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 Collocation solutions for the time fractional telegraph equation using cubic B-spline finite elements

2019 ◽  
Vol 57 (2) ◽  
pp. 131-144
Author(s):  
Orkun Tasbozan ◽  
Alaattin Esen
Keyword(s):  
Finite Elements ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Telegraph Equation ◽  
Numerical Results ◽  
Fractional Partial Differential Equations ◽  
Spline Collocation ◽  
B Spline ◽  
Fractional Telegraph Equation

Abstract In this study, we investigate numerical solutions of the fractional telegraph equation with the aid of cubic B-spline collocation method. The fractional derivatives have been considered in the Caputo forms. The L1and L2 formulae are used to discretize the Caputo fractional derivative with respect to time. Some examples have been given for determining the accuracy of the regarded method. Obtained numerical results are compared with exact solutions arising in the literature and the error norms L 2 and L ∞ have been computed. In addition, graphical representations of numerical results are given. The obtained results show that the considered method is effective and applicable for obtaining the numerical results of nonlinear fractional partial differential equations (FPDEs).

Travelling peakon and solitary wave solutions of modified Fornberg–Whitham equations with nonhomogeneous boundary conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
İhsan Çelikkaya
Keyword(s):  
Solitary Wave ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Test Problems ◽  
Von Neumann ◽  
Nonhomogeneous Boundary ◽  
Whitham Equations ◽  
Fourier Series Method ◽  
Nonhomogeneous Boundary Conditions ◽  
The Stability

Abstract In this study, the numerical solutions of the modified Fornberg–Whitham (mFW) equation, which describes immigration of the solitary wave and peakon waves with discontinuous first derivative at the peak, have been obtained by the collocation finite element method using quintic trigonometric B-spline bases. Although there are solutions of this equation by semi-analytical and analytical methods in the literature, there are very few studies on the solution of the equation by numerical methods. Any linearization technique has not been used while applying the method. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. To show the performance of the method, we have considered three test problems with nonhomogeneous boundary conditions having analytical solutions. The error norms L 2 and L ∞ are calculated to demonstrate the accuracy and efficiency of the presented numerical scheme.

A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method

2018 ◽  
Vol 29 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Ali Başhan ◽  
N. Murat Yağmurlu ◽  
Yusuf Uçar ◽  
Alaattin Esen
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Present Approach ◽  
Test Problems ◽  
Quadrature Method ◽  
Error Norm ◽  
B Spline

In the present paper, a novel perspective fundamentally focused on the differential quadrature method using modified cubic B-spline basis functions are going to be applied for obtaining the numerical solutions of the complex modified Korteweg–de Vries (cmKdV) equation. In order to test the effectiveness and efficiency of the present approach, three test problems, that is single solitary wave, interaction of two solitary waves and interaction of three solitary waves will be handled. Furthermore, the maximum error norm [Formula: see text] will be calculated for single solitary wave solutions to measure the efficiency and the accuracy of the present approach. Meanwhile, the three lowest conservation quantities will be calculated and also used to test the efficiency of the method. In addition to these test tools, relative changes of the invariants will be calculated and presented. In the end of these processes, those newly obtained numerical results will be compared with those of some of the published papers. As a conclusion, it can be said that the present approach is an effective and efficient one for solving the cmKdV equation and can also be used for numerical solutions of other problems.

THE BOUNDARY FACE METHOD WITH VARIABLE APPROXIMATION BY B-SPLINE BASIS FUNCTION

2012 ◽  
Vol 09 (01) ◽  
pp. 1240009 ◽  
Author(s):  
JINLIANG GU ◽  
JIANMING ZHANG ◽  
XIAOMIN SHENG
Keyword(s):  
Numerical Solutions ◽  
Basis Functions ◽  
Spline Functions ◽  
Well Performance ◽  
Geometric Information ◽  
B Spline ◽  
Numerical Tests ◽  
Boundary Integration ◽  
Spline Basis ◽  
Boundary Face

B-spline basis functions as a new approximation method is introduced in the boundary face method (BFM) to obtain numerical solutions of 3D potential problems. In the BFM, both boundary integration and variable approximation are performed in the parametric spaces of the boundary surfaces, therefore, keeps the exact geometric information of a body in which the problem is defined. In this paper, local bivariate B-spline functions are proposed to alleviate the influence of B-spline tensor product that will deteriorate the exactness of numerical results. Numerical tests show that the new method has well performance in both exactness and convergence.

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.

Analytical versus numerical solutions of the nonlinear fractional time–space telegraph equation

2021 ◽  
pp. 2150324
Author(s):  
Mostafa M. A. Khater ◽  
Dianchen Lu
Keyword(s):  
Analytical Solutions ◽  
Electromagnetic Waves ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Telegraph Equation ◽  
Scatter Matrix ◽  
B Spline ◽  
Stable Property ◽  
Time Space ◽  
Matrix Distribution

In this paper, the stable analytical solutions’ accuracy of the nonlinear fractional nonlinear time–space telegraph (FNLTST) equation is investigated along with applying the trigonometric-quantic-B-spline (TQBS) method. This investigation depends on using the obtained analytical solutions to get the initial and boundary conditions that allow applying the numerical scheme in an easy and smooth way. Additionally, this paper aims to investigate the accuracy of the obtained analytical solutions after checking their stable property through using the properties of the Hamiltonian system. The considered model for this study is formulated by Oliver Heaviside in 1880 to define the advanced or voltage spectrum of electrified transmission, with day-to-day distances from the electrified communication or the application of electromagnetic waves. The matching between the analytical and numerical solutions is explained by some distinct sketches such as two-dimensional, scatter matrix, distribution, spline connected, bar normal, filling with two colors plots.

scholarly journals Numerical investigation of the solutions of Schrödinger equation with exponential cubic B-spline finite element method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ozlem Ersoy Hepson ◽  
Idris Dag
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Time Discretization ◽  
Bound State ◽  
Test Problems ◽  
Spline Collocation ◽  
Error Norm ◽  
B Spline ◽  
Cubic Nonlinear Schrödinger Equation

AbstractIn this paper, we investigate the numerical solutions of the cubic nonlinear Schrödinger equation via the exponential cubic B-spline collocation method. Crank–Nicolson formulas are used for time discretization of the target equation. A linearization technique is also employed for the numerical purpose. Four numerical examples related to single soliton, collision of two solitons that move in opposite directions, the birth of standing and mobile solitons and bound state solution are considered as the test problems. The accuracy and the efficiency of the purposed method are measured by max error norm and conserved constants. The obtained results are compared with the possible analytical values and those in some earlier studies.

scholarly journals Double Laplace Decomposition Method and Finite Difference Method of Time-fractional Schrödinger Pseudoparabolic Partial Differential Equation with Caputo Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Mahmut Modanli ◽  
Bushra Bajjah
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Caputo Derivative ◽  
Decomposition Method ◽  
Linear Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Partial Differential ◽  
Laplace Decomposition Method

In this paper, an initial-boundary value problem for a one-dimensional linear time-dependent fractional Schrödinger pseudoparabolic partial differential equation with Caputo derivative of order α ∈ 0,1 is being considered. Two strong numerical methods are employed to acquire the solutions to the problem. The first method used is the double Laplace decomposition method where closed-form solutions are obtained for any α ∈ 0,1 . As the second method, the implicit finite difference scheme is applied to obtain the approximate solutions. To clarify the performance of these two methods, numerical results are presented. The stability of the problem is also investigated.

scholarly journals Numerical Solution of the coupled Burgers' equation by Trigonometric B-spline Collocation Method

2021 ◽  
Author(s):  
Yusuf UCAR ◽  
Nuri YAGMURLU ◽  
Mehmet YİĞİT
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
The Other ◽  
Test Problems ◽  
Spline Collocation ◽  
Unconditionally Stable ◽  
B Spline ◽  
B Splines ◽  
A New Technique ◽  
Coupled Burgers Equation

In the present study, the coupled Burgers’ equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which trigonometric cubic and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L₂ and L_{∞}. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

scholarly journals A modified cubic B-spline differential quadrature method for three-dimensional non-linear diffusion equations

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 453-463 ◽  
Author(s):  
Sumita Dahiya ◽  
Ramesh Chandra Mittal
Keyword(s):  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Initial Boundary ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Linear Diffusion ◽  
B Spline ◽  
Non Linear

AbstractThis paper employs a differential quadrature scheme for solving non-linear partial differential equations. Differential quadrature method (DQM), along with modified cubic B-spline basis, has been adopted to deal with three-dimensional non-linear Brusselator system, enzyme kinetics of Michaelis-Menten type problem and Burgers’ equation. The method has been tested efficiently to three-dimensional equations. Simple algorithm and minimal computational efforts are two of the major achievements of the scheme. Moreover, this methodology produces numerical solutions not only at the knot points but also at every point in the domain under consideration. Stability analysis has been done. The scheme provides convergent approximate solutions and handles different cases and is particularly beneficial to higher dimensional non-linear PDEs with irregularities in initial data or initial-boundary conditions that are discontinuous in nature, because of its capability of damping specious oscillations induced by high frequency components of solutions.

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