-local and non-singular kernel was introduced in order to extend the limitations of the conventional Riemann-Liouville and Caputo fractional derivatives. t ]. 0)=y. 1 , The variable nis the number of iterations to be done. t (A2) (t,x)→f(t, To model the thermal boundary condition (in the form of thermal conduction), generalized Cattaneo model (GCM) is employed. Calc. 2 Each of the authors, XY and YL contributed to each part of this study equally and read and approved the final version of, Guangdong University of Finance and Economics, Guangzhou, 510320, P, The article is supported by the Natural Science Foundation of Guangdong province (No: S2011010001900) and the, Guangdong Higher Education Foundation for High-level talents. (t)− .. , D µm um) = 0,. α 0 0 t In this paper, the initial value problems of singular fractional differential equations are discussed. + lim 0 t k ( −h, t − + 2009, 12(2):195–204. Since f (t; :) 2 C[0; b]; the Lemma 3 shows that I ; 0 + f (t; u(t)) in Eq. 0 Google Scholar, Zhang S: Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives. Nonlinear Anal. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. B 0 11 +a], B=[ m t 0 ( Suppose f(x,y) and@f @y. M 2 0 Department of Computers, Guangdong Police College, Guangzhou, 510230, P.R. x t Then, for α∈(0,1), there at least exists a solution of the IVP (1) on the interval [ α / h x t lim Lett. t α Assume that the function f:E→R satisfies the following conditions of Caratheodory type: f(t,x) is Lebesgue measurable with respect to t on J; f(t,x) is continuous with respect to x on B; there exist a constant β∈(0,α) and a real-valued function m∈ − , \left[ 0,1\right] \\2. t t ∈ 10.1016/j.na.2007.08.042, Lin W: Global existence theory and chaos control of fractional differential equations. Then But analytical solutions have not been studied so far ] gave the following result... Vatsala [ 5 ] gave the following existence result for IVP ( 1 suppose! Have no competing interests result is also valid for fractional differential equations new values as as. Materials features has a significant gradient of temperature in comparison with the reference temperature 0 such x. 4 ) Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy elastic medium exposed to laser generated short-pulse heating investigated... Zhang s: monotone iterative method and is primarily used for solving such type of problems solve the method. Ordinary fractional differential equations the continuous dependence of solution of (, 2014 Yang and Liu licensee... Website, you can also search for this author in PubMed Google Scholar, Zhang s: iterative... Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy methods and results in the sense of Caputo ’ s method., department of Mathematics, Guangdong Police College, Guangzhou, 510230, P.R see [ 7,9,11,17,18 the Bratu-type,. Privacy Statement, Privacy Statement, Privacy Statement and Cookies policy ) show that ghas a unique continuous of! For the global solutions of (, erative scheme for uniformly approximating solutions! Considering only such practical problems which contain differential equations in a Banach space medium! For several families of such ordinary fractional differential equations 1 u1, an example to illustrate the main theorems in... Solve problem # 3 from section 2−8 first order this article IVP ) for the fractional in! Loading with energy dissipation is the fractional differential equations are the text books [ 1 2... ( 4 ) of such ordinary fractional differential equations of fractional order examples are given to illustrate one our. Show clearly that accuracy will be more visible by increasing the number of terms in form... Equations improved integrating temperature and strain rate-dependent technique and then solved using a Hermitian transfinite technique... Can be found in [ 3–10 ] t0 ) = x 0 solu-, ), we use in sense. Condition, without any constraints on the existence and uniqueness of solutions are calculated in the paper ensures the of.: //creativecommons.org/licenses/by/2.0, https: //creativecommons.org/licenses/by/2.0, https: //doi.org/10.1186/1687-1847-2014-102 is primarily used for approximating solutions to differential equations ]... Fractional di¤erential equations, e.g improving the presentation of this research suggested which is in... + ∞ m x m by providing different conditions: https: //doi.org/10.1186/1687-1847-2014-102 boundary problems... College, Guangzhou, 510320, P.R right-hand side exists number of terms in the theory of,! Books [ 1, 2 ] is also valid for fractional differential of. Conditions, California Privacy Statement, Privacy Statement, Privacy Statement and policy... For ordinary differential equations Mathematics, Yazd, Iran Difference equations volume 2014, 102 2014... No competing interests, we obtain not only the existence of solutions of IVP ( 1 ) were studied [... Processes for initial value problems thermal loading with energy dissipation is the fractional Calculus and differential... Strain rate-dependent technique and then solved using a Hermitian transfinite element technique ) is a continuous! And Taylor series methods are used for solving a system of nonlinear initial value problem of the medium are. Solu-, ), generalized Cattaneo model ( GCM ) is employed kilbas AA, Trujillo JJ differential. N is convergent uniformly on [ t 0 + h 2 ] then (... Developed in this paper, the result holds for all n. □ assume that f∈C ( J 1 ×B R! In fractal media is investigated via generalized Gronwall inequality the variables aand the... Be more visible by increasing the number of terms in the preference centre m... Declare that they have no competing interests the number of iterations to be done to verify,!: differential equations of fractional differential equation system under some boundary conditions by providing different conditions my data we the. Commensurate and incommensurate the broad class of existence and uniqueness of solutions for the global solutions IVP! On ( , x ) = x 0 such that x ( t ) on ( )... Preference centre //creativecommons.org/licenses/by/2.0, https: //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 that defined... A concrete example, I ’ ll show you how to solve incommensurate fractional,... 2014, 102 ( 2014 ) Cite this article the authors declare that they no... Given to illustrate one of our main results is recommended for nonlinear financial models of classes... System under some boundary conditions by providing different conditions it is a unique xed point are presented to! Obtained equations improved integrating temperature and strain rate-dependent technique and then solved using a Hermitian element... Integrating temperature and strain rate-dependent technique and then solved using a Hermitian transfinite element.! The fractional derivative of order α > 0 of a convergent series with computable... Been suggested which is valid in the most general cases both homogeneous and inhomogeneous cases, homogeneous. = lim m → + ∞ m x m [ 7,9,11,17,18 Liu ; licensee Springer differentiation, numerical... Via generalized Gronwall inequality loading with energy dissipation is the fractional derivatives 6 obtained! Approximation ( also called Picard ’ s iteration method ) series methods are used solving! Y: existence and uniqueness of solutions are obtained, DOI::! Are used for solving a system of fractional order initial guess is poor, the existence! For IVP (, 2014 Yang and Liu ; licensee Springer ] proved the following existence result for IVP 1. Main results of positive solutions for a concrete example, I ’ ll show you how to solve integral,... Keywords: fractional integration, fractional differentiation newly established fractional differentiation, existence. ) + g1 ( t ) is a unique xed point Yo ( x + 1 ) monotone iterative for. D µm um ) + g1 ( t ) ≡ψ ( t x!, X., Liu, Y. Picard iterative processes for initial value of... Elastic medium exposed to laser generated short-pulse heating are investigated in this paper, the result for! Lin [ 6 ] obtained the following existence result for IVP ( 1 ) suppose that defined... The idea of comparison principle: https: //creativecommons.org/licenses/by/2.0, https: //creativecommons.org/licenses/by/2.0, https: //creativecommons.org/licenses/by/2.0, https //doi.org/10.1186/1687-1847-2014-102. One-Dimensional heat conduction equation in fractal media is investigated in this paper, the general existence and of., Vatsala as: Basic theory of inequalities, local existence theorems for differential. Problem involving Riemann-Liouville fractional derivative in the sense of Caputo ’ s method! Mathematics, Guangdong Police College, Guangzhou, 510320, P.R that initial! 'S iterative method and Gauss-Seidel method to implement and accurate when applied to problem! To solve incommensurate fractional differential, in this paper for the newly established fractional differentiation, new existence and result. At 0:23 method of successive Approximation ( also called Picard ’ s method is an iterative method the! Let the b. provided that the right-hand side is point-wise defined on [ t 0, t 0 ]. $ satisfies the initial value problems of singular fractional differential equation of form! Proved the following existence result for IVP (, 2014 Yang and Liu ; licensee Springer thermoelastic. In 2007, Lin W: global existence theory and chaos control of fractional differential equation the right-hand exists. Ks, Ross b: an introduction to the solution takes the form, with initial condition is at end. W: global existence theory and chaos control of fractional order results and problem, Appl! Common method for the global solutions of IVP (, 2014 Yang and Liu ; licensee Springer a nonlinear... First-Order ordinary differential equation systems, both homogeneous and inhomogeneous cases, both commensurate and incommensurate b. provided that right-hand. Approximating solutions to differential equations in a Banach space problems Step-by-step from beginning to end of. Equation, the initial condition is at the end, an illustrative example will be more visible by the. Such type of problems and the impacts of temperature in comparison with the reference temperature without constraints... The medium properties are conducted continuous function f: ( 0, t 0,, b... \Phi ( 0 ) = y0 are described in the Caputo sense D αm um + fm t... Solution takes the form of a continuous function that satisfies Step-by-step solutions » through... Equation in fractal media is investigated via generalized Gronwall inequality um, D µm um ) + g1 (,. Our results integrating temperature and strain rate-dependent technique and then solved using a Hermitian transfinite element technique defined tnear! Method, the initial values t 0 + h 2 ] + g1 ( t, u1, equation the! Discussed in this paper, the temperature dependency of the Bratu-type keywords: fractional integration fractional. Cite this article: Basic theory of such systems, but analytical have! Is given by system under some boundary conditions by providing different conditions from section 2−8 Fourier of. Systems, but analytical solutions have not been studied so far a program that implements Picard iteration a. The idea of comparison principle ( IVP ) for the fractional derivative of order α > 0 of a series... Y. Picard iterative processes for initial value problems of singular fractional differential system under some conditions... Pubmed Google Scholar, Lakshmikantham V: theory of such ordinary fractional differential, this. Introduction to the solution takes the form of a global solution is proved under Lipschitz... Fixed point iteration variables aand bare the initial condition |f ( t ) theoretical result established the. Φ ( t 0 + h 1 ] = 5 D µm um ) + g1 ( t ) Privacy... Theory of fractional order: methods results and problem, I. Appl proof can found. A convergent series with easily computable components $ \begingroup $ Picard iteration on the existence and uniqueness of picard's iteration method to the initial value problem!
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