Quartic B-spline Collocation Method Research Articles

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.

Viscous Effect on Solitary Kelvin Wave in Open Cylindrical Channel under Precession

Viscous effect is introduced into the system of Navier–Stokes equations, that were derived to study the solitary Kelvin mode in an open cylindrical channel that precesses. Accordingly, three new weakly nonlinear models were derived: Korteweg–de Vries-Burgers, and two new Benjamin–Bona–Mahony-Burgers. The first was solved analytically by discussing the phase solution and numerically using an implicite finite difference method to track the solution with time under diffusion effect. The second two models were solved numerically only using the Quartic B-Spline collocation method. By manipulating the scaling the first model included only the gravity force effect, and the second included both gravity and Coriolis forces. The numerical method is tested experimentally by comparing the velocity solutions with ADV signal extracted from the ADV measurements under bore conditions, and the amplitude solution with the solitary kelvin mode.

N-Dimensional quartic B-spline collocation method to solve different types of n-dimensional partial differential equations

In this article, we introduce a new structure to the quartic B-spline functions in n-dimensions to solve various mathematical problems of partial differential equations in n-dimensions. We use quartic B-spline functions in one dimension to generate multi-dimensional quartic B-spline functions as two and three-dimensional. We measure the efficiency and accuracy of the proposed collocation method as shown in five problems in two and three dimensions. We compare our new method with other different methods.

Development of a Direct Time Integration Method Based on Quartic B-spline Collocation Method

A new version of quartic B-spline direct time integration method for dynamic analysis of structures is presented. This procedure is derived based on uniform quartic B-spline piecewise polynomial approximations and collocation method, named Alpha Quartic B-spline time integration method. In this way, at first, the method is implemented to solve the governing differential equation of motion of single-degree-of-freedom systems, and later, the proposed method is generalized for multi-degree-of-freedom systems. Stability and accuracy analysis of the proposed algorithm have been investigated completely. In the proposed algorithm by using two collocation parameters α1 and α2, unconditional stability is achieved, but a local instability is created. The best values of these two parameters have been determined not only to maintain the stability, but also to ensure the desired accuracy. For accuracy analysis, dissipation and dispersion errors have been investigated for different cases of α’s. Finally, for the proposed method, a simple step-by-step algorithm was presented. The effectiveness and robustness of the proposed algorithm in solving linear dynamic problems are demonstrated in the numerical examples.

B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems

This paper is concerned with the construction and convergence analysis of two B-spline collocation methods for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The first method is based on uniform mesh, while the second method is based on non-uniform mesh. For the second method, we use a grading function to construct the non-uniform grid. We prove that the method based on uniform mesh is of second-order accuracy and the method based on non-uniform mesh is of fourth-order accuracy. Three nonlinear examples with derivative dependent source functions are considered to verify the performance and theoretical rate of convergence of present methods. Moreover, we consider some special cases of the problem under consideration in order to compare our methods with other existing methods. It is shown that our second method based on cubic B-spline basis functions has the same order of convergence as quartic B-spline collocation method [1]. Moreover, our methods yield more accurate results and are computationally attractive than the methods developed in [1–8]. The proposed methods are applied on three real-life problems, the first problem describes the distribution of radial stress on a rotationally shallow membrane cap, the second problem arises in the study of thermal explosion in cylindrical vessel and the third problem arises in astronomy.

Numerical Solution of Modified Forms of Camassa-Holm and Degasperis-Procesi Equations via Quartic B-Spline Collocation Method

In this paper, a collocation finite difference scheme based on Quartic B-spline function is developed for solving non-linear modified Camassa-Holm and Degasperis-Procesi equations. A finite difference scheme and Quartic B-spline function are used to discretize the time and spatial derivatives, respectively. The obtained numerical results are compared with the exact analytical solutions and some methods existing in literature. The numerical solutions of proposed non-linear equations are acquired without any linearization technique. The convergence of the method is proved of order \((\Delta t + h^2)\). The efficiency of the proposed scheme is demonstrated through illustrative examples. The presented scheme is realized to be a very reliable alternate method to some existing schemes for such physical problems.

Quartic B-Spline Collocation Method for Eighth Order Boundary Value Problems

Quartic B-Spline Collocation Method for Eighth Order Boundary Value Problems

B-spline collocation and quasi-interpolation methods for boundary layer flow and convection heat transfer over a flat plate

The B-spline collocation methods and a new ODEs solver based on B-spline quasi-interpolation are developed to study the problem of forced convection over a horizontal flat plate, numerically. The problem is a system of nonlinear ordinary differential equations which arises in boundary layer flow. A more accurate value of $$\sigma =f^{\prime \prime }(0)$$?=f?(0) obtained by applying quartic B-spline collocation method and utilized to solve the system of ODE. The results are shown to be precise as compared to the corresponding results obtained by Howarth.

Quartic B-Spline Collocation Method for Fifth Order Boundary Value Problems

finite element method involving collocation method with quartic B-splines as basis functions have been developed to solve fifth order boundary value problems. The fifth order and fourth order derivatives for the dependent variable are approximated by the central differences of third order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain. The proposed method is tested on four linear and two non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literature. ao(x)y (5) (x) + a1(x)y (4) (x)+ a2(x)y ''' (x) + a3(x)y '' (x)+ a4(x)y ' (x) + a5(x)y(x) = b(x), c < x < d (1) subject to the boundary conditions y(c) = A0 , y(d) = B0, y ' (c) = A1, y ' (d) = B1, y '' (c) = A2, (2) where A0 , B0 , A1 , B1 , A2 are finite real constants and a0(x), a1(x), a2(x), a3(x), a4(x), a5(x) and b(x) are all continuous functions defined on the interval (c,d).

Quartic B-spline collocation method for fifth order boundary value problems

In this paper, we present a new quartic B-spline collocation method for linear and nonlinear fifth order boundary value problems. The method is fourth order convergent. It develops not only the quartic spline approximation solution but also the higher order approximation derivatives. Numerical examples are performed and comparisons with some other methods are also made. Numerical results show our method has higher accuracy and efficiency.

A numerical technique for solution of the MRLW equation using quartic B-splines

Numerical scheme based on quartic B-spline collocation method is designed for the numerical solution of modified regularized long wave (MRLW) equation. Unconditional stability is proved using Von-Neumann approach. Performance of the method is checked through numerical examples. Using error norms L 2 and L ∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the new method is established through comparison with the existing techniques.

Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation

Abstract The collocation method using quartic B-splines is described for the numerical solutions of the Burgers’ equation. The effect of the quartic B-splines in the collocation method is sought. The same method is applied to the time split Burgers’ equation. Numerical comparison of results of both algorithms and some other published numerical results are done by studying three standard problems.