scholarly journals A New Variant of B-Spline for the Solution of Modified Fractional Anomalous Subdiffusion Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
M. S. Hashmi ◽  
Zainab Shehzad ◽  
Asifa Ashraf ◽  
Zhiyue Zhang ◽  
Yu-Pei Lv ◽  
...  
Keyword(s):  
Biological Sciences ◽  
Numerical Technique ◽  
Algebraic Equations ◽  
Von Neumann ◽  
B Spline ◽  
New Variant ◽  
Liouville Derivative ◽  
Data Points ◽  
Subdiffusion Equation ◽  
Von Neumann Stability Analysis

The objective of this paper is to present an efficient numerical technique for solving time fractional modified anomalous subdiffusion equation. Anomalous diffusion equation has its role in various branches of biological sciences. B-spline is a piecewise function to draw curves and surfaces, which maintain its degree of smoothness at the connecting points. B-spline provides an active process of approximation to the limit curve. In current attempt, B-spline curve is used to approximate the solution curve of time fractional modified anomalous subdiffusion equation. The process is kept simple involving collocation procedure to the data points. The time fractional derivative is approximated with the discretized form of the Riemann-Liouville derivative. The process results in the form of system of algebraic equations, which is solved using a variant of Thomas algorithm. In order to ensure the convergence of the procedure, a valid method named Von Neumann stability analysis is attempted. The graphical and tabular display of results for the illustrated examples is presented, which stamped the efficiency of the proposed algorithm.

scholarly journals Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Imtiaz Wasim ◽  
Muhammad Abbas ◽  
Muhammad Amin
Keyword(s):  
Collocation Method ◽  
Numerical Technique ◽  
Time Derivative ◽  
Fourier Method ◽  
Spatial Dimension ◽  
Test Problems ◽  
Spline Collocation ◽  
Von Neumann ◽  
B Spline ◽  
Spline Collocation Method

In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.

scholarly journals Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers’ equations numerically

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adel R. Hadhoud ◽  
H. M. Srivastava ◽  
Abdulqawi A. M. Rageh
Keyword(s):  
Operational Matrix ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Von Neumann ◽  
B Spline ◽  
Burgers Equations ◽  
Finite Difference Approximations ◽  
L1 Approximation ◽  
Spatial Derivatives ◽  
Numerical Approaches

AbstractThis paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval $[0, L]$ [ 0 , L ] . The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

scholarly journals Numerical Analysis WSGD Scheme for One- and Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation with Collocation Method via Fractional B-Spline

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mohammad Ramezani
Keyword(s):  
Diffusion Equation ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
B Spline ◽  
Computational Work ◽  
Distributed Order ◽  
Fractional Reaction

AbstractThe main propose of this paper is presenting an efficient numerical scheme to solve WSGD scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation. The proposed method is based on fractional B-spline basics in collocation method which involve Caputo-type fractional derivatives for $$0 < \alpha < 1$$ 0 < α < 1 . The most significant privilege of proposed method is efficient and quite accurate and it requires relatively less computational work. The solution of consideration problem is transmute to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. The finally, several numerical WSGD Scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation.

Solution of non-linear time fractional telegraph equation with source term using B-spline and Caputo derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdul Majeed ◽  
Mohsin Kamran ◽  
Noreen Asghar
Keyword(s):  
Caputo Derivative ◽  
Numerical Solutions ◽  
Linear Time ◽  
Initial Boundary ◽  
Telegraph Equation ◽  
Test Problems ◽  
Von Neumann ◽  
B Spline ◽  
Nonlinear Part ◽  
Spline Basis

Abstract This article focusses on the implementation of cubic B-spline approach to investigate numerical solutions of inhomogeneous time fractional nonlinear telegraph equation using Caputo derivative. L1 formula is used to discretize the Caputo derivative, while B-spline basis functions are used to interpolate the spatial derivative. For nonlinear part, the existing linearization formula is applied after generalizing it for all positive integers. The algorithm for the simulation is also presented. The efficiency of the proposed scheme is examined on three test problems with different initial boundary conditions. The influence of parameter α on the solution profile for different values is demonstrated graphically and numerically. Moreover, the convergence of the proposed scheme is analyzed and the scheme is proved to be unconditionally stable by von Neumann Fourier formula. To quantify the accuracy of the proposed scheme, error norms are computed and was found to be effective and efficient for nonlinear fractional partial differential equations (FPDEs).

scholarly journals Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets

2005 ◽  
Vol 2005 (1) ◽  
pp. 113-121 ◽  
Author(s):  
M. Lakestani ◽  
M. Razzaghi ◽  
M. Dehghan
Keyword(s):  
Integral Equations ◽  
Algebraic Equations ◽  
B Spline ◽  
Compactly Supported ◽  
Hammerstein Integral Equations ◽  
Spline Wavelets

Compactly supported linear semiorthogonal B-spline wavelets together with their dual wavelets are developed to approximate the solutions of nonlinear Fredholm-Hammerstein integral equations. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example.

scholarly journals The exponential cubic B-spline collocation method for the Kuramoto-Sivashinsky equation

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 853-861 ◽  
Author(s):  
Ozlem Ersoy ◽  
Idiris Dag
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Spline Approximation ◽  
Algebraic Equations ◽  
Spline Collocation ◽  
Fully Integrated ◽  
B Spline ◽  
Spline Collocation Method ◽  
Sivashinsky Equation ◽  
Good Agreement

In this study the Kuramoto-Sivashinsky (KS) equation has been solved using the collocation method, based on the exponential cubic B-spline approximation together with the Crank Nicolson. KS equation is fully integrated into a linearized algebraic equations. The results of the proposed method are compared with both numerical and analytical results by studying two text problems. It is found that the simulating results are in good agreement with both exact and existing numerical solutions.

A modified trigonometric cubic B-spline collocation technique for solving the time-fractional diffusion equation

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Neeraj Dhiman ◽  
M.J. Huntul ◽  
Mohammad Tamsir
Keyword(s):  
Diffusion Equation ◽  
Numerical Technique ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Dirichlet Boundary Conditions ◽  
Unconditionally Stable ◽  
Content Type ◽  
B Spline ◽  
Time Fractional Diffusion Equation ◽  
The Stability

Purpose The purpose of this paper is to present a stable and efficient numerical technique based on modified trigonometric cubic B-spline functions for solving the time-fractional diffusion equation (TFDE). The TFDE has numerous applications to model many real objects and processes. Design/methodology/approach The time-fractional derivative is used in the Caputo sense. A modification is made in trigonometric cubic B-spline (TCB) functions for handling the Dirichlet boundary conditions. The modified TCB functions have been used to discretize the space derivatives. The stability of the technique is also discussed. Findings The obtained results are compared with those reported earlier showing that the present technique gives highly accurate results. The stability analysis shows that the method is unconditionally stable. Furthermore, this technique is efficient and requires less storage. Originality/value The current work is novel for solving TFDE. This technique is unconditionally stable and gives better results than existing results (Ford et al., 2011; Sayevand et al., 2016; Ghanbari and Atangana, 2020).

Parametric Generation of Planing Hulls with NURBS Surfaces

2013 ◽  
Vol 57 (04) ◽  
pp. 241-261
Author(s):  
Francisco L. Perez-Arribas ◽  
Erno Peter-Cosma
Keyword(s):  
Maximum Deviation ◽  
Geometric Features ◽  
Surface Geometry ◽  
Orthogonal Projections ◽  
Parametric Generation ◽  
B Spline ◽  
Spline Curves ◽  
Hull Design ◽  
Data Points ◽  
Control Curves

This article presents a mathematical method for producing hard-chine ship hulls based on a set of numerical parameters that are directly related to the geometric features of the hull and uniquely define a hull form for this type of ship. The term planing hull is used generically to describe the majority of hard-chine boats being built today. This article is focused on unstepped, single-chine hulls. B-spline curves and surfaces were combined with constraints on the significant ship curves to produce the final hull design. The hard-chine hull geometry was modeled by decomposing the surface geometry into boundary curves, which were defined by design constraints or parameters. In planing hull design, these control curves are the center, chine, and sheer lines as well as their geometric features including position, slope, and, in the case of the chine, enclosed area and centroid. These geometric parameters have physical, hydrodynamic, and stability implications from the design point of view. The proposed method uses two-dimensional orthogonal projections of the control curves and then produces three-dimensional (3-D) definitions using B-spline fitting of the 3-D data points. The fitting considers maximum deviation from the curve to the data points and is based on an original selection of the parameterization. A net of B-spline curves (stations) is then created to match the previously defined 3-D boundaries. A final set of lofting surfaces of the previous B-spline curves produces the hull surface.

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