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The main aim of the paper is to develop a new B-splines collocation algorithm based on modified cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic Burgers’-type equations with Dirichlet boundary conditions.

A modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and an algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithm reduced the Burgers’ equations into a system of first-order nonlinear ordinary differential equations in time variable. Then, strong stability preserving Runge-Kutta 3rd order (SSP-RK3) scheme is used to solve the obtained system.

A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from to the schemes developed in Abbas et al. (2014) and Nazir et al. (2016), and the developed algorithms are free from linearization process and finite difference operators.

To the best knowledge of the authors, this technique is novel for solving nonlinear partial differential equations, and the new proposed technique gives better results than the results discussed in Ozis et al. (2003), Kutluay et al. (1999), Khater et al. (2008), Korkmaz and Dag (2011), Kutluay et al. (2004), Rashidi et al. (2009), Mittal and Jain (2012), Mittal and Jiwari (2012), Mittal and Tripathi (2014), Xie et al. (2008) and Kadalbajoo et al. (2005).

The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc.

Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the Thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed.

A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014; Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations.

To the best of the authors’ knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008; Batiha et al., 2007; Mittal and Bhatia, 2013; Jiwari, 2015).

The purpose of this paper is to present a modified form of trigonometric cubic B-spline (TCB) collocation method to solve nonlinear Fisher’s type equations. Taylor series expansion is used to linearize the nonlinear part of the problem. Five examples are taken for analysis. The obtained results are better than those obtained by some numerical methods as well as exact solutions. It is noted that the modified form of TCB collocation method is an economical and efficient technique to approximate the solution PDEs. The authors also carried out the stability analysis which proves that the method is unconditionally stable.

The authors present a modified form of TCB collocation method to solve nonlinear Fisher’s type equations. Taylor series expansion is used to linearize the nonlinear part of the problem. The authors also carried out the stability analysis.

The authors found that the proposed method results are better than those obtained by some numerical methods as well as exact solutions. It is noted that the modified form of TCB collocation method is an economical and efficient technique to approximate the solution PDEs.

The authors propose a new method, namely, modified form of TCB collocation method. In the authors’ best knowledge, aforesaid method is not proposed by any other author. The authors used this method to solve nonlinear Fisher’s type equations and obtained more accurate results than the results obtained by other methods.

The purpose of this paper is to present a stable and efficient numerical technique based on modified trigonometric cubic B-spline functions for solving the time-fractional diffusion equation (TFDE). The TFDE has numerous applications to model many real objects and processes.

The time-fractional derivative is used in the Caputo sense. A modification is made in trigonometric cubic B-spline (TCB) functions for handling the Dirichlet boundary conditions. The modified TCB functions have been used to discretize the space derivatives. The stability of the technique is also discussed.

The obtained results are compared with those reported earlier showing that the present technique gives highly accurate results. The stability analysis shows that the method is unconditionally stable. Furthermore, this technique is efficient and requires less storage.

The current work is novel for solving TFDE. This technique is unconditionally stable and gives better results than existing results (Ford et al., 2011; Sayevand et al., 2016; Ghanbari and Atangana, 2020).

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree.

The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program.

There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method.

A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids.

Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also.

ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

The 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.

The 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.

A 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.

The 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.

The purpose of this paper is to check the efficiency of isogeometric analysis (IGA) by comparing its results with classical finite element method (FEM), generalized finite element method (GFEM) and other enriched versions of FEM through numerical examples of free vibration problems.

Since its conception, IGA was widely applied in several problems. In this paper, IGA is applied for free vibration of elastic rods, beams and trusses. The results are compared with FEM, GFEM and the enriched methods, concerning frequency spectra and convergence rates.

The results show advantages of IGA over FEM and GFEM in the frequency error spectra, mostly in the higher frequencies.

Isogeometric analysis shows a feasible tool in structural analysis, with emphasis for problems that requires a high amount of vibration modes.

A collocation technique based on re-defined quintic B-splines over Crank-Nicolson is presented to solve the Fisher's type equation. We take three cases of aforesaid equation. The stability analysis and rate of convergence are also done.

The quintic B-splines are re-defined which are used for space integration. Taylor series expansion is applied for linearization of the nonlinear terms. The discretization of the problem gives up linear system of equations. A Gaussian elimination method is used to solve these systems.

Three examples are taken for analysis. The analysis gives guarantee that the present method provides much better results than previously presented methods in literature. The stability analysis and rate of convergence show that the method is unconditionally stable and quadratic convergent for Fisher's type equation. Moreover, the present method is simple and easy to implement, so it may be considered as an alternative method to solve PDEs.

This work is the original work of authors which is neither published nor submitted anywhere else for publication.

Numerical instability such as spurious oscillation is an important problem in the simulation of heat wave propagation. The purpose of this study is to propose a time discontinuous Galerkin isogeometric analysis method to reduce numerical instability of heat wave propagation in the medium subjected to heat sources, particularly heat impulse.

The essential vectors of temperature and the temporal gradients are assumed to be discontinuous and interpolated individually in the discretized time domain. The isogeometric analysis method is applied to use its property of smooth description of the geometry and to eliminate the mesh-dependency. An artificial damping scheme with proportional stiffness matrix is brought into the final discretized form to reduce the numerical spurious oscillations.

The numerical spurious oscillations in the simulation of heat wave propagation are effectively eliminated. The smooth description of geometry with spline functions solves the mesh-dependency problem and improves the numerical precision.

The time discontinuous Galerkin method is applied within the isogeometric analysis framework. The proposed method is effective in the simulation of the wave propagation problems subjecting to impulse load with numerical stability and accuracy.