Septic B-spline Research Articles

New Exact and Numerical Experiments for the Caudrey-Dodd-Gibbon Equation

In this study, an exact and a numerical method namely direct algebraic method and collocation finite element method are proposed for solving soliton solutions of a special form of fifth-order KdV (fKdV) equation that is of particular importance: Caudrey-Dodd-Gibbon (CDG) equation. For these aims, homogeneous balance method and septic B-spline functions are used for exact and numerical solutions, respectively. Next, it is proved by applying von-Neumann stability analysis that the numerical method is unconditionally stable. The error norms $L_{2}$ and $L_{\infty }$ have been computed to control proficiency and conservation properties of the suggested algorithm. The obtained numerical results have been listed in the tables. The graphs are modelled so that easy visualization of properties of the problem. Also, the obtained results indicate that our method is favourable for solving such problems.

A Novel Numerical Approach for Solving the Newell-Whitehead Equation

Numerical solutions of Newell-Whitehead equation are investigated by collocation method in this study. Since higher order functions produce better approximations, septic B-spline basis functions is used for analysis and approximation. Error norms are calculated for the adequacy and effectiveness of the current method. Unconditional stability is proved using Von-Neumann theory. The numerical results are obtained and the comparisons are presented in the tables. Additionally, simulations of all numerical results are plotted to show the numerical behavior of the solution. Numerical results make the method more convenient and systematically handle the nonlinear solution process. The numerical solutions found make the method attractive and reliable for the solution of Fitzhugh-Nagumo type equations.

Derivation of septic B-spline function in n-dimensional to solve n-dimensional partial differential equations

Abstract In this study, a new structure for the septic B-spline collocation algorithm in n-dimensional is presented as a continuation of generating B-spline functions in n-dimensional to solve mathematical models in n-dimensional. The septic B-spline collocation algorithm is displayed in three forms: one dimensional, two dimensional, and three dimensional. In various domains, these constructs are essential for solving mathematical models. The effectiveness and correctness of the suggested method are demonstrated using a few two- and three-dimensional test problems. The proposed new structure provides better results than other methods because it deals with a larger number of points than the field. To create comparisons, we use different numerical approaches accessible in the literature.

Septic B-spline Collocation Method for Numerical Solution of the Generalized Nonlinear Schrodinger Equation

Bu çalışma, genelleştirilmiş lineer olmayan Schordinger (GNLS) denkleminin yüksek doğruluklu sayısal çözümünü elde etmek içindir. Çalışmada iki farklı zaman parçalanması kullanılacaktır. İlk zaman parçalanması doğruluğu iki olan ve literatürde iyi bilinen Crank-Nicolson yöntemi, ikinci zaman parçalanması ise tek adımlı ve dördüncü mertebeden doğruluğa sahip yöntemdir. Zaman parçalanması için genel bir yöntem kullanıldıktan sonra konum parçalanması için ise septik B-spline fonksiyonların kullanıldığı kolokasyon yöntemi kullanılacaktır. Zaman ve konum parçalanması uygulandıktan sonra lineer olmayan bir denklem sistemi elde edilecektir. Denklem sistemi Matlab paket programı yardımıyla çözülürken öncelikle iç iterasyonlu bir lineerleştirme kullanılacak ve istenilen zaman kadarki çözümler iteratif olarak bulunacaktır Son olarak solitary dalgasının yayılımı test problemi kullanılarak önerilen metotlar test edilmiştir.

Numerical and dynamical behaviors of nonlinear traveling wave solutions of the Kudryashov–Sinelshchikov equation

In this article, a mathematical model symbolizing numerical solutions of the nonlinear Kudryashov–Sinelshchikov (KS) equation has been investigated. For this aim, we use collocation finite element method with septic B-spline functions and appropriate initial and boundary conditions on uniform mesh points. The proficiency and effectiveness of the presented method are examined in four test examples, consisting; behavior of a single soliton, collision of two solitons, Maxwellian initial condition and growth of a wavy hole. L2 and L∞ error norms are used to validate practicality and reliability of our numerical algorithm. In addition, two invariants are calculated to determine the conservation properties of the presented algorithm and the numerical results have been shown to be unconditionally stable. Obtained numerical results have been illustrated with tables and graphics for easy visualization of properties of the problem modeled. Numerical calculations to solve such problems prove the accuracy and effectiveness of the finite element method. In collocation method, using B-spline functions gives rise to a technique that needs only the unknown parameters at certain node points to create the solution. Collocation method has two excellent advantages: establishing method does not include integrations and the resulting matrix system is banded with small band width. Consequently, B-splines associate with the collocation provide a simple solution procedure of linear and nonlinear partial differential equations (PDEs). After that, we have analyzed error estimation for the Galerkin finite element method. Furthermore, dynamical properties of traveling waves of the KS equation are investigated in absence and presence of external periodic perturbation. In absence of perturbation, periodic traveling wave solution of the KS equation are studied. On the other hand, a collection of various quasiperiodic wave features of the KS equation is presented in presence of periodic perturbation. Therefore, it can be said that the numerical technique is effective and valid in obtaining numerical solutions of such PDEs. In addition, the study on dynamical properties of the KS equation provides qualitative behavior of the corresponding wave features.

A new perspective for analytical and numerical soliton solutions of the Kaup–Kupershmidt and Ito equations

In present study, it is aimed to implement the analytical method and a numerical approach based on finite element method to get approximate soliton solutions of two special forms of fifth-order KdV equation (fKdV) that are of particular importance: Kaup–Kupershmidt (K–K) and Ito equations. To reach these aims, the simplest equation method and septic B-spline collocation method are introduced. L2 and L∞ error norms are computed for single soliton solutions to demonstrate the proficiency and accuracy of the present method. The method is shown to be unconditionally stable by performing the von-Neumann stability analysis. The obtained numerical results have been displayed with tables. Additionally, to show the analytical and numerical behaviors of the single soliton, all figures are drawn in 2D and 3D. Analytical and numerical results ensure that the methods are more suitable and systematically handle the solution procedures of nonlinear equations arising in mathematical physics.

On the analytical and numerical approximations to the forced damped Gardner Kawahara equation and modeling the nonlinear structures in a collisional plasma

We perform a detailed study on the completely non-integrable forced damped Gardner/Extended Kawahara equation (FDEKE). Three techniques are introduced to determine abundance approximations to the proposed equation. In the first technique, the ansatz method is carried out for deriving some general formulas for the analytical approximations. In the second and third techniques, the FDEKE is analyzed numerically using both the septic B-spline collocation method and the method of lines. As a realistic model, the obtained approximations are employed for studying the properties of the periodic forced dissipative extended Kawahara solitary and cnoidal waves in a pair-ion plasma comprised of Maxwellian electrons and two fluid positive and negative ions. Both numerical and analytical approximations are graphically compared with each other. Also, the global maximum residual error L∞ for all obtained approximations is estimated for checking the accuracy of these approximations. Moreover, the obtained approximations can be applied for studying the features of the dissipative localized and periodic higher-order structures in optical fiber, ocean, sea, different models of plasma physics, and fluid mechanics.

A collocation algorithm based on septic B-splines and bifurcation of traveling waves for Sawada–Kotera equation

In present work, a mathematical model standing for the numerical solutions of the nonlinear Sawada–Kotera (S–K) equation has been examined. For this purpose, we have used collocation finite element method for the model problem. The solution for each unknown is written as a linear combination of time parameters and combination of the septic B-splines. Next, applying Von-Neumann theory, we demonstrate that the scheme is marginally stable. Error norms L2 and L∞ are computed for single soliton solutions to display the practicality and robustness the suggested scheme. The obtained numerical results have been illustrated with tables. Furthermore, to show the numerical behavior of the single soliton, some of the simulations are depicted in 2D and 3D. Present results show that the method procures high accurate solutions so present scheme will be useful for the other nonlinear scientific problems. After that, bifurcation behavior of traveling waves for the S-K equation has been examined depending on the velocity (v) of the traveling wave. It has been seen that velocity (v) of the traveling wave plays a key role for the existence of different kind of quasiperiodic motions of the nonlinear S-K equation. Coexisting quasiperiodic motions for traveling wave solutions of the equation have been presented.

English

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline finite elements method, Exponential cubic B-spline method, Septic B-spline technique, Quintic B-spline Galerkin method, and B-spline Galerkin method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper, we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

Numerical approximation for inverse problem of the Ostrovsky-Burgers equation

This article considers a nonlinear inverse problem of the Ostrovsky-Burgers equation by using noisy data. This equation was appeared in modeling internal waves in the ocean or surface waves in a shallow channel with an uneven bottom under the effects of the interfacial friction. Two B-Splines with different levels, the quintic B-spline and septic B-spline, are used to study this problem. For both B-splines, convergence analysis is calculated and results show that an excellent estimation of the unknown functions of the nonlinear inverse problem. Indeed, by comparing the numerical results, we showed that the accuracy and stability of the septic B-spline method is more than those for the quintic B-spline method.

Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation

This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modified Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation finite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms $L_{2}$, $L_{\infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation.

Numerical Solution of the Seventh Order Boundary Value Problems Using B-Spline Method

We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using seventh degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value problem. We obtain Septic B-Spline formulation and the Collocation B-spline Method is formulated as an approximation solution. We apply the presented method to solve an example of seventh-order boundary value problem which the results show that there is an agreement between approximate solutions and exact solutions. Resulting low absolute errors show that the presented numerical method is effective for solving high order boundary value problems. Finally, a general conclusion has been included.

Multiple Lump Novel and Accurate Analytical and Numerical Solutions of the Three-Dimensional Potential Yu–Toda–Sasa–Fukuyama Equation

The accuracy of novel lump solutions of the potential form of the three–dimensional potential Yu–Toda–Sasa–Fukuyama (3-Dp-YTSF) equation is investigated. These solutions are obtained by employing the extended simplest equation (ESE) and modified Kudryashov (MKud) schemes to explore its lump and breather wave solutions that characterizes the dynamics of solitons and nonlinear waves in weakly dispersive media, plasma physics, and fluid dynamics. The accuracy of the obtained analytical solutions is investigated through the perspective of numerical and semi-analytical strategies (septic B-spline (SBS) and variational iteration (VI) techniques). Additionally, matching the analytical and numerical solutions is represented along with some distinct types of sketches. The superiority of the MKud is showed as the fourth research paper in our series that has been beginning by Mostafa M. A. Khater and Carlo Cattani with the title “Accuracy of computational schemes”. The functioning of employed schemes appears their effectual and ability to apply to different nonlinear evolution equations.

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.

An Efficient Superconvergent Spline Collocation Algorithm for Solving Fourth Order Singularly Perturbed Problems

In this work, improvised quintic B-spline collocation method is used to solve a class of fourth order singularly perturbed boundary value problems. The proposed method is an extension of classical collocation technique. Posteriori corrections are made to quintic B-spline interpolant and its derivatives, which leads to formation of improvised quintic B-spline interpolant. The improvisation is done by forcing the spline interpolant to satisfy interpolatory and some special end conditions. These corrections are made to achieve optimal order of convergence, which is not possible by standard formation of splines. Convergence analysis is performed using Green’s function approach. Four test problems are solved to show the effectiveness of the proposed method and its improvement over classical quintic and septic B-spline collocation techniques. The technique is found to be stable even for extremely small values of perturbation parameter.

Approximate Simulations for the Non-linear Long-Short Wave Interaction System

This research paper studies the semi-analytical, and numerical solutions of on the nonlinear long-short wave interaction system which represents an optical field that does not alter through multiplication due to a sensitive balance between linear and nonlinear impacts in the elastic medium that is defined as the medium which can adjust in the figure as a consequence of a deforming strength and come back its shape to its original form when the force eliminates. In this medium, the wave is produced by vibrations that are a consequence of acoustic power and is known as a sound wave or acoustic wave. The Adomian decomposition method and cubic\,\&\, septic B-spline method are applied to the suggested system to obtain distinct types of solutions that are used to explain the novel physical properties of this system. These novels' features are described by different types of figures that show more physical properties of this model. Also, the convergence between the obtained solutions is discussed by some tables that show the values of absolute error between them.

Collocation Method for Solving the Generalized KdV Equation

In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including; single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, L2 and L∞ and invariants I1, I2 and I3 have been calculated. Our numerical results are compared with some of those available in the literature.

Numerical approximation for viscous Cahn–Hilliard equation via septic B-spline

In this paper, a numerical technique is proposed for solving the nonlinear viscous Cahn–Hilliard equation. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating septic B-spline approach to approximate the spatial derivative. The convergence of the method is considered, and the theoretical results are supported by numerical examples. We compare our numerical results with those given in the literature in terms of numerical accuracy. The numerical comparisons show that the presented method provides highly accurate results.

Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations

In this paper, a numerical solution of the coupled Burgers' equations based on septic B-spline collocation method is presented. The scheme is based on the Crank–Nicolson formulation for time integration and septic B-spline functions for space integration. The method has been showed unconditionally stable by using Von-Neumann technique. The efficiency of this method is demonstrated by applying two test problems. The obtained numerical results are found in a good agreement with the exact solution. This method is efficient, powerful, and economical. It can also applicable to other linear and nonlinear partial differential equations.

Septic B-spline method for solving nonlinear singular boundary value problems arising in physiological models

In this paper, we present a numerical method based on septic B-spline function for nonlinear singular second-order two-point boundary value problems, which depend on different physiological processes as thermal explosions problem and the steady state oxygen diffusion in a spherical cell with Michaelis–Menten uptake kinetics and distribution of heat sources in the human head. Septic B-spline method has a truncation error of O(h^8) and converges to the exact solution with O(h^6). The numerical problems show that our method is very effective. The resulting sets of differential equations are modified at the singular point and are treated by using septic B-spline for finding the numerical solution. The maximum absolute errors and the absolute residual errors are acceptable.