Cubic B-splines Research Articles

Solution of the foam-drainage equation with cubic B-spline hybrid approach

This work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and the L 1 formula, in integer and fractional cases, respectively. Then, the collocation approach together with cubic B-splines (CBS) basis are employed to estimate the unknown solution and its derivatives. With the help of these discretizations and Quasi-linearization, solving non-linear FD model transforms to the system of linear algebraic equations. The solution of the linear system approximates the CBS coefficients which further leads to the numerical solutions. Moreover, by Von Neumann stability it is proved that the proposed scheme is unconditionally stable. To evaluate the performance and accuracy of the technique, absolute error (AE), L 2, and L ∞ norms are presented. The obtained outcomes are also matched with some existing results in literature. It is noted from simulations that the proposed method gives quite accurate solutions.

Numerical Investigation of the Fractional Diffusion Wave Equation with the Mittag–Leffler Function

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ-weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O(h2+(Δt)2). Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques.

On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type

<abstract><p>The major objective of this scheme is to investigate both the existence and the uniqueness of a solution to an integro-differential equation of the second order that contains the Caputo-Fabrizio fractional derivative and integral, as well as the q-integral of the Riemann-Liouville type. The equation in question is known as the integro-differential equation of the Caputo-Fabrizio fractional derivative and integral. This equation has not been studied before and has great importance in life applications. An investigation is being done into the solution's continued reliance. The Schauder fixed-point theorem is what is used to demonstrate that there is a solution to the equation that is being looked at. In addition, we are able to derive a numerical solution to the problem that has been stated by combining the Simpson's approach with the cubic-b spline method and the finite difference method with the trapezoidal method. We will be making use of the definitions of the fractional derivative and integral provided by Caputo-Fabrizio, as well as the definition of the q-integral of the Riemann-Liouville type. The integral portion of the problem will be handled using trapezoidal and Simpson's methods, while the derivative portion will be solved using cubic-b spline and finite difference methods. After that, the issue will be recast as a series of equations requiring algebraic thinking. By working through this problem together, we are able to find the answer. In conclusion, we present two numerical examples and contrast the outcomes of those examples with the exact solutions to those problems.</p></abstract>

Simultaneous determination of the time‐dependent potential and force terms in a fourth‐order Rayleigh–Love equation

This paper considers an inverse coefficient problem of simultaneous determination of solely time‐dependent potential and force terms with the unknown longitudinal displacement from a Rayleigh–Love equation from two integral overdetermination measurements. Unique solvability of this theorem is investigated theoretically by using contraction principle. Although, the aforesaid inverse identification problem is ill‐posed but has a unique solution. We use the cubic B‐spline (CBS) collocation and Tikhonov regularization techniques to discretize the direct problem and to obtain stable as well as accurate solutions, respectively. The stability, for the discretized system of the direct problem, is also carried out by means of the von Neumann method. The acquired results demonstrate that accurate as well as stable solutions for the and are accessed for , when for both smooth and discontinuous potential and force terms. The stability analysis shows that the discretized system of the direct problem is unconditionally stable. Since the noisy data are introduced, the investigation and analysis model real circumstances where the practical quantities are naturally infested with noise. The technique establishes that accurate, as well as stable, solutions are obtained.

Reconstruction of a potential coefficient in the Rayleigh–Love equation with non-classical boundary condition

PurposeThe purpose of this paper is to reconstruct the potential numerically in the fourth-order Rayleigh–Love equation with boundary and nonclassical boundary conditions, from additional measurement.Design/methodology/approachAlthough, the aforesaid inverse identification problem is ill-posed but has a unique solution. The authors use the cubic B-spline (CBS) collocation and Tikhonov regularization techniques to discretize the direct problem and to obtain stable as well as accurate solutions, respectively. The stability, for the discretized system of the direct problem, is also carried out by means of the von Neumann method.FindingsThe acquired results demonstrate that accurate as well as stable solutions for the a(t) are accessed for {10–8, 10–7, 10–6, 10–5}, when p {0.01%, 0.1%} for both linear and nonlinear potential coefficient a(t). The stability analysis shows that the discretized system of the direct problem is unconditionally stable.Research limitations/implicationsSince the noisy data are introduced, the investigation and analysis model real circumstances where the practical quantities are naturally infested with noise.Practical implicationsThe acquired results demonstrate that accurate as well as stable solutions for the a(t) are accessed for {10–8, 10–7, 10–6, 10–5}, when p {0.01%, 0.1%} for both linear and nonlinear potential coefficient a(t). The stability analysis shows that the discretized system of the direct problem is unconditionally stable.Originality/valueThe potential term in the fourth-order Rayleigh–Love equation from additional measurement is reconstructed numerically, for the first time. The technique establishes that accurate, as well as stable solutions are obtained.

Numerical Solutions of Third-Order Time-Fractional Differential Equations Using Cubic B-Spline Functions

Numerous fields, including the physical sciences, social sciences, and earth sciences, benefit greatly from the application of fractional calculus (FC). The fractional-order derivative is developed from the integer-order derivative, and in recent years, real-world modeling has performed better using the fractional-order derivative. Due to the flexibility of B-spline functions and their capability for very accurate estimation of fractional equations, they have been employed as a solution interpolating polynomials for the solution of fractional partial differential equations (FPDEs). In this study, cubic B-spline (CBS) basis functions with new approximations are utilized for numerical solution of third-order fractional differential equation. Initially, the CBS functions and finite difference scheme are applied to discretize the spatial and Caputo time fractional derivatives, respectively. The scheme is convergent numerically and theoretically as well as being unconditionally stable. On a variety of problems, the validity of the proposed technique is assessed, and the numerical results are contrasted with those reported in the literature.

NUMERICAL INVESTIGATION OF THE NONLINEAR FRACTIONAL OSTROVSKY EQUATION

This research paper investigates the numerical solutions of the nonlinear fractional Ostrovsky equation through five recent numerical schemes (Adomian decomposition (AD), El Kalla (EK), Cubic B-Spline (CBS), extended Cubic B-Spline (ECBS), exponential Cubic B-Spline (ExCBS) schemes). We investigate the obtained computational solutions via the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods. This model is considered as a mathematical modification model of the Korteweg–de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions show the localized wave packet as a persistent and dominant feature. The accuracy of the obtained numerical solutions is investigated by calculating the absolute error between the exact and numerical solutions. Many sketches are given to illustrate the matching between the exact and numerical solutions.

Cubic spline solutions of the ninth order linear and non-linear boundary value problems

A lot of numerical formulations of physical phenomena contain 9th-order BVPs. The presented probe intends to consider the spline solutions of the 9th-order boundary value problems using Cubic B Spline(CBS). Ninth order boundary value problems arise in the study of laminar viscous flow in a semi-porous channel, astrophysics, hydrodynamic & hydro-magnetic stability. The derived technique is exceptionally useful and is appropriate for such kinds of linear and non-linear boundary value problems. Cubic B Spline(CBS) is successfully applied to mathematical models. The outcomes are contrasted to those presented in the literature, revealing that the introduced technique leads to an advisable estimation of the exact solution.

Computational and numerical simulations of nonlinear fractional Ostrovsky equation

This article investigates the accuracy of the obtained analytical solutions of the nonlinear fractional Ostrovsky (NLFO.) equation. The researched solutions are obtained by exploiting the well-known generalized Tanh–function (GTF.) method with the Atengana– conformable fractional (ACF.) derivative in the wave transformation. Five–numerical schemes (Adomian decomposition (AD), El Kalla (EK), cubic B-Spline (CBS), extended cubic B-Spline (ECBS), and exponential cubic B-Spline (ExCBS)) handle these solutions to check their solutions. The investigated fractional model is a general form of the KdV equation; however, the KdV equation’s solutions replace this effect with radiating inertia gravity waves. This model also describes a weak description of nonlinear ocean wave processes considering Earth rotation. The obtained analytical and numerical results are sketched through Mathematica 12 in different plot types to explain the waves’ dynamical behavior. Additionally, the computational obtained solutions’ stability is investigated. Finally, the paper’s contribution and obtained results’ novelty are demonstrated by comparing our solutions with those that have been obtained in previous scientific articles.

New Cubic B-Spline Approximation for Solving Linear Two-Point Boundary-Value Problems

In this study, we introduce a new cubic B-spline (CBS) approximation method to solve linear two-point boundary value problems (BVPs). This method is based on cubic B-spline basis functions with a new approximation for the second-order derivative. The theoretical new approximation for a second-order derivative and the error analysis have been successfully derived. We found that the second-order new approximation was O(h3) accurate. By using this new second-order approximation, the proposed method was O(h5) accurate. Four numerical problems consisting of linear ordinary differential equations and trigonometric equations with different step sizes were performed to validate the accuracy of the proposed methods. The numerical results were compared with the least squares method, finite difference method, finite element method, finite volume method, B-spline interpolation method, extended cubic B-spline interpolation method and the exact solutions. By finding the maximum errors, the results consistently showed that the proposed method gave the best approximations among the existing methods. We also found that our proposed method involved simple implementation and straightforward computations. Hence, based on the results and the efficiency of our method, we can say that our method is reliable and a promising method for solving linear two-point BVPs.

Five semi analytical and numerical simulations for the fractional nonlinear space-time telegraph equation

The accuracy of analytical obtained solutions of the fractional nonlinear space–time telegraph equation that has been constructed in (Hamed and Khater in J. Math., 2020) is checked through five recent semi-analytical and numerical techniques. Adomian decomposition (AD), El Kalla (EK), cubic B-spline (CBS), extended cubic B-spline (ECBS), and exponential cubic B-spline (ExCBS) schemes are used to explain the matching between analytical and approximate solutions, which shows the accuracy of constructed traveling wave solutions. In 1880, Oliver Heaviside derived the considered model to describe the cutting-edge or voltage of an electrified transmission. The matching between solutions has been explained by plotting them in some different sketches.

On semi analytical and numerical simulations for a mathematical biological model; the time-fractional nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equation

Abstract Through five latest numerical schemes (Adomian decomposition (AD), El Kalla (EK), cubic B - spline (CBS), expanded Cubic B-Spline (ECBS), exponential cubic B - spline (ExCBS), this manuscript examines semi-analytical and numerical solutions of the time-fractional nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equation. Using the Caputo–Fabrizio fractional derivative and expanded Riccati - expansion process in Hamed et al.(2020) [1], developed computational solutions are investigated to determine the sufficient conditions for the implementation of the above-suggested schemes. In combustion theory, mathematical biology, and other study fields, the quasi-linear model is parabolic in simulating specific reaction-diffusion systems. The model’s solution represents the proliferation of a favored gene, and moving waves are pursued by nonlinear interaction. By measuring the absolute error between the exact and numerical solutions, the obtained numerical solutions’ consistency is examined. To explain the correspondence between the exact and numerical solutions, several sketches are given.

Computational study of the convection-diffusion equation using new cubic B-spline approximations

<abstract> This paper introduces an efficient numerical procedure based on cubic B-Spline (CuBS) with a new approximation for the second-order space derivative for computational treatment of the convection-diffusion equation (CDE). The time derivative is approximated using typical finite differences. The key benefit of the scheme is that the numerical solution is obtained as a smooth piecewise continuous function which empowers one to find approximate solution at any desired position in the domain. Further, the new approximation has considerably increased the accuracy of the scheme. A stability analysis is performed to assure that the errors do not magnify. Convergence analysis of the scheme is also discussed. The scheme is implemented on some test problems and the outcomes are contrasted with those of some current approximating techniques from the literature. It is concluded that the offered scheme is equitably superior and effective. </abstract>

New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

PurposeThe purpose of this paper is to present a new cubic B-spline (CBS) approximation technique for the numerical treatment of coupled viscous Burgers’ equations arising in the study of fluid dynamics, continuous stochastic processes, acoustic transmissions and aerofoil flow theory.Design/methodology/approachThe system of partial differential equations is discretized in time direction using the finite difference formulation, and the new CBS approximations have been used to interpolate the solution curves in the spatial direction. The theoretical estimation of stability and uniform convergence of the proposed numerical algorithm has been derived rigorously.FindingsA different scheme based on the new approximation in CBS functions is proposed which is quite different from the existing methods developed (Mittal and Jiwari, 2012; Mittal and Arora, 2011; Mittal and Tripathi, 2014; Raslan et al., 2017; Shallal et al., 2019). Some numerical examples are presented to validate the performance and accuracy of the proposed technique. The simulation results have guaranteed the superior performance of the presented algorithm over the existing numerical techniques on approximate solutions of coupled viscous Burgers’ equations.Originality/valueThe current approach based on new CBS approximations is novel for the numerical study of coupled Burgers’ equations, and as far as we are aware, it has never been used for this purpose before.

Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells

This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied physics, fluid dynamics, and applied mathematics. The recommended method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 8th order BVPs using Cubic-B spline but it also describes the estimated derivatives of 1st order to 8th order of the analytic solution. The strategy is effectively applied to numerical examples and the outcomes are compared with the existing results. The method proposed in this paper provides better approximations to the exact solution.

Equatorial spread F statistics and empirical representation for IRI: A regional model for the Brazilian longitude sector

The empirical representation of the equatorial spread F (ESF) statistics in the IRI scheme requires well established distribution statistics of ESF occurrence and intensity as a function of local time, season/month, latitude and solar and magnetic activity levels. We present here a regional model for the quiet-time spread F distribution in the Brazilian longitude sector. In view of the well known dependence of spread F occurrence on magnetic declination angle, and the fact that the declination angle varies rapidly from the west coast (Peruvian sector) to the east coast (Brazilian sector) of south America the present model can be said to be valid for the latter sector. 13 years of spread F data simultaneously collected (during 1978–1990) over the equatorial site Fortaleza and low latitude site Cachoeira Paulista are used in this model. Only spread F data that is believed to be related to plasma bubble developments is used in the model. The data were first grouped into solar flux range bins representing low, medium, and high solar activity levels, represented, respectively, by F10.7 ≤ 100; 100 < F10.7 < 180; and F10.7 ≥ 180. Spread F percentage occurrence as a function of (nocturnal) local time for each of the 12 months in each solar flux range was calculated, and form the database for the model. Cubic-B spline fits of the data in local time, latitude, season/month and solar flux nodes constitute the structure of the present model. The model confirms many characteristic features of the spread F statistics already known as well as brings out some new outstanding features for the Brazilian sector. Among the results to be highlighted are: The spread F onset and peak occurrence get delayed in local time with increasing distance from the equator, indicating the plasma bubble origin for the low latitude ionogram spread F traces; The plasma bubble occurrence as well as the vertical rise velocity increase with the increase in solar flux; They attain larger values in summer months (centered around December) than in equinoctial months (March and September). The latitudinal variation in spread F, though based only on two-station data sets in this study, looks compatible with the latitude variation of ion density fluctuations observed by the AE-E satellite. The model will be made available to interested users.