98 results on '"Caputo derivative"'
Search Results
2. A Novel Semi-Analytical Scheme to Deal with Fractional Partial Differential Equations (PDEs) of Variable-Order.
- Author
-
Kheybari, Samad, Alizadeh, Farzaneh, Darvishi, Mohammad Taghi, Hosseini, Kamyar, and Hincal, Evren
- Subjects
- *
CHEBYSHEV polynomials , *COLLOCATION methods , *PARTIAL differential equations , *FRACTIONAL differential equations , *PROBLEM solving - Abstract
This article introduces a new numerical algorithm dedicated to solving the most general form of variable-order fractional partial differential models. Both the time and spatial order of derivatives are considered as non-constant values. A combination of the shifted Chebyshev polynomials is used to approximate the solution of such equations. The coefficients of this combination are considered a function of time, and they are obtained using the collocation method. The theoretical aspects of the method are investigated, and then by solving some problems, the efficiency of the method is presented. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations.
- Author
-
Kumar, Saurabh, Gupta, Vikas, and Zeidan, Dia
- Subjects
- *
MATRICES (Mathematics) , *PARTIAL differential equations , *ALGEBRAIC equations , *NONLINEAR differential equations , *NEWTON-Raphson method , *COLLOCATION methods - Abstract
In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. An efficient fourth order Hermite spline collocation method for time fractional diffusion equation describing anomalous diffusion in two space variables.
- Author
-
Priyanka, Sahani, Saroj, and Arora, Shelly
- Subjects
HEAT equation ,SPLINES ,ALGEBRAIC equations ,SOBOLEV spaces ,COLLOCATION (Linguistics) ,COLLOCATION methods ,SPLINE theory - Abstract
Anomalous diffusion of particles in fluids is better described by the fractional diffusion models. A robust hybrid numerical algorithm for a two-dimensional time fractional diffusion equation with the source term is presented. The well-known L1 scheme is considered for semi-discretization of the diffusion equation. To interpolate the semi-discretized equation, orthogonal collocation with bi-quintic Hermite splines as the basis is chosen for the smooth solution. Quintic Hermite splines interpolate the solution as well as its first and second order derivatives. The technique reduces the proposed problem to an algebraic system of equations. Stability analysis of the implicit scheme is studied using H ~ 1 m -norm defined in Sobolev space. The optimal order of convergence is found to be of order O (h 4) in spatial direction and is of order O (Δ t) 2 - α in the temporal direction where h is the step size in space direction and Δ t is the step size in time direction and α is the fractional order of the derivative. Numerical illustrations have been presented to discuss the applicability of the proposed hybrid numerical technique to the problems having fractional order derivative. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Collocation-based numerical simulation of fractional order Allen–Cahn equation.
- Author
-
Choudhary, Renu and Kumar, Devendra
- Subjects
- *
FINITE differences , *COMPUTER simulation , *NUMERICAL analysis , *EQUATIONS , *COLLOCATION methods - Abstract
This article looks for a reliable numerical technique to solve the Allen–Cahn equation using the Caputo time-fractional derivative. The fractional derivative semi-discretization approach using finite differences of the second order is shown first. The cubic B-spline collocation method is used to get a full discretization. We prove the conditional stability and convergence of the suggested approach. The technique's effectiveness is demonstrated with numerical examples using two test problems. Numerical analysis confirms the approach's efficiency and the method's continued correctness. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. Spectral collocation methods for fractional multipantograph delay differential equations*.
- Author
-
Shi, Xiulian, Wang, Keyan, and Sun, Hui
- Subjects
- *
COLLOCATION methods , *VOLTERRA equations , *INTEGRAL equations , *FRACTIONAL differential equations , *DELAY differential equations - Abstract
In this paper, we propose and analyze a spectral collocation method for the numerical solutions of fractional multipantograph delay differential equations. The fractional derivatives are described in the Caputo sense. We present that some suitable variable transformations can convert the equations to a Volterra integral equation defined on the standard interval [−1, 1]. Then the Jacobi–Gauss points are used as collocation nodes, and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence analysis of the proposed method is investigated in the infinity norm and weighted L2 norm. To perform the numerical simulations, some test examples are investigated, and numerical results are presented. Further, we provide the comparative study of the proposed method with some existing numerical methods. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
7. Numerical Solution of Advection–Diffusion Equation of Fractional Order Using Chebyshev Collocation Method.
- Author
-
Ali Shah, Farman, Kamran, Boulila, Wadii, Koubaa, Anis, and Mlaiki, Nabil
- Subjects
- *
NUMERICAL solutions to equations , *ADVECTION-diffusion equations , *COLLOCATION methods , *DISCRETIZATION methods - Abstract
This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The primary motivation for using the Laplace transform is its ability to avoid the classical time-stepping scheme and overcome the adverse effects of time steps on numerical accuracy and stability. Our method comprises three primary steps: (i) reducing the time-dependent equation to a time-independent equation via the Laplace transform, (ii) employing the Chebyshev spectral collocation method to approximate the solution of the transformed equation, and (iii) numerically inverting the Laplace transform. We discuss the convergence and stability of the method and assess its accuracy and efficiency by solving various problems in two dimensions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
8. Collocation-Based Approximation for a Time-Fractional Sub-Diffusion Model.
- Author
-
Lätt, Kaido, Pedas, Arvet, Soots, Hanna Britt, and Vikerpuur, Mikk
- Subjects
- *
FRACTIONAL differential equations , *COLLOCATION methods , *ORDINARY differential equations , *INTEGRAL equations - Abstract
We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
9. A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations.
- Author
-
Moi, Sandip, Biswas, Suvankar, and Sarkar, Smita Pal
- Subjects
- *
VOLTERRA equations , *COLLOCATION methods , *NUMERICAL solutions to equations , *NUMERICAL analysis , *INTEGRO-differential equations , *INTEGRAL operators - Abstract
A linear fractional-order weakly singular fuzzy Volterra integro-differential equation has been examined. In this case, the Caputo fractional-order derivative has been considered. A new type of spectral collocation method based on the Lagrange interpolation basis polynomial has been studied and modified for the equation. In the spectral collocation technique, it is necessary to choose collocation points to find the numerical solution of the equation. We have chosen the collocation points based on the Chebyshev extreme points or Gauss–Lobatto–Chebyshev points of order N. We have used the fractional Gauss–Jacobi quadrature method to approximate the fractional integral terms of the proposed equation. Also, the integral operators have been approximated by the Gauss quadrature rule. A theorem has been given to demonstrate that there exists a unique solution for the proposed equation. In addition, Banach's fixed point principle has been applied in the proof of the existence and uniqueness theorem. The convergence analysis of the proposed numerical technique is given in the form of some lemmas and theorems. Some numerical experiments have been performed to verify the proposed method. Five different kinds of errors have been computed and compared to do the error analysis. Also, these kinds of error analysis have been examined by analyzing the result in the form of different graphs and tables. The numerical results of the proposed technique have been compared with an existing method, Adomian decomposition method (ADM). [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
10. Collocation Based Approximations for a Class of Fractional Boundary Value Problems.
- Author
-
Soots, Hanna Britt, Latt, Kaido, and Pedas, Arvet
- Subjects
- *
BOUNDARY value problems , *CAPUTO fractional derivatives , *INTEGRAL equations , *COLLOCATION methods , *INTEGRO-differential equations - Abstract
A boundary value problem for fractional integro-differential equations with weakly singular kernels is considered. The problem is reformulated as an integral equation of the second kind with respect to z = Dα Capy, the Caputo fractional derivative of y of order α, with 1 < α < 2, where y is the solution of the original problem. Using this reformulation, the regularity properties of both y and its Caputo derivative z are studied. Based on this information a piecewise polynomial collocation method is developed for finding an approximate solution zN of the reformulated problem. Using zN, an approximation yN for y is constructed and a detailed convergence analysis of the proposed method is given. In particular, the attainable order of convergence of the proposed method for appropriate values of grid and collocation parameters is established. To illustrate the performance of our approach, results of some numerical experiments are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
11. Numerical method for solving two‐dimensional of the space and space–time fractional coupled reaction‐diffusion equations.
- Author
-
Hadhoud, Adel R., Rageh, Abdulqawi A. M., and Agarwal, Praveen
- Subjects
- *
COLLOCATION methods , *RIESZ spaces , *REACTION-diffusion equations , *ORDINARY differential equations , *ALGEBRAIC equations , *SPACETIME , *FRACTIONAL differential equations - Abstract
This paper proposes the shifted Legendre Gauss–Lobatto collocation (SL‐GLC) scheme to solve two‐dimensional space‐fractional coupled reaction–diffusion equations (SFCRDEs). The proposed method is implemented by expressing the function and its spatial fractional derivatives as a finite expansion of shifted Legendre polynomials. Then the expansion coefficients are determined by reducing the SFCRDEs with their initial and boundary conditions to a system of ordinary differential equations for these coefficients. Subsequently, we applied the proposed method to discretize the temporal and spatial variables to convert the two‐dimensional spacetime fractional coupled reaction–diffusion equations (STFCRDEs) to a system of algebraic equations. Some results regarding the error estimation are obtained. Several examples are discussed to validate the capability and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
12. A Numerical Study Based on Haar Wavelet Collocation Methods of Fractional-Order Antidotal Computer Virus Model.
- Author
-
Zarin, Rahat, Khaliq, Hammad, Khan, Amir, Ahmed, Iftikhar, and Humphries, Usa Wannasingha
- Subjects
- *
COMPUTER viruses , *FIXED point theory , *CONTROLLER area network (Computer network) , *COMPUTER simulation , *COLLOCATION methods , *VIRAL transmission - Abstract
Computer networks can be alerted to possible viruses by using kill signals, which reduces the risk of virus spreading. To analyze the effect of kill signal nodes on virus propagation, we use a fractional-order SIRA model using Caputo derivatives. In our model, we show how a computer virus spreads in a vulnerable system and how it is countered by an antidote. Using the Caputo operator, we fractionalized the model after examining it in deterministic form. The fixed point theory of Schauder and Banach is applied to the model under consideration to determine whether there exists at least one solution and whether the solution is unique. In order to calculate the approximate solution to the model, a general numerical algorithm is established primarily based on Haar collocations and Broyden's method. In addition to being mathematically fast, the proposed method is also straightforward and applicable to different mathematical models. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
13. Numerical Scheme with Convergence Analysis and Error Estimate for Variable Order Weakly Singular Integro-Differential Equation.
- Author
-
Yadav, Poonam, Singh, B. P., Alikhanov, Anatoly A., and Singh, Vineet Kumar
- Subjects
MATRICES (Mathematics) ,INTEGRO-differential equations ,ALGEBRAIC equations ,DIFFERENTIAL equations ,COLLOCATION methods - Abstract
This paper represents a new application of Legendre wavelet and interpolating scaling function to discuss the approximate solution of variable order integro-differential equation having weakly singular kernel. So far, this technique has been used to solve variable order integro differential equation. In this paper, it is extended to solve variable order integro differential equation with weakly singular kernel. For this purpose, we derive the operational matrices of Legendre wavelets and interpolating scaling function. The resulting operational matrices along with the collocation method transform the original problem into a system of algebraic equation. By solving this system, the approximate solution is obtained. The convergence and error estimate of the presented method have been rigorously investigated. We also discuss the numerical stability of the method. The numerical result of some inclusive examples has been provided through a table and graph for both basis functions that support the robustness and desired precision of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
14. Second-order convergent scheme for time-fractional partial differential equations with a delay in time.
- Author
-
Choudhary, Renu, Kumar, Devendra, and Singh, Satpal
- Subjects
- *
DELAY differential equations , *PARTIAL differential equations , *TRANSPORT equation , *COLLOCATION methods - Abstract
This paper aims to construct an effective numerical scheme to solve convection-reaction-diffusion problems consisting of time-fractional derivative and delay in time. First, the semi-discretization process is given for the fractional derivative using a finite-difference scheme with second-order accuracy. Then the cubic B-spline collocation method is employed to get the full discretization. We prove that the suggested scheme is conditionally stable and convergent. Two numerical examples are incorporated to verify the effectiveness of the algorithm. Numerical investigations support the proposed method's accuracy and show that the method solves the problem efficiently. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
15. BERNSTEIN COLLOCATION TECHNIQUE FOR VOLTERRA-FREDHOLM FRACTIONAL ORDER INTEGRO-DIFFERENTIAL EQUATIONS.
- Author
-
OYEDEPO, T., AJILEYE, G., AYINDE, A. M., and OTAIDE, I. J.
- Subjects
ALGEBRAIC equations ,MATRIX inversion ,COLLOCATION methods ,INTEGRO-differential equations ,LINEAR equations - Abstract
In this study, we solve Fractional Volterra-Fredholm Integro-Differ ential Equations (FVFIDEs) using the Bernstein Collocation Technique (BCT). The approach breaks the problem down into a set of linear algebraic equations, which are then resolved by matrix inversion to get the unknown constants. The accuracy and effectiveness of the procedure are demonstrated using numerical examples in tables and figures. The outcomes demonstrate that the strategy worked better in terms of increasing accuracy and necessitating less strenuous labour. [ABSTRACT FROM AUTHOR]
- Published
- 2023
16. Numerical treatments of the nonlinear coupled time‐fractional Schrödinger equations.
- Author
-
Hadhoud, Adel R., Agarwal, Praveen, and Rageh, Abdulqawi A. M.
- Subjects
- *
CAPUTO fractional derivatives , *SCHRODINGER equation , *COLLOCATION methods , *NONLINEAR Schrodinger equation , *DISCRETIZATION methods - Abstract
In this article, we focused on solving numerically the coupled nonlinear Schrödinger equations (NLFSEs) with Caputo fractional derivative in time. The discrete schemes are constructed by using the trigonometric B‐spline collocation method and the non‐polynomial B‐spline method for space discretization respectively, while the L1‐formula is applied in time discretization. The Von Neumann approach is applied to examine the stability of the proposed methods. For validating the accuracy and efficiency of the presented schemes, numerical tests are compared with the exact solution. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
17. Compatibility of the Paraskevopoulos's algorithm with operational matrices of Vieta–Lucas polynomials and applications.
- Author
-
Talib, Imran, Noor, Zulfiqar Ahmad, Hammouch, Zakia, and Khalil, Hammad
- Subjects
- *
NONLINEAR differential equations , *POLYNOMIALS , *ORDINARY differential equations , *COLLOCATION methods , *JACOBI method , *FRACTIONAL differential equations - Abstract
In this study, the numerically stable operational matrices are proposed to approximate the Caputo fractional-order derivatives by introducing an algorithm. The proposed operational matrices are named fractional Vieta–Lucas differentiation matrices that are constructed by using the basis of shifted Vieta–Lucas polynomials (VLPs) and Caputo-fractional derivatives. In addition, we numerically solve the multi-order linear and nonlinear fractional-order differential equations by introducing a new numerical algorithm that is based on Paraskevopoulos's algorithm together with the newly proposed operational matrices of shifted VLPs. The applicability of Paraskevopoulos's algorithm was previously studied with the Adomian decomposition technique to solve fractional-order ordinary differential equations. We extend its applicability to the operational matrices technique. However, to the best of our knowledge, no previous study has reported that discusses the applicability of Paraskevopoulos's algorithm with the operational matrices of shifted VLPs. To demonstrate the advantages of the newly proposed numerical algorithm, the multi-order linear and nonlinear Caputo fractional-order differential equations are solved numerically. The solutions of the first, third, fourth, and fifth examples obtained by using the proposed algorithm are compared with the solutions obtained otherwise by using various numerical approaches including stochastic approach, Taylor matrix method, Bessel collocation method, shifted Jacobi collocation method, spectral Tau method, and Chelyshkov collocation method. It is shown that the proposed numerical algorithm and the fractional Vieta–Lucas differentiation matrices are highly efficient in solving all the aforementioned examples. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
18. A new efficient algorithm based on fifth-kind Chebyshev polynomials for solving multi-term variable-order time-fractional diffusion-wave equation.
- Author
-
Sadri, Khadijeh and Aminikhah, Hossein
- Subjects
- *
CHEBYSHEV polynomials , *ALGEBRAIC equations , *SOBOLEV spaces , *EQUATIONS , *OPERATOR equations , *COLLOCATION methods - Abstract
An algorithm based on a class of the Chebyshev polynomials family called the fifth-kind Chebyshev polynomials (FCPs) is introduced to solve the multi-term variable-order time-fractional diffusion-wave equation (MVTFD-WE). Appeared fractional derivative operators in these equations are of the Caputo type. Coupling FCPs and the collocation method leads to reduce the MVTFD-WE to a system of algebraic equations. The convergence of the proposed scheme is investigated in a weighted Sobolev space via obtaining error bounds for approximate solutions which shows the method error tends to zero if the number of terms of the series solution is selected sufficiently large. The applicability and efficiency of the suggested method are examined through several illustrative examples. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
19. Numerical study of variable order model arising in chemical processes using operational matrix and collocation method.
- Author
-
Kashif, Mohd, Singh, Manpal, Som, Tanmoy, and Craciun, Eduard-Marius
- Subjects
CHEMICAL models ,NEWTON-Raphson method ,CHEMICAL processes ,COLLOCATION methods ,CAPUTO fractional derivatives ,NONLINEAR equations - Abstract
This article introduces the fractional variable order (VO) Gray–Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton's iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray–Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem. • Numerical solution of generalized Gray–Scott model using a new operational matrix and spectral collocation method. • Validation of efficiency and effectiveness of the proposed method by applying it to numerical examples. • Discussed the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. An efficient numerical method for nonlinear fractional differential equations based on the generalized Mittag‐Leffler functions and Lagrange polynomials.
- Author
-
Li, Yu and Zhang, Yanming
- Subjects
- *
NONLINEAR differential equations , *COLLOCATION methods , *POLYNOMIALS , *INTEGRAL equations , *FRACTIONAL differential equations - Abstract
In this paper, an efficient numerical method is developed for solving a class of nonlinear fractional differential equations. The main idea is to transform the nonlinear fractional differential equations into a system of integral equations involved the generalized Mittag‐Leffler functions, and then to discretize the integral equations by the technique of exponential integrators and collocation methods with the polynomials of Lagrange basis. The convergence of this method is proven. The linear stability analysis of this method is carried out, and the stability region is derived. Finally, numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
21. On numerical simulations of time fractional Phi-four equation using Caputo derivative.
- Author
-
Kamran, Mohsin, Majeed, Abdul, and Li, Jing
- Subjects
PARTIAL differential equations ,COMPUTER simulation ,NUCLEAR physics ,COLLOCATION methods ,EQUATIONS - Abstract
The fractional model which played an essential role in nuclear and particle physics used to describe the nuclear element interaction is the Phi-four model. This manuscript aims to scrutinize the new numerical solution of the nonlinear time fractional Phi-four equation subject to nonhomogeneous initial-boundary conditions by means of cubic-B-spline collocation method (CBSCM). The main advantage of cubic B-spline method over existing techniques is that it efficiently provides a piecewise-continuous, closed form solution and it is simpler and easy to apply to many problems involving partial differential equations. In this approach the fractional differential equation is converted into system of equations. The non-integer derivative " α " is considered in Caputo sense. The discretization of Caputo derivative is done using L1 formula, while B-spline basis functions are used for the interpolation of spatial derivative. The applicability of the proposed scheme is examined on two test problems. The influence of different parameters is studied and captured graphically and numerically. The proposed scheme is proved to be unconditionally stable. Moreover, the error norms are computed to quantify the accuracy. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
22. A new operational matrix based on Müntz–Legendre polynomials for solving distributed order fractional differential equations.
- Author
-
Pourbabaee, Marzieh and Saadatmandi, Abbas
- Subjects
- *
ALGEBRAIC equations , *POLYNOMIALS , *LINEAR differential equations , *COLLOCATION methods , *NONLINEAR equations , *FRACTIONAL differential equations , *LINEAR systems - Abstract
Our main aim in this work is to find the operational matrix of fractional derivative and the operational matrix of distributed order fractional derivative for the Müntz–Legendre polynomials (MLPs). The operational matrix approach with the tau method or collocation method is applied to reduce the solution of the linear/nonlinear distributed order fractional differential equations (DFDEs) to a system of linear/nonlinear algebraic equations. Moreover, seven numerical examples are included to show the validity and applicability of the suggested methods. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
23. Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel.
- Author
-
Sadri, Khadijeh, Hosseini, Kamyar, Baleanu, Dumitru, Ahmadian, Ali, and Salahshour, Soheil
- Subjects
- *
ALGEBRAIC equations , *FUNCTIONAL equations , *LINEAR equations , *INTEGRO-differential equations , *COLLOCATION methods , *CHEBYSHEV polynomials , *LINEAR systems - Abstract
The shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
24. Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations.
- Author
-
Pourbabaee, Marzieh and Saadatmandi, Abbas
- Subjects
CHEBYSHEV polynomials ,FRACTIONAL differential equations ,COLLOCATION methods ,ALGEBRA ,NONLINEAR equations - Abstract
This work presents a new approximation approach to solve the linear/nonlinear distributed order fractional differential equations using the Chebyshev polynomials. Here, we use the Chebyshev polynomials combined with the idea of the collocation method for converting the distributed order fractional differential equation into a system of linear/nonlinear algebraic equations. Also, fractional differential equations with initial conditions can be solved by the present method. We also give the error bound of the modified equation for the present method. Moreover, four numerical tests are included to show the effectiveness and applicability of the suggested method. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
25. An indirect convergent Jacobi spectral collocation method for fractional optimal control problems.
- Author
-
Yang, Yin, Zhang, Jiaqi, Liu, Huan, and O. Vasilev, Aleksandr
- Subjects
- *
COLLOCATION methods , *CAPUTO fractional derivatives , *ALGEBRAIC equations , *JACOBI polynomials , *DYNAMICAL systems , *JACOBI method - Abstract
In this paper, we present a novel indirect convergent Jacobi spectral collocation method for fractional optimal control problems governed by a dynamical system including both classical derivative and Caputo fractional derivative. First, we present some necessary optimality conditions. Then we suggest a new Jacobi spectral collocation method to discretize the obtained conditions. By the proposed method, we get a system of algebraic equations by solving of which we can approximate the optimal solution of the main problem. Finally, we present a convergence analysis for our method and solve three numerical examples to show the efficiency and capability of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
26. Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost.
- Author
-
Kumar, Nitin and Mehra, Mani
- Subjects
- *
COLLOCATION methods , *BOUNDARY value problems , *FRACTIONAL differential equations , *NONLINEAR dynamical systems , *FRACTIONAL integrals , *CALCULUS of variations - Abstract
This paper exhibits a numerical method for solving general fractional optimal control problems involving a dynamical system described by a nonlinear Caputo fractional differential equation, associated with a fractional Bolza cost composed as the aggregate of a standard Mayer cost and a fractional Lagrange cost given by a Riemann–Liouville fractional integral. By using the Lagrange multiplier within the calculus of variations and by applying integration by part formula, the necessary optimality conditions are derived in terms of a nonlinear two‐point fractional‐order boundary value problem. An operational matrix of fractional order right Riemann–Liouville integration is proposed and by utilizing it, the obtained two‐point fractional‐order boundary value problem is reduced into the solution of an algebraic system. An L2‐error estimate in the approximation of unknown variable by Legendre wavelet is derived and in the last, illustrative examples are included to demonstrate the applicability of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
27. Collocation method for solving nonlinear fractional optimal control problems by using Hermite scaling function with error estimates.
- Author
-
Kumar, Nitin and Mehra, Mani
- Subjects
ERROR functions ,COLLOCATION methods ,NONLINEAR programming ,TAYLOR'S series ,BERNSTEIN polynomials ,MATRIX functions - Abstract
Summary: This article presents an efficient numerical method for solving fractional optimal control problems (FOCPs) by utilizing the Hermite scaling function operational matrix of fractional‐order integration. The proposed technique is applied to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. The NLP solver is then used to solve FOCP. Furthermore, the L2‐error estimates in the approximation of unknown variables and the approximation of block pulse operational matrix of fractional‐order integration are derived and illustrative examples are included to demonstrate the applicability of the proposed method. Moreover, the results are compared with the Haar wavelet collocation method, hybrid of block‐pulse and Taylor polynomials method, Bernstein polynomials method, and the Boubaker hybrid function method to show the superiority of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
28. Local meshless differential quadrature collocation method for time-fractional PDEs.
- Author
-
Ahmad, Imtiaz, Siraj-ul-Islam, Mehnaz, and Zaman, Sakhi
- Subjects
DIFFERENTIAL quadrature method ,RADIAL basis functions ,FRACTIONAL differential equations ,NUMERICAL analysis ,BURGERS' equation ,PARTIAL differential equations ,COLLOCATION methods - Abstract
This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of α∈(0,1) and α∈(1,2). To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
29. Fractional retarded differential equations and their numerical solution via a multistep collocation method.
- Author
-
Maleki, Mohammad and Davari, Ali
- Subjects
- *
DELAY differential equations , *FRACTIONAL differential equations , *NUMERICAL solutions to differential equations , *COLLOCATION methods , *SOBOLEV spaces - Abstract
In this paper, we consider the nonlinear fractional retarded differential equations (FRDE). We extend the results of the existence and uniqueness of the solution, the propagation of derivative discontinuities and the dependence of the solution on the parameters of the equation. Next, we develop an efficient multistep collocation method for solving this type of equations. The proposed scheme is especially suited for FRDEs with piecewise smooth solutions, due to its essential feature of local approximations on subintervals. The stability of the scheme is accessed, and the convergence analysis is studied for functions in appropriate Sobolev spaces. Numerical results confirm the spectral accuracy and the stability of the proposed method for large domain calculations. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
30. THE NUMERICAL SOLUTION OF THE TIME-FRACTIONAL NON-LINEAR KLEIN-GORDON EQUATION VIA SPECTRAL COLLOCATION METHOD.
- Author
-
Yin YANG, Xingfa YANG, Jindi WANG, and Jie LIU
- Subjects
- *
COLLOCATION methods , *KLEIN-Gordon equation - Abstract
In this paper, we consider the numerical solution of the time-fractional non-linear Klein-Gordon equation. We propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of Jacobi interpolation polynomial for this equation. A rigorous error analysis is provided for the spectral methods to show both the errors of approximate solutions and the errors of approximate derivatives of the solutions decaying exponentially in infinity-norm and weighted L2-norm. Numerical tests are carried out to confirm the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
31. Bernoulli Operational Matrix of Fractional Derivative for Solution of Fractional Differential Equations.
- Author
-
Belgacem, Rachid, Bokhari, Ahmed, and Amir, Abdessamad
- Subjects
BERNOULLI equation ,FRACTIONAL differential equations ,NUMERICAL solutions to differential equations ,BERNOULLI polynomials ,ALGEBRAIC equations ,COLLOCATION methods - Abstract
The aim of this paper is to present a numerical method based on Bernoulli polynomials for numerical solutions of fractional differential equations(FDEs). The Bernoulli operational matrix of fractional derivatives[31] is derived and used together with tau and collocation methods to reduce the FDEs to a system of algebraic equations. Hence, the solutions obtained using this method give good approximations. Illustrative examples are included to demonstrate the validity and applicability of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
32. Fractional-order Legendre-collocation method for solving fractional initial value problems.
- Author
-
Al-Mdallal, Qasem M. and Abu Omer, Ahmed S.
- Subjects
- *
LEGENDRE'S functions , *INITIAL value problems , *COLLOCATION methods , *FRACTIONAL differential equations , *NONLINEAR systems - Abstract
In this paper, we present a numerical algorithm for solving second-order fractional initial value problems. This numerical algorithm is based on a fractional Legendre-collocation spectral method. The governing fractional differential equation is converted into a nonlinear system of algebraic equations. The error analysis of the proposed numerical algorithm is presented. Comparisons with other numerical methods shows that the proposed algorithm is more accurate and simpler to implement. Several examples are discussed to illustrate the efficiency and accuracy of the present scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
33. Numerical Solution of Fractional Neutral Functional Differential Equations by A shifted Chebyshev Computational Matrix.
- Author
-
Kouhkani, S. and Koppelaar, H.
- Subjects
NUMERICAL solutions to differential equations ,CHEBYSHEV systems ,COLLOCATION methods ,ALGEBRAIC equations ,CAPUTO fractional derivatives - Abstract
In this article, we develop a direct solution technique for solving Fractional Neutral Functional-Differential Equations (FNFDEs) using a matrix method based upon the shifted Chebyshev tau and shifted Chebyshev collocation method. The fractional derivatives are described in the Caputo sense. The main characteristic behind the approach using this technique is that it reduces the problems to a system of algebraic equations. The results reveal that the proposed method is very effective and simple. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
34. An efficient computational technique for solving a fractional-order model describing dynamics of neutron flux in a nuclear reactor.
- Author
-
Roul, Pradip, Rohil, Vikas, Espinosa-Paredes, Gilberto, and Obaidurrahman, K.
- Subjects
- *
NEUTRON flux , *NEUTRON diffusion , *DEUTERIUM oxide , *NUCLEAR reactors , *COLLOCATION methods , *PROBLEM solving - Abstract
This paper focuses on the development of a computational technique for numerical solution of the time-fractional neutron diffusion (FND) equation, which describes dynamics of neutron flux in a nuclear reactor. The time-fractional derivatives are discretized by means of L1 scheme, while the space derivative is discretized by employing a collocation method based on quartic B-spline (QUBS) basis functions. Numerical experiments are presented to illustrate the performance of proposed technique. Results reveal that the proposed technique yields a reliable approximation to the solution of the underlying model problem and has O (k + h 4) − order convergence, with k and h representing the step sizes in time and space directions, respectively. It is important to point out that the numerical method for the underlying problem has not yet been discussed in the literature. The effect of fractional order derivative on the behavior of neutron flux is investigated. • A time-fractional neutron diffusion model is considered. • A high accuracy numerical method is developed for solving this model problem. • Neutron fluxes were computed for three materials water, heavy water and graphite. • Influence of fractional order derivative on the neutron flux profile is investigated. • The numerical experiments show the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. An efficient scheme for solving a system of fractional differential equations with boundary conditions.
- Author
-
Hatipoglu, Veysel, Alkan, Sertan, and Secer, Aydin
- Subjects
- *
MATHEMATICAL equivalence , *FRACTIONAL differential equations , *DIFFERENTIAL equations , *COLLOCATION methods , *CAPUTO fractional derivatives - Abstract
In this study, the sinc collocation method is used to find an approximate solution of a system of differential equations of fractional order described in the Caputo sense. Some theorems are presented to prove the applicability of the proposed method to the system of fractional order differential equations. Some numerical examples are given to test the performance of the method. Approximate solutions are compared with exact solutions by examples. Some graphs and tables are presented to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
36. Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems.
- Author
-
Pedas, Arvet, Tamme, Enn, and Vikerpuur, Mikk
- Subjects
- *
NUMERICAL solutions to differential equations , *COLLOCATION methods , *FUNCTIONAL analysis , *CALCULUS of variations - Abstract
We construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form ( D ∗ α y ) ( t ) = f ( t , y ( t ) ) with Caputo type fractional derivatives D ∗ α y of order α > 0 . Using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. To illustrate the reliability of the proposed algorithms two numerical examples are given. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
37. Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method.
- Author
-
Morgado, Maria Luísa, Rebelo, Magda, Ferrás, Luis L., and Ford, Neville J.
- Subjects
- *
ERROR analysis in mathematics , *COLLOCATION methods , *CHEBYSHEV series , *NUMERICAL analysis , *APPROXIMATION theory , *FRACTIONAL differential equations - Abstract
In this work we present a new numerical method for the solution of the distributed order time-fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
38. 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
39. Spline collocation for fractional weakly singular integro-differential equations.
- Author
-
Pedas, Arvet, Tamme, Enn, and Vikerpuur, Mikk
- Subjects
- *
COLLOCATION methods , *INTEGRO-differential equations , *SPLINE theory , *BOUNDARY value problems , *CAPUTO fractional derivatives , *MATHEMATICAL reformulation - Abstract
We consider a class of boundary value problems for linear fractional weakly singular integro-differential equations which involve Caputo-type derivatives. Using an integral equation reformulation of the boundary value problem, we first study the regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the boundary value problem by suitable non-polynomial approximations is discussed. Optimal global convergence estimates are derived and a super-convergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
40. A novel method for solving second order fractional eigenvalue problems.
- Author
-
Reutskiy, S.Yu.
- Subjects
- *
FRACTIONAL calculus , *EIGENVALUES , *PROBLEM solving , *FRACTIONAL differential equations , *ANALYTICAL solutions , *COLLOCATION methods - Abstract
The paper presents a new numerical method for solving eigenvalue problems for fractional differential equations. It combines two techniques: the method of external excitation (MEE) and the backward substitution method (BSM). The first one is a mathematical model of physical measurements when a mechanical, electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies. The BSM consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. The numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
41. The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions.
- Author
-
Parvizi, Maryam and Eslahchi, M. R.
- Subjects
- *
JACOBI polynomials , *STOCHASTIC convergence , *CONTROL theory (Engineering) , *INTEGRAL equations , *MATHEMATICS theorems , *COLLOCATION methods - Abstract
In this paper, we apply the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions. Due to existence of integral boundary conditions, after reformulation of this equation in the integral form, the method is proposed for solving the obtained integral equation. Also, the convergence and stability analysis of the proposed method are studied in two main theorems. Furthermore, the optimum degree of convergence in the L2 norm is obtained for this method. Furthermore, some numerical examples are presented in order to illustrate the performance of the presented method. Finally, an application of the model in control theory is introduced. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
42. Cubic B-spline collocation method and its application for anomalous fractional diffusion equations in transport dynamic systems.
- Author
-
Sayevand, K., Yazdani, A., and Arjang, F.
- Subjects
- *
BOUNDARY value problems , *DYNAMICAL systems , *NUMERICAL solutions to integral equations , *NUMERICAL analysis , *COLLOCATION methods - Abstract
In this paper, we approximate the solution of the initial and boundary value problems of anomalous second- and fourth-order sub-diffusion equations of fractional order. The fractional derivative is used in the Caputo sense. To solve these equations, we will use a numerical method based on B-spline basis functions and the collocation method. It will be shown that the proposed scheme is unconditionally stable and convergent. Three numerical examples are adopted to demonstrate the performance of the proposed scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
43. Numerical solutions for fractional differential equations by Tau-Collocation method.
- Author
-
Allahviranloo, T., Gouyandeh, Z., and Armand, A.
- Subjects
- *
FRACTIONAL differential equations , *COLLOCATION methods , *MATHEMATICAL transformations , *ALGEBRAIC equations , *NONLINEAR systems , *MATRICES (Mathematics) , *MATHEMATICAL forms , *STOCHASTIC convergence - Abstract
The main purpose of this paper is to provide an efficient numerical approach for multi-order fractional differential equations based on a Tau-Collocation method. To do this, multi-order fractional differential equations transformed into a system of nonlinear algebraic equations in matrix form. Thus, by solving this system unknown coefficients are obtained. The fractional derivatives are described in the Caputo sense. The rate of convergence for the proposed method is established in the L w p norm. Some numerical example is also provided to illustrate our results. The results reveal that the method is very effective and simple. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
44. A backward euler orthogonal spline collocation method for the time-fractional Fokker- Planck equation.
- Author
-
Fairweather, Graeme, Zhang, Haixiang, Yang, Xuehua, and Xu, Da
- Subjects
- *
FOKKER-Planck equation , *PARTIAL differential equations , *COLLOCATION methods , *SPLINE theory , *CAPUTO fractional derivatives - Abstract
We formulate and analyze a novel numerical method for solving a time-fractional Fokker-Planck equation which models an anomalous subdiffusion process. In this method, orthogonal spline collocation is used for the spatial discretization and the time-stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence of the method are considered, and the theoretical results are supported by numerical examples, which also exhibit superconvergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1534-1550, 2015 [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
45. Modified spline collocation for linear fractional differential equations.
- Author
-
Kolk, Marek, Pedas, Arvet, and Tamme, Enn
- Subjects
- *
COLLOCATION methods , *LINEAR equations , *FRACTIONAL differential equations , *INITIAL value problems , *NUMERICAL solutions to differential equations , *INTEGRAL equations , *MATHEMATICAL reformulation - Abstract
We propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. Theoretical results are verified by some numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
46. Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations.
- Author
-
Mokhtary, P.
- Subjects
- *
EXPONENTIAL functions , *STOCHASTIC convergence , *LEGENDRE'S functions , *COLLOCATION methods , *SET theory , *FRACTIONAL differential equations , *INTEGRO-differential equations - Abstract
In this paper, Legendre Collocation method, an easy-to-use variant of the spectral methods for the numerical solution of a class of fractional integro-differential equations (FIDE’s), is researched. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by using Legendre Gauss quadrature formula. An efficient convergence analysis of the proposed method is given and rate of convergence is established in the L 2 -norm. Due to the fact that the solutions of FIDE’s usually have a weak singularity at origin, we use a variable transformation to change the original equation into a new equation with a smooth solution. We prove that after this regularization technique, numerical solution of the new equation by adopting the Legendre collocation method has exponentially rate of convergence. Numerical results are presented which clarify the high accuracy of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
47. A nonpolynomial collocation method for fractional terminal value problems.
- Author
-
Ford, N.J., Morgado, M.L., and Rebelo, M.
- Subjects
- *
POLYNOMIALS , *COLLOCATION methods , *FRACTIONAL calculus , *BOUNDARY value problems , *PROBLEM solving , *STOCHASTIC convergence - Abstract
In this paper we propose a nonpolynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α , 0 < α < 1 . The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a nonpolynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
48. Discrete-time orthogonal spline collocation method with application to two-dimensional fractional cable equation.
- Author
-
Zhang, Haixiang, Yang, Xuehua, and Han, Xuli
- Subjects
- *
DISCRETE systems , *ORTHOGONAL systems , *SPLINE theory , *COLLOCATION methods , *TOPOLOGY - Abstract
Discrete-time orthogonal spline collocation (OSC) methods are presented for the two-dimensional fractional cable equation, which governs the dynamics of membrane potential in thin and long cylinders such as axons or dendrites in neurons. The proposed scheme is based on the OSC method for space discretization and finite difference method for time, which is proved to be unconditionally stable and convergent with the order O ( τ min ( 2 − γ 1 , 2 − γ 2 ) + h r + 1 ) in L 2 -norm, where τ , h and r are the time step size, space step size and polynomial degree, respectively, and γ 1 and γ 2 are two different exponents of fractional derivatives with 0 < γ 1 , γ 2 < 1 . Numerical experiments are presented to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the method described herein, and super-convergence phenomena at the partition nodes is also exhibited, which is a characteristic of the OSC methods, namely, the rates of convergence in the maximum norm at the partition nodes in u x and u y are approximately h r + 1 in our numerical experiment. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
49. SPECTRAL COLLOCATION METHOD FOR MULTI-ORDER FRACTIONAL DIFFERENTIAL EQUATIONS.
- Author
-
GHOREISHI, F. and MOKHTARY, P.
- Subjects
FRACTIONAL differential equations ,CAPUTO fractional derivatives ,COLLOCATION methods ,STOCHASTIC convergence ,MATHEMATICAL regularization - Abstract
In this paper, the spectral collocation method is investigated for the numerical solution of multi-order Fractional Differential Equations (FDEs). We choose the orthogonal Jacobi polynomials and set of Jacobi Gauss--Lobatto quadrature points as basis functions and grid points respectively. This solution strategy is an application of the matrix-vector-product approach in spectral approximation of FDEs. The fractional derivatives are described in the Caputo type. Numerical solvability and an efficient convergence analysis of the method have also been discussed. Due to the fact that the solutions of fractional differential equations usually have a weak singularity at origin, we use a variable transformation method to change some classes of the original equation into a new equation with a unique smooth solution such that, the spectral collocation method can be applied conveniently. We prove that after this regularization technique, numerical solution of the new equation has exponential rate of convergence. Some standard examples are provided to confirm the reliability of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
50. An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis.
- Author
-
Joujehi, A. Soltani, Derakhshan, M.H., and Marasi, H.R.
- Subjects
- *
FRACTIONAL differential equations , *PARTIAL differential equations , *FLUID mechanics , *NONLINEAR equations , *KLEIN-Gordon equation , *SINE-Gordon equation , *COLLOCATION methods - Abstract
The fundamental purpose of this paper is to study the numerical solution of multi-term time fractional nonlinear Klein–Gordon equation, using regularized beta functions and fractional order Bernoulli wavelets. First, the exact formulas for the fractional integrals of the fractional order Bernoulli wavelets were obtained. Using properties of the regularized beta functions and their operational matrices the operational matrices of the fractional order Bernoulli wavelets were calculated. Through new operational matrices and appropriate collocation points, the time fractional nonlinear Klein–Gordon equation were transformed to a system of nonlinear algebraic equations. The convergence analysis and error bound of the proposed method were then performed. A sufficient number of numerical simulations were considered to show the effectiveness and validity of the presented numerical method and its theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.