81 results
Search Results
2. The RBF partition of unity method for a 2D time-fractional parabolic equation.
- Author
-
Raeisi, Banafsheh, Darani, Mohammadreza Ahmadi, and Fardi, Mojtaba
- Subjects
- *
PARTITION of unity method , *LINEAR systems , *EQUATIONS , *FRACTIONAL differential equations - Abstract
This paper proposes a method to solve a 2D time-fractional parabolic equation on arbitrary domains using an RBF partition of unity procedure based on a non-uniform time-stepping framework. The proposed method involves partitioning the main domain into overlapping subdomains, computing local approximations on each subdomain, and joining them together by PU weights to obtain a global approximation. This approach enhances stability, reduces computational costs, and replaces an ill-conditioned linear system with several small well-conditioned linear systems. To handle the weak singularity near the initial time, a time-graded mesh is used to obtain a non-uniform time-stepping scheme, which increases the temporal convergence rate. The convergence analysis of the method is examined to obtain an error bound for the local approximation. Numerical tests are conducted to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. On the convergence rate of the splitting-up scheme for rough partial differential equations.
- Author
-
He, Yuchen
- Subjects
- *
PARTIAL differential equations , *LINEAR differential equations , *FRACTIONAL differential equations , *STOCHASTIC partial differential equations , *CAHN-Hilliard-Cook equation - Abstract
In this paper, the splitting-up scheme for rough partial differential equations (RPDE) is introduced. The convergence rate of the splitting-up scheme for weak solutions to semilinear partial differential equations with linear rough noises is given. An application to stochastic partial differential equations with fractional Browian motions is considered. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. Adaptive meshless numerical method of solving 2D variable order time fractional mobile-immobile advection-diffusion equations.
- Author
-
Du, Hong and Chen, Zhong
- Subjects
- *
ADVECTION-diffusion equations , *HILBERT space , *SPLINE theory , *FRACTIONAL differential equations - Abstract
This paper inspires us to propose an adaptive meshless numerical method of solving a 2D variable order time fractional mobile-immobile advection-diffusion equation on arbitrary domain. The mentioned method has three main contributions. (i) The adaptive meshless method could decrease amounts of computations effectively. (ii) On arbitrary domain, we successfully apply Legendre multiwaves in reproducing kernel Hilbert space (RKHS) defined on rectangle domain to approximate the exact solution. (iii) We skillfully build a special spline space, convergence order is obtained in view of convergence theories of spline space. Numerical experiments further demonstrate the validity of the method. • The adaptive meshless method could decrease amounts of computations effectively. • On arbitrary domain, we apply Legendre multiwaves to approximate the exact solution using some approximation theorems. • We successfully give the error estimate in the case of low regularity of the function. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
5. Vibration analysis of variable fractional viscoelastic plate based on shifted Chebyshev wavelets algorithm.
- Author
-
Dang, Rongqi, Yang, Aiming, Chen, Yiming, Wei, Yanqiao, and Yu, Chunxiao
- Subjects
- *
IRON & steel plates , *FRACTIONAL differential equations , *MATHEMATICAL errors , *DESIGN protection , *ALGORITHMS , *WAVELET transforms - Abstract
In this paper, new and effective methods are provided for modeling and numerical simulation of viscoelastic plate, respectively. Viscoelastic plate is modeled by a variable fractional derivative model with better fitting effect and is numerically analyzed directly in time domain using shifted Chebyshev wavelets algorithm for the first time. A governing equation with three independent variables is established. The feasibility and accuracy of proposed algorithm are verified by convergence analysis and error estimations of mathematical example. Under different conditions, the ternary variable fractional differential equation of plate is solved numerically by shifted Chebyshev wavelets algorithm directly in time domain. The vibrations of plate are analyzed under different loads, loading times and temperatures. Numerical solutions for the stress of plate with two materials are also calculated and analyzed under the same load. The displacement of plate increases with increasing load, loading time and temperature. The bending resistance of PET plate is better than that of polyurea plate. These results are consistent with the existing references and actual practice. Therefore, the variable fractional model and shifted Chebyshev wavelets algorithm are accurate for the study of plate. They can be applied to complex problems in engineering. All obtained conclusions can provide theoretical basis for the protection and design of load-bearing structures in engineering. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
6. A computationally efficient strategy for time-fractional diffusion-reaction equations.
- Author
-
Garrappa, Roberto and Popolizio, Marina
- Subjects
- *
REACTION-diffusion equations , *FINITE differences , *SYLVESTER matrix equations , *PARTIAL differential equations , *FRACTIONAL differential equations , *ADVECTION-diffusion equations - Abstract
An efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
7. Approximate controllability for fractional semilinear parabolic equations.
- Author
-
Huang, Yong, Liu, Zhenhai, and Wen, Ching-Feng
- Subjects
- *
PARABOLIC operators , *FRACTIONAL differential equations , *CONTROLLABILITY in systems engineering , *LINEAR operators , *EQUATIONS , *NONLINEAR analysis - Abstract
In this paper, we deal with the control systems described by a large class of fractional semilinear parabolic equations. Firstly, we reformulate the fractional parabolic equations into abstract fractional differential equations associated with a semigroup on an appropriate Banach space. Secondly, we introduce a suitable concept on a mild solution for this kind of fractional parabolic equations and present the existence and uniqueness of mild solution by utilizing the theory of semigroup of linear operator, nonlinear analysis method and fixed point theorem. Then, the approximate controllability of the fractional semilinear parabolic equations is formulated and proved. At the end of the paper, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
8. Tensorized low-rank circulant preconditioners for multilevel Toeplitz linear systems from high-dimensional fractional Riesz equations.
- Author
-
Zhang, Lei, Zhang, Guo-Feng, and Liang, Zhao-Zheng
- Subjects
- *
LINEAR systems , *FAST Fourier transforms , *PARTIAL differential equations , *FRACTIONAL differential equations , *EQUATIONS , *CIRCULANT matrices , *LOW-rank matrices - Abstract
The solution of d -dimensional fractional partial differential equations (FPDEs) is challenging in numerical computation. This paper proposes a low-rank preconditioned conjugate gradient (PCG) method for discretizations with a low-rank right-hand side stemming from the d -dimensional fractional Riesz equation. At each iteration step, a combination of low-rank in Tensor Train (TT) format technique and Fast Fourier Transform (FFT) is used to accelerate multilevel Toeplitz matrix-vector multiplications. Furthermore, a low-rank circulant preconditioner is constructed based on the spectrum of the coefficient matrix. The reported numerical results demonstrate the competitiveness of the new method compared with a state-of-the-art matrix-free approach based on FFT. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
9. Rothe method and numerical analysis for a new class of fractional differential hemivariational inequality with an application.
- Author
-
Weng, Yun-hua, Chen, Tao, Li, Xue-song, and Huang, Nan-jing
- Subjects
- *
NUMERICAL analysis , *FRACTIONAL differential equations , *BANACH spaces , *VARIATIONAL inequalities (Mathematics) , *APPROXIMATION error , *SURJECTIONS , *DIFFERENTIAL inequalities - Abstract
The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
10. Boundary value problems of fractional Fokker–Planck equations.
- Author
-
Aleroev, Temirkhan S., Aleroeva, Hedi T., Huang, Jianfei, Tamm, Mikhail V., Tang, Yifa, and Zhao, Yue
- Subjects
- *
BOUNDARY value problems , *FRACTIONAL differential equations , *FOKKER-Planck equation , *SEPARATION of variables , *EIGENFUNCTIONS - Abstract
This paper is devoted to solving boundary value problems for important fractional differential equations of the Fokker–Planck family, in particular, to studying fractional differential equation for advection–dispersion. The consideration is carried out by the separation of variables (the Fourier method). Most part of this paper is devoted to justification of this method, to proof of a basis of the system of eigenfunctions for the basic equation (Aleroev et al., 2015) for modeling the random walk of a point particle which starts to move at the origin of coordinates in t = 0 on a self-similar fractal set. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
11. A-posteriori error estimations of the GJF-Petrov–Galerkin methods for fractional differential equations.
- Author
-
Mao, Wenting, Chen, Yanping, and Wang, Huasheng
- Subjects
- *
FRACTIONAL differential equations , *INITIAL value problems , *SOBOLEV spaces - Abstract
In this paper, we develop an efficient spectral GJF-Petrov–Galerkin algorithm and the postprocessed method to solve a class of fractional initial value problems. The main part of these algorithms is to use a special set of general Jacobi functions (GJFs) to form the trial space and test space. We give a rigorous error analysis in non-uniformly weighted Sobolev spaces and obtain optimal error estimates. In particular, the postprocessing technique is used to construct the postprocessed method. In addition, we derive its superconvergence estimates and define a-posteriori error estimators that are asymptotically exact. Numerical experiments are included to support the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
12. Wavelet approximation scheme for distributed order fractional differential equations.
- Author
-
Kumar, Yashveer, Singh, Somveer, Srivastava, Nikhil, Singh, Aman, and Singh, Vineet Kumar
- Subjects
- *
FRACTIONAL differential equations , *LINEAR differential equations , *ALGEBRAIC equations , *DIFFERENTIAL operators , *LINEAR orderings , *BIORTHOGONAL systems , *WAVELET transforms - Abstract
This paper is concerned with the study of wavelet approximation scheme based on Legendre and Chebyshev wavelets for finding the approximate solutions of distributed order linear differential equations. The operational matrix for distributed order fractional differential operator is derived for Legendre and Chebyshev wavelets basis. Furthermore, the obtained operational matrix along with Gauss Legendre quadrature formula and standard Tau method are utilized to reduce the distributed order linear differential equations into the system of linear algebraic equations. For the better understanding of the method, numerical algorithms are also provided for the considered problems. In order to verify the desired accuracy of the proposed method, five test examples are included and numerical experiments confirm the theoretical results and illustrate applicability and efficiency of the proposed method. Also, the convergence analysis, error bounds, error estimate and numerical stability of presented method via Legendre and Chebyshev wavelets are investigated. Moreover, comparison of the numerical results obtained by the proposed approach is provided with the results of existing method. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
13. A weighted finite difference method for subdiffusive Black–Scholes model.
- Author
-
Krzyżanowski, Grzegorz, Magdziarz, Marcin, and Płociniczak, Łukasz
- Subjects
- *
FINITE difference method , *BLACK-Scholes model , *FRACTIONAL differential equations , *CAPUTO fractional derivatives - Abstract
In this paper we focus on the subdiffusive Black–Scholes (B–S) model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We find the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank–Nicolson (C–N) scheme. The proposed method has 2 − α order of accuracy with respect to time where α ∈ (0 , 1) is the subdiffusion parameter, and 2 with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some numerical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
14. A fast preconditioned iterative method for two-dimensional options pricing under fractional differential models.
- Author
-
Chen, Xu, Ding, Deng, Lei, Siu-Long, and Wang, Wenfei
- Subjects
- *
PARTIAL differential equations , *FRACTIONAL differential equations , *FINITE differences , *FASTING , *LEVY processes , *KRYLOV subspace , *FRACTIONAL programming - Abstract
In recent years, fractional partial differential equation (FPDE) has been widely applied in options pricing problems, which better explains many important empirical facts of financial markets. However, the vast majority of the literatures focus on pricing the single asset option under the FPDE framework. In this paper, a two-dimensional FPDE governing the valuation of rainbow options is established, where two underlying assets are assumed to follow independent exponential Lévy processes, and its boundary conditions are determined by solving one-dimensional FPDEs. A second-order accurate finite difference scheme is proposed to discretize the two-dimensional FPDE. Given the block Toeplitz with Toeplitz block structure of the coefficient matrix, a fast preconditioned Krylov subspace method is developed for solving the resulting linear system with O (N log N) computational complexity per iteration, where N is the matrix size. The proposed preconditioner accelerates the convergence of the iterative method with theoretical analysis. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed numerical methods. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
15. Quenching study of two-dimensional fractional reaction–diffusion equation from combustion process.
- Author
-
Xu, Qinwu and Xu, Yufeng
- Subjects
- *
REACTION-diffusion equations , *COMBUSTION , *FINITE differences , *GALERKIN methods , *FRACTIONAL differential equations - Abstract
The quenching phenomenon and its physical characters are critical issues in the study of combustion process. In this paper, a two-dimensional temporal fractional combustion model with Caputo derivative is considered. We design an ingenious adaptive finite difference discontinuous Galerkin method to solve this fractional combustion model. To catch quenching moment more accurately, an explicit bisection adaptive strategy is proposed for discretization of fractional derivative. Moreover, discontinuous Galerkin method is employed for spatial discretization to settle large gradient changes and non-smoothness property of solution in adjacent area of quenching moment. The physical properties of combustion process, such as positivity, monotonicity and stability, are theoretically or numerically investigated. Finally, we carry out systematic numerical experiments for such abstract combustor with different shapes and various parameter settings, which illustrate that the dynamics of such model depends on the intensity of initial inputs, the order of fractional derivative and the area of spatial domain. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
16. Distributed order equations as boundary value problems
- Author
-
Ford, N.J. and Morgado, M.L.
- Subjects
- *
DISTRIBUTED algorithms , *BOUNDARY value problems , *DIFFERENTIAL equations , *APPROXIMATION theory , *NUMERICAL analysis , *EXISTENCE theorems , *UNIQUENESS (Mathematics) - Abstract
Abstract: There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the solution depends, among other parameters, on Caputo-type initial conditions. In this paper we discuss the existence and uniqueness of solutions and we propose a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
17. A compact integrated RBF method for time fractional convection–diffusion–reaction equations.
- Author
-
Qiao, Yuanyang, Zhao, Jianping, and Feng, Xinlong
- Subjects
- *
TRANSPORT equation , *STEREOLITHOGRAPHY , *RADIAL basis functions , *FRACTIONAL differential equations , *EQUATIONS , *COMPACTING - Abstract
Abstract In this paper, a local compact integrated radial basis function (CIRBF) method is proposed to solve the time fractional convection–diffusion–reaction equations. The proposed CIRBF scheme derives high accuracy and fast convergence rate with grid refinement. The spatial derivatives are discretized by combining with the integrated RBF approximation and compact approximation in a stencil. The nodal function values and the nodal second derivatives are used to establish the relation between the physical space and the RBF weight space. The time discretization is replaced by the second order shifted Grünwald scheme. Moreover, the unconditional stability and convergence of the proposed method are shown. Finally, some numerical examples on 2D and 3D domains are carried out to verify the accuracy and efficiency of the proposed algorithm for the fractional differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
18. The well-posedness for fractional nonlinear Schrödinger equations.
- Author
-
Peng, Li, Zhou, Yong, and Ahmad, Bashir
- Subjects
- *
FRACTIONAL differential equations , *SCHRODINGER equation , *CAUCHY problem , *HARMONIC analysis (Mathematics) , *EXISTENCE theorems , *CAPUTO fractional derivatives - Abstract
Abstract In this paper we apply the tools of harmonic analysis to study the Cauchy problem for time fractional Schrödinger equations. The existence and a sharp decay estimate for solutions of the given problem in two different spaces are addressed. Some fundamental properties of operators appearing in the solution of the problem are also discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
19. Global weak solutions to a spatio-temporal fractional Landau–Lifshitz–Bloch equation.
- Author
-
Ayouch, Chahid, Essoufi, El-Hassan, and Tilioua, Mouhcine
- Subjects
- *
BLOCH equations , *GALERKIN methods , *SPATIOTEMPORAL processes , *COMMUTATION (Electricity) , *UNIQUENESS (Mathematics) - Abstract
Abstract The present paper deals with global existence of weak solutions of a time-space fractional Landau–Lifshitz–Bloch equation involving the weak Caputo derivative and a fractional Laplacian. We use Faedo–Galerkin method with some commutator estimates in order to prove global existence of weak solutions for the model. The uniqueness is also discussed in a special one dimensional case. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
20. Ground state solutions of Pohoz̆aev type for fractional Choquard equations with general nonlinearities.
- Author
-
Luo, Huxiao
- Subjects
- *
GROUND state (Quantum mechanics) , *NONLINEAR theories , *FRACTIONAL differential equations , *FOURIER transforms , *PSEUDODIFFERENTIAL operators - Abstract
Abstract In this paper, we study the fractional Choquard equation (− Δ) s u + u = (| x | − μ ∗ F (u)) f (u) , in R N , where N ≥ 3 , 0 < s < 1 , 0 < μ < min { N , 4 s } , and f ∈ C (R , R) satisfies the general Berestycki–Lions conditions. Combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz̆aev type for the above equation. The result improves some ones in Shen et al. (2016). [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
21. High-order time stepping Fourier spectral method for multi-dimensional space-fractional reaction–diffusion equations.
- Author
-
Alzahrani, S.S. and Khaliq, A.Q.M.
- Subjects
- *
HEAT equation , *NONLINEAR systems , *NUMERICAL analysis , *FRACTIONAL differential equations , *APPROXIMATION theory - Abstract
Abstract This paper introduces a high-order time stepping scheme, that is based on using Fourier spectral in space and a fourth-order diagonal Padé approximation to the matrix exponential function for solving multi-dimensional space-fractional reaction–diffusion equations. The resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non-linear system at each time step while maintaining the stability of the scheme. The non-locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices. This scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator. It also attains spectral convergence for multiple spatial dimensions. The stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions, which provides an indication of the stability of the method. The convergence analysis is performed empirically to show that the scheme is fourth-order accurate in time, as expected. Numerical experiments on reaction–diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space-fractional reaction–diffusion equations and to validate the effectiveness of the scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
22. Existence results for boundary value problems of nonlinear fractional differential equations
- Author
-
Chai, Guoqing
- Subjects
- *
BOUNDARY value problems , *NONLINEAR differential equations , *FIXED point theory , *MATHEMATICAL mappings , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
Abstract: In this paper, we consider the existence of solutions for the nonlinear fractional differential equation with the boundary value conditions where and are the standard Caputo derivative with . By using the contraction mapping principle and the Schauder fixed point theorem, some existence results are obtained. In addition, Lemma 2.6 in this paper is a valuable tool in seeking solvability of the fractional differential equations. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
23. Invariant analysis and conservation laws for the time fractional [formula omitted]-dimensional Zakharov–Kuznetsov modified equal width equation using Lie group analysis.
- Author
-
Saha Ray, S.
- Subjects
- *
INVARIANTS (Mathematics) , *CONSERVATION laws (Mathematics) , *LIE groups , *FRACTIONAL differential equations , *LAGRANGE equations - Abstract
Abstract In this paper, the invariance properties of the time fractional (2 + 1) -dimensional Zakharov–Kuznetsov modified equal width (ZK-MEW) equation have been investigated using the Lie group analysis method. Lie point symmetries of the time fractional (2 + 1) -dimensional ZK-MEW equation have been derived by using the Lie group analysis method of fractional differential equations. Using the Lie symmetry analysis, the vector fields and the symmetry reduction of this equation are obtained. It is shown that the time fractional (2 + 1) -dimensional ZK-MEW equation can be transformed to an equation with Erdélyi–Kober fractional derivative. Finally using new conservation theorem with formal Lagrangian, the new conserved vectors are well constructed with a detailed derivation, which constitutes the conservation analysis for the time fractional (2 + 1) -dimensional ZK-MEW equation. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
24. Exponential integrator methods for systems of non-linear space-fractional models with super-diffusion processes in pattern formation.
- Author
-
Iyiola, O.S. and Wade, B.A.
- Subjects
- *
FRACTIONAL differential equations , *NONLINEAR theories , *LAPLACIAN operator , *RIESZ spaces , *STOCHASTIC convergence , *ALGORITHMS - Abstract
Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this paper, we propose an exponential integrator method for space fractional models as an attractive and easy-to-code alternative for other existing second-order exponential integrator methods. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. One of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The scheme is established to be second-order convergent; and proven to be robust for nonlinear space fractional reaction–diffusion problems involving non-smooth initial data. Our approach is exhibited by solving a system of two-dimensional problems which exhibits pattern formation and has applications in cell-division. Empirically, super-diffusion processes are displayed by investigating the effect of the fractional power of the underlying Laplacian operator on the pattern formation found in these models. Furthermore, the superiority of our method over competing second order ETD schemes, BDF2 scheme, and IMEX schemes is demonstrated. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
25. On the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity.
- Author
-
Liang, Sihua, Repovš, Dušan, and Zhang, Binlin
- Subjects
- *
SCHRODINGER equation , *ELECTROMAGNETIC fields , *KIRCHHOFF'S theory of diffraction , *NONLINEAR analysis , *FRACTIONAL differential equations - Abstract
In this paper, we consider the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity ε 2 s M ( [ u ] s , A ε 2 ) ( − Δ ) A ε s u + V ( x ) u = | u | 2 s ∗ − 2 u + h ( x , | u | 2 ) u , x ∈ R N , u ( x ) → 0 , as | x | → ∞ , where ( − Δ ) A ε s is the fractional magnetic operator with 0 < s < 1 , 2 s ∗ = 2 N ∕ ( N − 2 s ) , M : R 0 + → R + is a continuous nondecreasing function, V : R N → R 0 + and A : R N → R N are the electric and magnetic potentials, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ε < E ; and (ii) for any m ∗ ∈ N , has m ∗ pairs of solutions if ε < E m ∗ , where E and E m ∗ are sufficiently small positive numbers. Moreover, these solutions u ε → 0 as ε → 0 . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
26. Auxiliary equation method for time-fractional differential equations with conformable derivative.
- Author
-
Akbulut, Arzu and Kaplan, Melike
- Subjects
- *
FRACTIONAL differential equations , *DERIVATIVES (Mathematics) , *NUMERICAL solutions to differential equations , *ORDINARY differential equations , *INTEGERS - Abstract
In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
27. On fractional Schrödinger equation with periodic and asymptotically periodic conditions.
- Author
-
Zhang, Wen, Zhang, Jian, and Mi, Heilong
- Subjects
- *
FRACTIONAL differential equations , *SCHRODINGER equation , *ASYMPTOTIC expansions , *NONLINEAR theories , *LAPLACIAN matrices , *EXISTENCE theorems - Abstract
In this paper, we study the following fractional Schrödinger equation ( − Δ ) s u + V ( x ) u = f ( x , u ) , x ∈ R N , where ( − Δ ) s ( s ∈ ( 0 , 1 ) ) denotes the fractional Laplacian. Under some weaker conditions on the nonlinearity, we obtain the existence of solutions for the above problem in periodic case and asymptotically periodic case via variational methods, respectively. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
28. On the time-fractional Schrödinger equation: Theoretical analysis and numerical solution by matrix Mittag-Leffler functions.
- Author
-
Garrappa, Roberto, Moret, Igor, and Popolizio, Marina
- Subjects
- *
FRACTIONAL differential equations , *COMPUTATIONAL complexity , *DISCRETIZATION methods , *OPERATOR theory , *SCHRODINGER equation , *MATRIX functions - Abstract
This paper presents a deep analysis of a time-dependent Schrödinger equation with fractional time derivative. After the discretization of the spatial operator the equation is reformulated in terms of a special system of fractional differential equations; it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations. The main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations (possibly with high frequency as it is typical with Schrödinger equations) and a persisting memory (as a consequence of the fractional order); moreover, an accurate spatial discretization gives rise to systems of large to very large size, involving a noteworthy computational complexity. By means of a theoretical analysis the exact solution is split into two or three terms (depending on the order of the fractional derivative), thus to face the numerical computation by different and suitably selected methods: direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memory and a step-by-step strategy, in conjunction with matrix function, for the oscillating term. In both cases, Krylov subspace methods are employed for the computation of matrix functions and convergence results are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
29. Hybrid Laplace transform and finite difference methods for pricing American options under complex models.
- Author
-
Ma, Jingtang, Zhou, Zhiqiang, and Cui, Zhenyu
- Subjects
- *
LAPLACE transformation , *FINITE difference method , *ORDINARY differential equations , *FRACTIONAL differential equations , *MARKOV processes - Abstract
In this paper, we propose a hybrid Laplace transform and finite difference method to price (finite-maturity) American options, which is applicable to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump–diffusion (HEJD), Markov regime switching models, and the finite moment log stable (FMLS) models. We first apply Laplace transforms to free boundary partial differential equations (PDEs) or fractional partial differential equations (FPDEs) governing the American option prices with respect to time, and obtain second order ordinary differential equations (ODEs) or fractional differential equations (FDEs) with free boundary, which is named as the early exercise boundary in the American option pricing. Then, we develop an iterative algorithm based on finite difference methods to solve the ODEs or FDEs together with the unknown free boundary values in the Laplace space. Both the early exercise boundary and the prices of American options are recovered through inverse Laplace transforms. Numerical examples demonstrate the accuracy and efficiency of the method in CEV, HEJD, Markov regime switching models and the FMLS models. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
30. Multi-dimensional [formula omitted]-fractional diffusion–wave equation and some properties of its fundamental solution.
- Author
-
Boyadjiev, Lyubomir and Luchko, Yuri
- Subjects
- *
WAVE equation , *FRACTIONAL differential equations , *CONTINUOUS functions , *FRACTIONAL calculus , *RANDOM walk generating functions , *PROBABILITY density function - Abstract
In this paper, a multi-dimensional α -fractional diffusion–wave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order α / 2 and the Riesz space-fractional derivative of the order α so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the α -fractional diffusion–wave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the α -fractional diffusion–wave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
31. Oscillation criteria for impulsive partial fractional differential equations.
- Author
-
Raheem, A. and Maqbul, Md.
- Subjects
- *
FRACTIONAL differential equations , *OSCILLATION theory of difference equations , *DIRICHLET problem - Abstract
In this paper, we established some sufficient conditions for oscillation of solutions of a class of impulsive partial fractional differential equations with forcing term subject to Robin and Dirichlet boundary conditions by using differential inequality method. As an application, we included an example to illustrate the main result. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
32. High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation.
- Author
-
Liu, Yang, Zhang, Min, Li, Hong, and Li, Jichun
- Subjects
- *
DISCONTINUOUS functions , *GALERKIN methods , *APPROXIMATION theory , *FRACTIONAL differential equations , *CAPUTO fractional derivatives - Abstract
In this paper, a high-order local discontinuous Galerkin (LDG) method combined with weighted and shifted Grünwald difference (WSGD) approximation is developed and discussed for a Caputo time-fractional subdiffusion equation. The time fractional derivative of order α , 0 < α < 1 , is approximated by a third-order method based on the idea of WSGD operator, while the spatial operator is approximated by the LDG method. Some useful lemmas are first introduced and proved, then the analysis of stability and optimal error estimate O ( Δ t 3 + h k + 1 ) are obtained for the LDG method. Extensive numerical results using P k , k = 0 , 1 , 2 , 3 , elements are presented to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
33. Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems.
- Author
-
Zhou, Yong and Zhang, Lu
- Subjects
- *
MULTIPLICITY (Mathematics) , *EXISTENCE theorems , *FRACTIONAL differential equations , *HAMILTONIAN systems , *LIOUVILLE'S theorem - Abstract
In this paper, by the critical point theory, we consider the existence and multiplicity of solutions for the following fractional differential equation t D ∞ α ( − ∞ D t α u ( t ) ) + L ( t ) u ( t ) = ∇ W ( t , u ( t ) ) , t ∈ R , where α ∈ ( 1 2 , 1 ] , − ∞ D t α and t D ∞ α are left and right Liouville–Weyl fractional derivatives of order α on the whole axis R respectively, u ∈ R n , L ( t ) is positive definite symmetric matrix for all t ∈ R and W : R × R n → R is a suitably chosen function. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
34. Numerical schemes of the time tempered fractional Feynman–Kac equation.
- Author
-
Deng, W.H. and Zhang, Z.J.
- Subjects
- *
NUMERICAL analysis , *FRACTIONAL differential equations , *FEYNMAN integrals , *COMPUTATIONAL complexity , *FINITE element method , *FINITE difference method - Abstract
This paper focuses on providing the computation methods for the backward time tempered fractional Feynman–Kac equation, being one of the models recently proposed in Wu et al. (2016). The discretization for the tempered fractional substantial derivative is derived, and the corresponding finite difference and finite element schemes are designed with well established stability and convergence. The performed numerical experiments show the effectiveness of the presented schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
35. Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis.
- Author
-
Yang, Yin, Chen, Yanping, Huang, Yunqing, and Wei, Huayi
- Subjects
- *
COLLOCATION methods , *FRACTIONAL differential equations , *WAVE equation , *STOCHASTIC convergence , *INTERPOLATION algorithms - Abstract
In this paper, we consider the numerical solution of the time-fractional diffusion-wave equation. Essentially, the time fractional diffusion-wave equation differs from the standard diffusion-wave equation in the time derivative term. We propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of Jacobi interpolation polynomial for this equation. The convergence of the method is rigorously established. Numerical tests are carried out to confirm the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
36. Global existence and asymptotic behavior for a time fractional reaction–diffusion system.
- Author
-
Alsaedi, Ahmed, Kirane, Mokhtar, and Lassoued, Rafika
- Subjects
- *
EXISTENCE theorems , *ASYMPTOTIC efficiencies , *FRACTIONAL differential equations , *MATHEMATICAL regularization , *LINEAR equations - Abstract
This paper is concerned with the existence of global in time solutions of a time fractional reaction–diffusion system with time fractional derivatives. Furthermore, the large time behavior of bounded solutions is investigated. Our method of proof relies on a maximal regularity result for fractional linear reaction–diffusion equations that has been derived by Bajlekova (2001). [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
37. Weak solutions of the time-fractional Navier–Stokes equations and optimal control.
- Author
-
Zhou, Yong and Peng, Li
- Subjects
- *
NAVIER-Stokes equations , *FRACTIONAL differential equations , *OPTIMAL control theory , *FRACTAL analysis , *GALERKIN methods - Abstract
In this paper, we deal with the Navier–Stokes equations with the time-fractional derivative of order α ∈ ( 0 , 1 ) , which can be used to simulate anomalous diffusion in fractal media. We firstly give the concept of the weak solutions and establish the existence criterion of weak solutions by means of Galerkin approximations in the case that the dimension n ≤ 4 . Moreover, a complete proof of the uniqueness is given when n = 2 . At last we give a sufficient condition of optimal control pairs. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
38. Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem.
- Author
-
Li, Zhiyuan, Luchko, Yuri, and Yamamoto, Masahiro
- Subjects
- *
FRACTIONAL differential equations , *INVERSE problems , *CAPUTO fractional derivatives , *EIGENFUNCTIONS , *LAPLACE transformation - Abstract
This paper deals with an initial–boundary value problem for a distributed order time-fractional diffusion equation with the fractional derivative in the Caputo sense. The method of the eigenfunctions expansion in combination with the Laplace transform is first employed to prove the uniqueness and existence of the solution to the initial–boundary value problem and then to show its analyticity in time. As an application of the analyticity of the solution, a uniqueness result for an important inverse problem of determination of the weight function in the distributed order derivative contained in the time-fractional diffusion equation from one interior point observation of its solution is obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
39. Element free Galerkin approach based on the reproducing kernel particle method for solving 2D fractional Tricomi-type equation with Robin boundary condition.
- Author
-
Dehghan, Mehdi and Abbaszadeh, Mostafa
- Subjects
- *
GALERKIN methods , *BOUNDARY value problems , *FRACTIONAL differential equations , *REPRODUCING kernel (Mathematics) , *FINITE element method - Abstract
The traditional element free Galerkin (EFG) approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares (MLS) approximation. In the current paper, we propose a new version of the EFG method based on the shape functions of reproducing kernel particle method (RKPM). In other words, based on the developed approach in Han and Meng (2001) the fractional Tricomi-type equation will be solved using the new technique. The fractional derivative has been introduced in the Caputo’s sense and is approximated by a finite difference plan of order O ( τ 3 − α ) , 1 < α < 2 . We use the EFG-RKPM to discrete the spatial direction. We illustrate some numerical results on non-rectangular domains. The unconditional stability and convergence of the new technique have been proved. Numerical examples display the theoretical results and the efficiency of the proposed approach. Also, the numerical results are compared with the finite element method (FEM) and EFG-MLS procedure. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
40. Two-dimensional Legendre wavelet method for travelling wave solutions of time-fractional generalized seventh order KdV equation.
- Author
-
Saha Ray, S. and Gupta, A.K.
- Subjects
- *
LEGENDRE'S functions , *WAVELETS (Mathematics) , *FRACTIONAL differential equations , *FRACTIONAL integrals , *NUMERICAL analysis , *HOMOTOPY theory - Abstract
In this paper, a new method based on the Legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve time-fractional seventh order KdV (sKdV) equation. Two-dimensional Legendre wavelet method is applied to compute the numerical solution of nonlinear time-fractional sKdV equation. The approximate solutions of nonlinear time fractional sKdV equation thus obtained by Legendre wavelet method are compared with the exact solutions as well as homotopy analysis method (HAM). The present scheme is very simple, effective and convenient for obtaining numerical solution of the sKdV equation. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
41. Identification and regularization for unknown source for a time-fractional diffusion equation.
- Author
-
Tuan, Nguyen Huy, Kirane, Mokhtar, Hoan, Luu Vu Cam, and Long, Le Dinh
- Subjects
- *
MATHEMATICAL regularization , *IDENTIFICATION (Statistics) , *FRACTIONAL differential equations , *INVERSE problems , *FOURIER transforms - Abstract
In this paper, we consider the inverse problem of determining a source in a time fractional diffusion equation where data are given at a fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate. Two numerical examples are presented to illustrate the validity and effectiveness of our method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
42. Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations.
- Author
-
Bhrawy, A.H. and Zaky, M.A.
- Subjects
- *
SCHRODINGER equation , *FRACTIONAL differential equations , *FRACTIONAL quantum mechanics , *ORDINARY differential equations , *NONLINEAR partial differential operators - Abstract
As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena. In this paper, we develop an exponentially accurate Jacobi–Gauss–Lobatto collocation (J–GL-C) method to solve the variable-order fractional Schrödinger equations in one dimension (1D) and two dimensions (2D). In this method, the aforementioned problem is reduced to a system of ordinary differential equations (ODEs) in the time variable. As a result, we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations, one based on the implicit Runge–Kutta (IRK) method of fourth order and the other based on Jacobi–Gauss–Radau collocation (J–GR-C) method. The validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1D and 2D. The convergence of the methods is graphically analyzed. The results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable-order nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
43. A fast second-order accurate method for a two-sided space-fractional diffusion equation with variable coefficients.
- Author
-
Feng, L.B., Zhuang, P., Liu, F., Turner, I., Anh, V., and Li, J.
- Subjects
- *
FRACTIONAL differential equations , *CRANK-nicolson method , *PARTITIONS (Mathematics) , *FINITE difference method , *LIOUVILLE'S theorem - Abstract
In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficients on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann–Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank–Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the second-order accuracy of O ( τ 2 + h 2 ) . Furthermore, we develop a fast accurate iterative method for the Crank–Nicolson scheme, which only requires storage of O ( m ) and computational cost of O ( m log m ) while retaining the same accuracy and approximation property as Gauss elimination, where m = 1 / h is the partition number in space direction. Finally, several numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
44. Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid.
- Author
-
Ming, Chunying, Liu, Fawang, Zheng, Liancun, Turner, Ian, and Anh, Vo
- Subjects
- *
ANALYTICAL solutions , *FRACTIONAL differential equations , *VISCOELASTIC materials , *FLUID dynamics , *SEPARATION of variables , *SHEARING force , *ORDINARY differential equations - Abstract
This paper derives analytical solutions for a class of new multi-term fractional-order partial differential equations, which include the terms for spatial diffusion, time-fractional diffusion (multi-term) and reaction. These models can be used to describe the nonlinear relationship between the shear stress and shear rate of generalized viscoelastic Oldroyd-B fluid and Burgers fluid. By using a modified separation of variables method, the governing fractional-order partial differential equations are transformed into a set of fractional-order ordinary differential equations. Mikusiński-type operational calculus is then employed to obtain the exact solutions of the linear fractional ordinary differential equations with constant coefficients. The solutions are expressed in terms of multivariate Mittag-Leffler functions. Different situations for the unsteady flows of generalized Oldroyd-B fluid and Burgers fluid due to a moving plate are considered via examples. Integral representations of the solutions are presented. It is shown that the presented results reduce to the corresponding results for classical Navier–Stokes, Oldroyd-B, Maxwell and second-grade fluids as special cases. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
45. Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well.
- Author
-
Yang, Liu and Liu, Zhisu
- Subjects
- *
FRACTIONAL differential equations , *MULTIPLICITY (Mathematics) , *SCHRODINGER equation , *PERTURBATION theory , *POTENTIAL well , *NUMERICAL solutions to differential equations - Abstract
In this paper, we investigate the following fractional Schrödinger equation with sublinear perturbation and steep potential well { ( − △ ) s u + λ V ( x ) u = f ( x , u ) + α ( x ) | u | ν − 2 u in R N , u ∈ H s ( R N ) , where 0 < s < 1 , 2 s < N , λ > 0 , 1 < ν < 2 , f ∈ C ( R N × R ) is of subcritical growth. By using variational methods, we prove that such a class of equations possess at least two nontrivial solutions. Moreover, the phenomenon of concentration of solutions is explored as well. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
46. A stable numerical method for multidimensional time fractional Schrödinger equations.
- Author
-
Hicdurmaz, Betul and Ashyralyev, Allaberen
- Subjects
- *
FRACTIONAL differential equations , *SCHRODINGER equation , *STABILITY theory , *MATHEMATICAL transformations , *NUMERICAL solutions to boundary value problems - Abstract
In this paper, the stability analysis is presented for a first order difference scheme applied to a nonhomogeneous time fractional Schrödinger differential equation. Based on the z-transform method, stability theorems are proved for the abstract case. The stability results are applied on initial–boundary value problems for multidimensional time fractional Schrödinger differential equations. Theoretical findings are validated by numerical experiments on one and two-dimensional time fractional Schrödinger differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
47. Existence and approximation of solution to neutral fractional differential equation with nonlocal conditions.
- Author
-
Chadha, Alka and Pandey, D.N.
- Subjects
- *
FRACTIONAL differential equations , *EXISTENCE theorems , *APPROXIMATION theory , *HILBERT space , *INTEGRAL equations , *FRACTIONAL integrals - Abstract
This paper is concerned with the approximation of the solution for neutral fractional differential equation with nonlocal conditions in an arbitrary separable Hilbert space H . We study an associated integral equation and then, consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal neutral fractional integral equation onto finite dimensional space. The sufficient condition for the existence and uniqueness of solutions to every approximate integral equation is derived by using analytic semigroup and Banach fixed point theorem. We demonstrate convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Moreover, we consider the Faedo–Galerkin approximations of the solution and demonstrate some convergence results. An example is also provided to illustrate the discussed abstract theory. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
48. Solving the fractional nonlinear Bloch system using the multi-step generalized differential transform method.
- Author
-
Abuteen, Eman, Momani, Shaher, and Alawneh, Ahmad
- Subjects
- *
NONLINEAR systems , *PROBLEM solving , *BLOCH equations , *NUCLEAR magnetic resonance , *NUMERICAL solutions to differential equations - Abstract
In this paper, the multi-step differential transform method is employed for the first time to solve a time-fractional nonlinear Bloch system. The nonlinear Bloch equation is known to govern the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance (NMR). This nonlinear Bloch equation is formed from a system of nonlinear ordinary differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The results obtained are in good agreement with the ones in the open literature and it is shown that the technique introduced here is robust, efficient and easy to implement. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
49. The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations.
- Author
-
Duan, Jun-Sheng, Chaolu, Temuer, Rach, Randolph, and Lu, Lei
- Subjects
- *
MATHEMATICAL decomposition , *STOCHASTIC convergence , *ACCELERATION (Mechanics) , *FRACTIONAL calculus , *NONLINEAR differential equations , *APPROXIMATION theory - Abstract
In this paper, we present the Adomian decomposition method and its modifications combined with convergence acceleration techniques, such as the diagonal Padé approximants and the iterated Shanks transforms, to solve nonlinear fractional ordinary differential equations. Two nonlinear numeric examples demonstrate that either the diagonal Padé approximants or the iterated Shanks transforms can efficiently extend the effective convergence region of the decomposition series solution. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
50. Nonlinear impulsive problems for fractional differential equations and Ulam stability
- Author
-
Wang, JinRong, Zhou, Yong, and Fec˘kan, Michal
- Subjects
- *
NONLINEAR differential equations , *IMPULSIVE differential equations , *FRACTIONAL calculus , *CAUCHY problem , *BOUNDARY value problems , *FIXED point theory - Abstract
Abstract: In this paper, the first purpose is treating Cauchy problems and boundary value problems for nonlinear impulsive differential equations with Caputo fractional derivative. We introduce the concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively. By using a new fixed point theorem, we obtain many new existence, uniqueness and data dependence results of solutions via some generalized singular Gronwall inequalities. The second purpose is discussing Ulam stability for impulsive fractional differential equations. Some new concepts in stability of impulsive fractional differential equations are offered from different perspectives. Some applications of our results are also provided. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.