scholarly journals Block Backward Differentiation Formulas for Fractional Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
T. A. Biala ◽  
S. N. Jator
Keyword(s):  
Numerical Approximation ◽  
Initial Value ◽  
One Dimensional ◽  
Numerical Tests ◽  
Numerical Integrators ◽  
Backward Differentiation Formulas ◽  
Collocation Approach ◽  
Fractional Differential ◽  
Large Systems ◽  
Differentiation Formulas

This paper concerns the numerical approximation of Fractional Initial Value Problems (FIVPs). This is achieved by constructing k-step continuous BDFs. These continuous schemes are developed via the interpolation and collocation approach and are used to obtain the discrete k-step BDF and (k-1) additional methods which are applied as numerical integrators in a block-by-block mode for the integration of FIVP. The properties of the methods are established and regions of absolute stability of the methods are plotted in the complex plane. Numerical tests including large systems arising form the semidiscretization of one-dimensional fractional Burger’s equation show that the methods are highly accurate and efficient.

The backward problem of parabolic equations with the measurements on a discrete set

2020 ◽  
Vol 28 (1) ◽  
pp. 137-144 ◽  
Author(s):  
Jin Cheng ◽  
Yufei Ke ◽  
Ting Wei
Keyword(s):  
Parabolic Equations ◽  
Numerical Approximation ◽  
Cross Validation ◽  
Continuation Method ◽  
Measurement Data ◽  
Initial Value ◽  
One Dimensional ◽  
Backward Problem ◽  
The One ◽  
Validation Rule

AbstractThe backward problems of parabolic equations are of interest in the study of both mathematics and engineering. In this paper, we consider a backward problem for the one-dimensional heat conduction equation with the measurements on a discrete set. The uniqueness for recovering the initial value is proved by the analytic continuation method. We discretize this inverse problem by a finite element method to deduce a severely ill-conditioned linear system of algebra equations. In order to overcome the ill-posedness, we apply the discrete Tikhonov regularization with the generalized cross validation rule to obtain a stable numerical approximation to the initial value. Numerical results for three examples are provided to show the effect of the measurement data.

scholarly journals Derivation of Diagonally Implicit Block Backward Differentiation Formulas for Solving Stiff Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Iskandar Shah Mohd Zawawi ◽  
Zarina Bibi Ibrahim ◽  
Khairil Iskandar Othman
Keyword(s):  
Initial Value Problems ◽  
Maximum Error ◽  
Computational Time ◽  
Initial Value ◽  
Stiff Initial Value Problems ◽  
Fully Implicit ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Order Four ◽  
Differentiation Formulas

The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two, order three, and order four are derived for solving stiff initial value problems (IVPs). The stability properties of the derived methods are investigated. The implementation of the method using Newton iteration is also discussed. The performance of the proposed methods in terms of maximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF) and fully implicit block extended backward differentiation formulas (FIBEBDF). The numerical results show that the proposed method outperformed both existing methods.

scholarly journals Applications of Optimal Spline Approximations for the Solution of Nonlinear Time-Fractional Initial Value Problems

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 249
Author(s):  
Enza Pellegrino ◽  
Francesca Pitolli
Keyword(s):  
Numerical Methods ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Real Life ◽  
Collocation Methods ◽  
Approximation Properties ◽  
Initial Value ◽  
Numerical Tests ◽  
Fractional Differential ◽  
Life Phenomena

Nonlinear fractional differential equations are widely used to model real-life phenomena. For this reason, there is a need for efficient numerical methods to solve such problems. In this respect, collocation methods are particularly attractive for their ability to deal with the nonlocal behavior of the fractional derivative. Among the variety of collocation methods, methods based on spline approximations are preferable since the approximations can be represented by local bases, thereby reducing the computational load. In this paper, we use a collocation method based on spline quasi-interpolant operators to solve nonlinear time-fractional initial value problems. The numerical tests we performed show that the method has good approximation properties.

scholarly journals A Family of High Order One-Block Methods for the Solution of Stiff Initial Value Problems

2019 ◽  
pp. 1-14
Author(s):  
I. J. Ajie ◽  
K. Utalor ◽  
P. Onumanyi
Keyword(s):  
Numerical Methods ◽  
Initial Value Problems ◽  
High Order ◽  
Numerical Implementation ◽  
Initial Value ◽  
Stiff Initial Value Problems ◽  
The Family ◽  
Block Methods ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

In this paper, we construct a family of high order self-starting one-block numerical methods for the solution of stiff initial value problems (IVP) in ordinary differential equations (ODE). The Reversed Adams Moulton (RAM) methods, Generalized Backward Differentiation Formulas (GBDF) and Backward Differentiation Formulas (BDF) are used in the constructions. The E-transformation is applied to the triples and a family of self-starting methods are obtained. The family is for . The numerical implementation of the methods on some stiff initial value problems are reported to show the effectiveness of the methods. The computational rate of convergence tends to the theoretical order as h tends to zero.

scholarly journals Block Hybrid k-Step Backward Differentiation Formulas for Large Stiff Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. N. Jator ◽  
E. Agyingi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stiff Systems ◽  
Partial Differential ◽  
Differentiation Formula ◽  
Block Scheme ◽  
Backward Differentiation Formulas ◽  
Large Systems ◽  
Grid Points ◽  
Differentiation Formulas

This paper presents a generalized high order block hybrid k-step backward differentiation formula (HBDF) for solving stiff systems, including large systems resulting from the semidiscretization parabolic partial differential equations (PDEs). A block scheme in which two off-grid points are specified by the zeros of the second degree Chebyshev polynomial of the first kind is examined for convergence, L and A stabilities. Numerical simulations that illustrate the accuracy of a Chebyshev based method are given for selected stiff systems and partial differential equations.

scholarly journals Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
I. S. M. Zawawi ◽  
Z. B. Ibrahim ◽  
F. Ismail ◽  
Z. A. Majid
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Initial Value ◽  
First Order ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Differentiation Formulas ◽  
A Performance

This paper focuses on the derivation of diagonally implicit two-point block backward differentiation formulas (DI2BBDF) for solving first-order initial value problem (IVP) with two fixed points. The method approximates the solution at two points simultaneously. The implementation and the stability of the proposed method are also discussed. A performance of the DI2BBDF is compared with the existing methods.

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