B-spline Scheme Research Articles

Wave propagation analysis in the modified nonlinear time fractional Harry Dym equation: Insights from Khater II method and B-spline schemes

This study aims to investigate the modified nonlinear time fractional Harry Dym equation using analytical and numerical techniques. The modified nonlinear time fractional Harry Dym equation is a generalization of the classical Harry Dym equation, which describes the propagation of nonlinear waves in a variety of physical systems. The conformable fractional derivative is used to define the time fractional derivative in the equation, which provides a natural and straightforward approach. The Khater II method, a powerful analytical technique, is employed to obtain approximate solutions for the equation. Additionally, three numerical schemes, namely, Cubic-B-spline, Quantic-B-spline and Septic-B-spline schemes, are developed and implemented to solve the equation numerically. The numerical results are compared with other numerical solutions to assess the accuracy and efficiency of the proposed schemes. The physical meaning of the modified nonlinear time fractional Harry Dym equation is discussed in detail, and its relation to other nonlinear evolution equations is highlighted. The results of this study provide new insights into the behavior of nonlinear waves in physical systems and contribute to a better understanding of the physical characterizations of the modified nonlinear time fractional Harry Dym equation.

A family of [formula omitted] four-point stationary subdivision schemes with fourth-order accuracy and shape-preserving properties

The four-point interpolatory scheme and the cubic B-spline are examples of the most well-known stationary subdivision procedures. They are based on the space of cubic polynomials and have their respective strengths and weaknesses. In this regard, the purpose of this study is to introduce a new type of subdivision scheme that integrates the advantages of both the four-point and the cubic B-spline schemes. The proposed scheme achieves approximation order ‘four’ and an improved smoothness C2, while keeping the same support of the basic limit function as the four-point scheme. Moreover, under some mild conditions, the new scheme has the properties of monotonicity and convexity preservation. Whereas most high-order shape-preserving schemes are non-linear and rather computationally complicated, the proposed scheme is linear and stationary. Several numerical examples are provided to illustrate the efficiency of our subdivision algorithm.

Numerical solution of the Whitham-Broer-Kaup shallow water equation by quartic B-spline collocation method

This paper introduces a novel quartic B-spline collocation method to address the coupled Whitham–Broer–Kaup (WBK) problem. The WBK problem is a topic of interest in the study of nonlinear wave phenomena and has applications in various fields, including fluid dynamics, plasma physics, and nonlinear optics. The method combines spatial quartic B-spline scheme discretization, and Crank–Nicolson temporal discretization. It is unconditionally stable as proven by the Von-Neumann technique. Numerical examples demonstrate the method’s superior accuracy compared to existing solutions. Error analysis employs and norms, while the method exhibits high computational efficiency. The nonlinearity is managed through Rubin-Graves linearization. Comparisons with prior approaches highlight its efficiency, stability, adaptability to complex problems. The quartic B-spline method is well-suited for simulating fluid flow phenomena in shallow water scenarios, offering high accuracy and low computational cost.

A high-accuracy computational technique based on $$L2-1_{\sigma }$$ and B-spline schemes for solving the nonlinear time-fractional Burgers’ equation

A high-accuracy computational technique based on $$L2-1_{\sigma }$$ and B-spline schemes for solving the nonlinear time-fractional Burgers’ equation

A robust fitted numerical scheme for singularly perturbed parabolic reaction–diffusion problems with a general time delay

The present paper deals with the class of time-delayed, singularly perturbed parabolic reaction–diffusion problems. In the x−t plane, parabolic boundary layers appear on the two lateral sides of the domain when a small parameter is multiplied by the second-order space derivative. A fitted operator-based numerical method is developed to solve the considered problem, and its detailed analysis is done. To discretize the spatial domain, an exponentially fitted cubic B-spline scheme is used, and for the discretization of the time derivative, we use the implicit Euler scheme on the uniform mesh. We improve accuracy in the temporal direction using Richardson’s extrapolation method, which results in second-order parameter-uniform convergence. The stability and uniform convergence analysis of the scheme are studied. The present scheme gives a more accurate solution than existing methods in the literature. To ensure that the established numerical scheme is applicable, two test examples are carried out. The obtained numerical results support the estimated value in theory.

Abundant and accurate computational wave structures of the nonlinear fractional biological population model

In this paper, the generalized exponential (GExp) method has been employed to construct novel solitary wave solutions of the nonlinear fractional biological population (FBP) model. This model is used to demonstrate the relation of the population with deaths and births. Many novel traveling wave solutions have been formulated in distinct forms such as exponential, hyperbolic and trigonometric forms. These solutions have been explained in three different axes. The first axis is plotting them in their three optional (real, imaginary and absolute value), the second axis is handling these solutions for constructing the requested conditions for applying trigonometric quintic B-spline (TQBS) scheme. The second one determines the accuracy of the obtained analytical solutions by showing the error’s value between the analytical and numerical solutions. At the same time, the third one is comparing our analytical and numerical solutions, which have recently been published that explain the paper’s contribution and novelty.

Prediction of the Hydrodynamic Forces for a Ship Oscillating in Calm Water by an Improved Higher Order Rankine Panel Method

This paper presents a frequency-domain Rankine source method based on a biquadratic B-spline scheme with an improved radiation mechanism. The improved radiation mechanism, based on the introduction of spatially varying Rayleigh artificial damping in addition to the simplified Seto’s radiation boundary conditions, is considered for modeling radiation of generated waves at various τ conditions, where τ=ωU/g including the undercritical condition (τ<0.25); this condition is present when a ship undergoes slow translation or low oscillatory frequency. In evaluations, the proposed method yields accurate solutions for unsteady flows produced by an oscillating, translationally moving submerged singularity. The radiation problem induced by a RIOS bulker is solved to have the resultant added mass and damping coefficients for further comparisons with the experimental data and the public numerical prediction by a simplified combined method at a wide τ region.

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.

Study on the Improved Radiation Boundary Conditions Based on the Quadratic B-spline Rankine Panel Method

This study presents a higher-order Rankine source method that is based on a biquadratic B-spline scheme with an improved radiation boundary condition. This method solves flow problems under all conditions, where , including the undercritical condition ( < 0.25); this condition is present when a ship undergoes slow translational motion. The proposed method is thus more flexible than the many Rankine-source models in the literature that are applicable only to the overcritical condition ( > 0.25). Specifically, the improved radiation boundary condition is derived from upstream Seto radiation boundary conditions and by incorporating the Rayleigh damping distribution over a free surface; the improved condition is explicit and efficient for solving flow problems. In evaluations, the proposed method yields accurate solutions for unsteady flows, which are produced by an oscillating, translationally moving submerged singularity. The method is also compared against asymptotic solutions and against analytic solutions obtained using a three-dimensional translating-pulsating-source Green function.

A Variant Cubic Exponential B-Spline Scheme with Shape Control

This paper presents a variant scheme of the cubic exponential B-spline scheme, which, with two parameters, can generate curves with different shapes. This variant scheme is obtained based on the iteration from the generation of exponentials and a suitably chosen function. For such a scheme, we show its C2-convergence and analyze the effect of the parameters on the shape of the generated curves and also discuss its convexity preservation. In addition, a non-uniform version of this variant scheme is derived in order to locally control the shape of the generated curves. Numerical examples are given to illustrate the performance of the new schemes in this paper.

Numerical simulations of Zakharov’s (ZK) non-dimensional equation arising in Langmuir and ion-acoustic waves

The trigonometric quintic B-spline scheme is used in this research paper to research Zakharov’s (ZK) nonlinear dimensional equation’s numerical solution. The ZK model’s solutions explain the relationship between the high-frequency Langmuir and the low-frequency ion-acoustic waves with many applications in optical fiber, coastal engineering, and fluid mechanics of electromagnetic waves, plasma physics, and signal processing. Three recent computational schemes (the expanded [Formula: see text]-expansion method, generalized Kudryashov method, and modified Khater method) have recently been used to investigate this model’s moving wave solution. Many innovative solutions have been established in this paper to determine the original and boundary conditions that allow numerous numerical schemes to be implemented. Here, the trigonometric quintic B-spline method is used to analyze the precision of the collected analytical solutions. To illustrate the precision of the numerical and computational solutions, distinct drawings are depicted.

Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

The analytical and numerical solutions of the (2+1) dimensional, Fisher-Kolmogorov-Petrovskii-Piskunov ((2+1) D-Fisher-KPP) model are investigated by employing the modified direct algebraic (MDA), modified Kudryashov (MKud.), and trigonometric-quantic B-spline (TQBS) schemes. This model, which arises in population genetics and nematic liquid crystals, describes the reaction–diffusion system by traveling waves in population genetics and the propagation of domain walls, pattern formation in bi-stable systems, and nematic liquid crystals. Many novel analytical solutions are constructed. These solutions are used to evaluate the requested numerical technique’s conditions. The numerical solutions of the considered model are studied, and the absolute value of error between analytical and numerical is calculated to demonstrate the matching between both solutions. Some figures are represented to explain the obtained analytical solutions and the match between analytical and numerical results. The used schemes’ performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations.

An efficient numerical scheme for the simulation of time-fractional nonhomogeneous Benjamin-Bona-Mahony-Burger model

The Benjamin-Bona-Mahony-Burger (BBM-Burger) equation is important for explaining the unidirectional propagation of long waves in nonlinear dispersion systems. This manuscript proposes an algorithm based on cubic B-spline basis functions to study the nonhomogeneous time fractional model of BBM-Burger via Caputo derivative. The discretization of fractional derivative is achieved by L1 formula, while the temporal and spatial derivatives are interpolated by means of Crank-Nicolson and forward finite difference scheme together with B-spline basis functions. The performance of the Cubic B-spline scheme (CBS) is examined by three test problems with homogeneous initial and boundary conditions. The obtained results are found to be in good agreement with the exact solutions. The behaviour of travelling wave is studied and presented in the form of tables and graphics for various values of α and t. A linear stability analysis, based on the von Neumann scheme, shows that the CBS is unconditionally stable. Moreover, the accuracy of the scheme is quantified by computing error norms.

Computational and approximate solutions of complex nonlinear Fokas–Lenells equation arising in optical fiber

This manuscript uses the generalized Khater (GK) method and the trigonometric quintic B-spline (TQBS) scheme to study the calculations and approximate solutions of complex nonlinear Fokas–Lenells (FL) equations. This model describes the propagation of short pulses in optical fibers. Many novel computing solutions have been obtained. The absolute, real, and imaginary values of some solutions are plotted in two three-dimensional and density graphs to explain the dynamic behavior of short pulses in the fiber. The use of constructed analytical solutions to evaluate initial and boundary conditions allows the application of numerical solutions to study the accuracy of our novel computational techniques. The performance of both methods demonstrates the ability, effectiveness, and ability to apply them to different forms of nonlinear evolution equations to check the accuracy of analytical and numerical solutions.

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.

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

Subdivision schemes play a vital role in curve modeling. The curves produced by the class of 2 n + 2 -point ternary scheme (Deslauriers and Dubuc (1989)) interpolate the given data while the curves produced by a class of 2 n + 2 -point ternary B-spline schemes approximate the given data. In this research, we merge these two classes to introduce a consolidated and unified class of combined subdivision schemes with two shape control parameters in order to grow versatility for overseeing valuable necessities. However, the proposed class of subdivision schemes gives optimal smoothness in the final shapes, yet we can increase its smoothness by using a proposed general formula in form of its Laurent polynomial. The theoretical analysis of the class of subdivision schemes is done by using various mathematical tools and using their coding in the Maple environment. The graphical analysis of the class of schemes is done in the Maple environment by writing the codes based on the recursive mathematical expressions of the class of subdivision schemes.

Strong Langmuir turbulence dynamics through the trigonometric quintic and exponential B-spline schemes

<abstract> In this manuscript, two recent numerical schemes (the trigonometric quintic and exponential cubic B-spline schemes) are employed for evaluating the approximate solutions of the nonlinear Klein-Gordon-Zakharov model. This model describes the interaction between the Langmuir wave and the ion-acoustic wave in a high-frequency plasma. The initial and boundary conditions are constructed via a novel general computational scheme. ^{[<xref ref-type="bibr" rid="b1">1</xref>]} has used five different numerical schemes, such as the Adomian decomposition method, Elkalla-expansion method, three-member of the well-known cubic B-spline schemes. Consequently, the comparison between our solutions and that have been given in ^{[<xref ref-type="bibr" rid="b1">1</xref>]}, shows the accuracy of seven recent numerical schemes along with the considered model. The obtained numerical solutions are sketched in two dimensional and column distribution to explain the matching between the computational and numerical simulation. The novelty, originality, and accuracy of this research paper are explained by comparing the obtained numerical solutions with the previously obtained solutions. </abstract>

Solution of Modified Equal Width Equation Using Quartic Trigonometric-Spline Method

Using a quartic trigonometric B-spline (QTB-S) scheme, the numerical solution is obtained for modified equal width equation (MEW eq.). The approach based on finite difference scheme with the help of Crank-Nicolson formulation. The finite difference scheme is used for time integration and QTB-S function for space integration. Performance and accuracy of the scheme is validated through testing two problems by using conserved laws and L∞ and L2 error norms.

On the Analytical and Numerical Solutions in the Quantum Magnetoplasmas: The Atangana Conformable Derivative (1+3)-ZK Equation with Power-Law Nonlinearity

In this research paper, our work is connected with one of the most popular models in quantum magnetoplasma applications. The computational wave and numerical solutions of the Atangana conformable derivative (1+3)-Zakharov-Kuznetsov (ZK) equation with power-law nonlinearity are investigated via the modified Khater method and septic-B-spline scheme. This model is formulated and derived by employing the well-known reductive perturbation method. Applying the modified Khater (mK) method, septic B-spline scheme to the (1+3)-ZK equation with power-law nonlinearity after harnessing suitable wave transformation gives plentiful unprecedented ion-solitary wave solutions. Stability property is checked for our results to show their applicability for applying in the model’s applications. The result solutions are constructed along with their 2D, 3D, and contour graphical configurations for clarity and exactitude.

Abundant distinct types of solutions for the nervous biological fractional FitzHugh\u2013Nagumo equation via three different sorts of schemes

The dynamical attitude of the transmission for the nerve impulses of a nervous system, which is mathematically formulated by the Atangana–Baleanu (AB) time-fractional FitzHugh–Nagumo (FN) equation, is computationally and numerically investigated via two distinct schemes. These schemes are the improved Riccati expansion method and B-spline schemes. Additionally, the stability behavior of the analytical evaluated solutions is illustrated based on the characteristics of the Hamiltonian to explain the applicability of them in the model’s applications. Also, the physical and dynamical behaviors of the gained solutions are clarified by sketching them in three different types of plots. The practical side and power of applied methods are shown to explain their ability to use on many other nonlinear evaluation equations.