B-spline Differential Quadrature Method Research Articles

Fractal patterns in fluid flows: A radial basis pseudospectral analysis of CuO-EG water nano, Saffman dusty, and plain Newtonian fluids in immiscible layers of a horizontal channel

Nano and dusty fluids provide new possibilities to improve energy transfer efficiencies. Motivated by immiscible fluids duct flow framework in the filtration of hydrocarbon materials, the objective of this manuscript is to present the numerical study of three immiscible fluids namely, copper oxide-ethylene glycol (CuO-EG) water nano, Saffman dusty, and plain Newtonian fluids in the horizontal channel. For the formulation of the mathematical model, the dusty fluid is positioned between the immiscible layers of the nanofluid and Newtonian fluid. The interface between the immiscible fluids is continuous and non-deformable in nature. The flow is driven by an externally time-dependent pressure gradient. The novel radial basis function pseudospectral (RBFPS) method is employed to solve the governed partial differential equations, and the resulting outcomes are compared with cubic B-spline differential quadrature and finite difference method. The impact of several crucial fluid parameters, comprising the volume percentage of CuO nanoparticles, the dust particle concentration parameter, and various others on velocity profiles, are presented graphically. It is observed that despite being present only in the lower zone, the CuO nanoparticles have an impact on the flow velocities in other zones also. The fluid and particle velocities decrease as the volume fraction of CuO increases.

Numerical simulation of one-dimensional nonlinear Schrodinger equation using PSO with exponential B-spline

Particle swarm optimization (PSO) is a widely adopted, rapid, efficient, and computer-friendly technique for optimization in various fields, including computer science, engineering, and mathematical modelling. In this paper, the nonlinear Schrodinger (NLS) equation has been solved numerically, which leads to soliton-type solutions, by utilizing the “Exponential cubic B-spline differential quadrature method (Expo-MCB-DQM) with PSO”. The exponential cubic B-spline basis functions involve a parameter that must be assigned a value to find the numerical solutions. Until now, the value of a parameter had been determined by the hit-and-trial approach, which leads to unstable results. By employing the PSO technique, the optimal value of the parameter is determined, which enhances the accuracy of the numerical solutions. This study introduces an innovative combination of Expo-MCB-DQM with PSO, which is an innovative combination that optimizes the parameters in the basis function, leading to improved outcomes. To validate the authenticity and effectiveness of this methodology, it has been applied to two problems to present the numerical results in comparison with the exact solution. Additionally, the paper provides insightful discussions on the significant application of the NLS equation in optical fiber communication.

Numerical Simulation and Dynamics of Burgers’ Equation Using the Modified Cubic B-Spline Differential Quadrature Method

In the present work, a numerical approach using the Crank–Nicolson scheme along with the modified cubic B-spline differential quadrature (CN-MCDQ) method is proposed to find the numerical approximations to Burgers’ equation. After applying the well-known Crank–Nicolson technique, Burgers’ equation is solved in this study by using the differential quadrature approach to approximate the derivatives that lead to a system of equations to be solved. When compared to other methods for obtaining numerical solutions, the proposed method is shown to be efficient and easy to implement while still providing accurate results. The obtained results are in agreement with the earlier available approaches and are even better in comparison in terms of less domain partition. Three test problems were used to evaluate the methodology, and the results are tabulated and graphically shown below.

Numerical study of generalized 2-D nonlinear Benjamin–Bona–Mahony–Burgers equation using modified cubic B-spline differential quadrature method

The objective of this work is to present the modified cubic B-spline differential quadrature (MCBSDQ) method for the numerical study of the generalized 2-D nonlinear Benjamin–Bona–Mahony–Burgers (BBMB) equation. A detailed description of the proposed method is provided. Formulation of the generalized BBMB equation is done using the proposed technique. The method’s stability analysis is discussed. The accuracy and efficiency are shown through numerical examples. The numerical results of this study have been compared with those of other existing methods in the literature. The proposed method has excellent agreement with other methods which are used to solve the generalized 2-D nonlinear BBMB equation.

A Numerical Approximation of 2D Coupled Burgers’ Equation Using Modified Cubic Trigonometric B-Spline Differential Quadrature Method

In the present paper, trigonometric B-spline DQM is applied to get the approximated solution of coupled 2D non-linear Burgers’ equation. This technique, named modified cubic trigonometric B-spline DQM, has been used to obtain accurate and effective numerical approximations of the above-mentioned partial differential equation. For checking the compatibility of results, different types of test examples are discussed. A comparison is done between <i>L2</i> and <i>L</i><sub>∞</sub> error norms with the previous, present results and with the exact solution. The resultant set of ODEs has been solved by employing the SSP RK 43 method. It is observed that the obtained results are improved compared to the previous numerical results in the literature.

Numerical solution of Generalized Burger–Huxley & Huxley’s equation using Deep Galerkin neural network method

In this paper, a deep learning algorithm based on Deep Galerkin method (DGM) is presented for the approximate solution of the generalized Burgers–Huxley equation (gBHE), and generalized Huxley’s equation (gHE). In this method, a deep neural network (DNN) is used for approximating the solution without generating mesh grid, which satisfies the differential operator, boundary and initial conditions. DNN is trained on randomly selected batches of time and space points, thus helping to avoid forming a mesh. Adam optimizer is used for optimizing the parameters of the DNN. Further, the convergence of the cost function and convergence of the neural network to the exact solution is demonstrated. This method shows very encouraging results which have been compared with recent methods such as: A fourth order improved numerical scheme (FDS4), Adomain-decomposition method (ADM), Modified cubic B-spline differential quadrature method (MCB-DQM), Variational iteration method (VIM), and others.

Numerical solutions of nonlinear Schrodinger equation with applications in optical fiber communication

Optical soliton is a nonlinear wave with huge information or data which travels over larger distance in optical fibers. This paramount application of optical soliton forced many mathematicians and physicians to focus on it which initiates to study the soliton solutions of mathematical model in optical fiber communication characterized by nonlinear Schrodinger equation (NLSE) along with its important applications. Nonlinear Schrodinger equation represents propagation of light pulse through an optical fiber. In this paper NLSE is solved numerically by using exponential cubic B-spline differential quadrature method. In this method leave-one out cross validation strategy is being used to find optimal value of parameter p which enhances accuracy and efficiency of its results which are near to exact solutions and approximate to available numerical solutions in the literature.

Soliton solution of coupled Korteweg-de Vries equation by quintic UAH tension B-spline differential quadrature method

The concern of this paper is to acquire the numerical solution of Coupled Korteweg-de Vries (CKdV) equation describing the behaviour of shallow water surfaces in terms of solitons. A new regime quintic Uniform Algebraic Hyperbolic (QUAH) tension B-spline with a very well-known method called Differential Quadrature Method (DQM) is used. Over the past few years, the idea of consuming DQM to solve partial differential equations numerically has acknowledged much consideration all over the science community. Initially, the QUAH tension B-spline is used as a basis function in DQM to solve the weighting coefficients. After calculating weighting coefficients, the basis function converts the Initial boundary value problem into an ordinary differential equation, which is further solved by using a strong preserving Runge-Kutta 43 regime. Then, to check the accuracy and amendment of the used method, different test problems are solved numerically as well as error norms L2 and L∞ are calculated. Calculated results are shown graphically and in tabular form for quick and easy access. An obtained result gives the insight that the proposed technique is effective and efficient for elucidating the variety of complex partial differential equations.

An Algorithm developed by implementing Modified Cubic Hyperbolic B-Spline based Differential Quadrature Method on non-linear Burgers’ equation

Present paper proposed a novel regime has been proposed named “Modified Cubic Hyperbolic B-Spline Differential Quadrature Method.” In the present method, a cubic Hyperbolic B-spline is used as a test function in the “Differential quadrature method.” This method has been employed to test the effectiveness and feasibility of approximation of the 1-D Burgers’ equation. In order to check the compatibility of the proposed method, a variety of test problems are discussed. A comparison is made between L2 and L∞ error norms by considering some examples already present in literature. The corresponding set of Ordinary Differential Equations is solved by using the Strong-Stability-Preserving Runge-Kutta (43) method, i.e. (SSP-RK43) method. It has been observed that obtained results have excellent compatibility with numerical approximations discussed in literature. Stability of developed regime is addressed by eigen values method.”

Numerical solution of coupled 1D Burgers' equation by Non-Uniform Algebraic-Hyperbolic B-spline Differential Quadrature Method

Present paper deals with the numerical solution of coupled 1D Burgers’ equation by implementing the Non-Uniform Algebraic Hyperbolic (NUAH) B-spline Differential Quadrature Method. In the present paper, the spatial variable discretization is done using NUAH B-spline, and the obtained system of ODE is dealt with using the SSP-RK43 scheme. To get the improvised results, the concept of modified cubic NUAH B-spline is incorporated. To test the effectiveness and accuracy of the proposed scheme, numerical examples are discussed. Stability analysis of the proposed scheme is investigated by the matrix stability analysis method. The present regime is worthwhile to deal with some complex natured PDEs, where finding the analytical solution is cumbersome.

Numerical Study of Interface Tracking for the Unsteady Flow of Two Immiscible Micropolar and Newtonian Fluids Through a Horizontal Channel with an Unstable Interface

The dynamics of the interaction between immiscible fluids is relevant to numerous complex flows in nature and industry, including lubrication and coating processes, oil extraction, physicochemical separation techniques, etc. One of the most essential components of immiscible flow is the fluid interface, which must be consistently monitored. In this article, the unsteady flow of two immiscible fluids i.e., an Eringen micropolar and Newtonian liquid is considered in a horizontal channel. Despite the no-slip and hyper-stick shear stress condition at the channel edge, it is accepted that the liquid interface is dynamic, migrating from one position to the next and possibly get absolute change; as a result, The CS (continuum surface) model is integrated with the single moment equation based on the VOF (volume of fluid) approach to trace the interface. The immiscible fluids are considered to flow under three applied pressure gradients (constant, decaying, and periodic) and flow is analyzed under seamless shear stress over the entire interface. The modified cubic b-spline differential quadrature method (MCB-DQM) is used to solve the modeled coupled partial differential equations for the fluid interface evolution. The advection and tracking of the interface with time, wave number, and amplitude are illustrated through graphs. It is observed that the presence of micropolar parameters affects the interface with time. The novelty of the current study is that previous studies (which considered the smooth and unstable movement of the micropolar fluid, the steady stream of two immiscible fluids, and interface monitoring through different modes) are extended and generalized to consider the time-dependent flow of two immiscible fluids namely Eringen micropolar and Newtonian with a moving interface in a horizontal channel. For the decaying pressure gradient case, which requires more time to achieve the steady-state, the peak of the waves resembles those for the constant pressure gradient case. The interface becomes steady for a more extensive time when a constant pressure gradient is applied. The interface becomes stable quickly with time as the micropolar parameter is decreased for the constant pressure gradient case i.e., weaker micropolar fluids encourage faster stabilization of the interface. With periodic pressure gradient, the interface takes more time to stabilize, and the crest of the waves is significantly higher in amplitude compared to the constant and decaying pressure cases. The simulations demonstrate the excellent ability of MCB-DQM to analyze complex interfacial immiscible flows.

Numerical approximation of 1D and 2D non-linear Schrödinger equations by implementing modified cubic Hyperbolic B-spline based DQM

In present study, Modified Cubic Hyperbolic B-spline Differential Quadrature Method is constructed for getting the numerical approximations of 1D and 2D non-linear Schrödinger equations. Modified cubic Hyperbolic B-spline is implemented as basis function to compute the required weighting coefficients regarding DQM. Thereafter, by spatial discretization, obtained system of ordinary differential equations is obtained which is tackled using Strong Stability Preserving-Runge–Kutta 43 scheme. Accuracy of the proposed scheme is tested by comparing the exact and numerical approximations. L 2 and L ∞ errors are computed for different parameters.

Numerical study of time dependent flow of immiscible Saffman dusty (fluid-particle suspension) and Eringen micropolar fluids in a duct with a modified cubic B-spline Differential Quadrature method

Immiscible flows arise in many diverse applications in mechanical, chemical, and environmental engineering. Such flows involve interfacial conditions and often feature mass (species) diffusion. Motivated by applications in non-Newtonian duct processing, in the present article a comprehensive mathematical model and computational simulation with the modified cubic B-spline-Differential Quadrature method (MCB-DQM) is described for the unsteady flow of two immiscible fluids - dusty (fluid-particle suspension) and Eringen micropolar fluids - through horizontal channels. Mass transfer is invoked due to particle concentration effects in the dusty fluid. The stable liquid-liquid interface is considered between two immiscible fluids. Fluids are considered to flow under three different pressure gradients- constant, decaying, and periodic pressure gradient and the flow characteristics are scrutinized for each case. The coupled partial differential equations are solved with the MCB-DQM under physically realistic boundary conditions. Linear velocity, micro-rotation (Eringen angular velocity) is visualized graphically for the effects of the key hydrodynamic and solutal parameters i. e. Reynolds number, particle concentration parameter, Eringen micropolar material parameter, volume fraction parameter, pressure gradient, time, viscosity ratio, and density ratio. The simulations extend the current understanding of two-fluid interfacial duct hydrodynamics and mass transfer and are relevant to chemical engineering separation processing systems.

Numerical simulation of interface tracking between two immiscible micropolar and dusty fluids

The development of the interfacetracking between two immiscible liquids is one of the most challenging problems in fluid dynamics. The simulation of the lubrication process, oil extraction, and the physicochemical mechanism that govern the behaviour of the uncertain conduct of interface is a key element in “gas-oil” or “oil–water” natural gas-oil reservoirs systems and should be tracked for optimal recovery. Motivated by the interface evolution process, in this article, an unsteady flow of two immiscible Eringen micropolar and Dusty (fluid-particle suspension) fluids is considered through a horizontal channel and the functioning of the interface is analysed. Although the channel wall is hyper-stick and no-slip, it is believed that the fluid–fluid interface is unsteady and can also be deformed from one location to another; thus, the single momentum equation is coupled for monitoring the interface profiles using the VOF (Volume of fluid) technique. The modelled coupled- partial differential equations are numerically solved by the modified B-spline differential quadrature method. The interface profile under the influence of different parameters, i.e., Froude, Capillary, wave and Reynolds number, amplitude, the ratio of viscosities, and time is analysed. Three applied pressure gradients namely constant Ge = 10, periodic Ge = 10Sin(t), and decaying Ge = e-t are considered with various constant values of other hydrodynamic parameters. It's been noted that thetopologydevelops with time asthe pulsing pattern repeats quicker for a longer period before becoming stable. The flow's qualitative characteristics remain preserved, the interface begins to move vertically as the amplitude is elevated. A higher interfacial drift is seen in the applied periodic pressure gradient case as compared to the constant and decaying pressure gradient scenario. The tracking does take high time to be stable in the presence of the micropolar parameter with all three applied pressure gradient cases.

A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.

Modification of quintic B-spline differential quadrature method to nonlinear Korteweg-de Vries equation and numerical experiments

In this manuscript, a novel modification of quintic B-spline differential quadrature method is given. By this modification, the obtained algebraic system do not have any ghost points. Therefore, the equation system is solvable and doesn't need any additional equation. Another advantage of this modification is the fact that the obtained numerical results are better than classical methods. Finite difference method is used as another component of the algorithm to obtain numerical solution of Korteweg-de Vries (KdV) equation which is the important model of nonlinear phenomena. Nine effective experiments namely single soliton, double solitons, interaction of two solitons, splitting to the solitons, interaction of three-, four-, and five solitons, evolution of solitons via Maxwellian initial condition and train of solitons are given with comparison of error norms and invariants. All of the given results with earlier studies show that the present scheme has improved classical common methods. Rate of convergence for single soliton and double solitons problems are reported.

A numerical regime for 1-D Burgers’ equation using UAT tension B-spline differential quadrature method

Present work deals with the numerical solution of 1D nonlinear Burgers’ equation. In this article, modified cubic uniform algebraic trigonometric tension B-spline is implemented as the basis function. Modified cubic UAT tension B-spline is incorporated in the differential quadrature method to fetch the values of weighting coefficients, as finding the weighting coefficients is the main key in differential quadrature method. After the spatial discretization of the equations, the reduced system of ordinary differential equations is obtained, which is tackled by employing the SSP-RK43 scheme. Accuracy of the present regime is verified by implementing notion of and error norms. On making comparisons with the earlier outcomes, it is noticed that present regime has produced better results, as well as is easy to implement. Main outcome of this work lies in findings of the better numerical approximations of some linear and nonlinear partial differential equations, specifically where the analytical solutions do not exist.

Numerical approximation of coupled 1D and 2D non-linear Burgers’ equations by employing Modified Quartic Hyperbolic B-spline Differential Quadrature Method

In this paper, the numerical solution of coupled 1D and coupled 2D Burgers' equation is provided with the appropriate initial and boundary conditions, by implementing "modified quartic Hyperbolic B-spline DQM". In present method, the required weighting coefficients are computed using modified quartic Hyperbolic B-spline as a basis function. These coupled 1D and coupled 2D Burgers' equations got transformed into the set of ordinary differential equations, tackled by SSPRK43 scheme. Efficiency of the scheme and exactness of the obtained numerical solutions is declared with the aid of 8 numerical examples. Numerical results obtained by modified quartic Hyperbolic B-spline are efficient and it is easy to implement

Study of One Dimensional Hyperbolic Telegraph Equation Via a Hybrid Cubic B-Spline Differential Quadrature Method

Hybrid cubic B-spline differential quadrature method (in brief, HCB-DQM) have been proposed for the study of one space dimensional time dependent partial differential equations. Specially, we have applied the proposed method for 1D telegraph equation (1D TE). The HCB-DQM is based on DQM with hybrid cubic-B-splines as base functions, and so, the 1D TE is reduced to a system of first order ordinary differential equations (ODEs) which is solved via SSP-RK43 scheme. The accurateness and efficiency of the proposed method is tested for four examples of 1D TE via $$L_2$$ and $$L_\infty $$ errors. It is demonstrated that proposed HCB-DQM produces stable solutions for 1D telegraph equation. The evaluated solutions are compared with the exact results. Also, we compare the evaluated results with existing recently published results. It is seen that the produced results are agreed well with the exact solutions. Easy to implementation, economically and straightforwardness is main advantage of HCB-DQM.

Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method

In this paper, quintic B-spline differential quadrature method (QBDQM) has been used to obtain the numerical approximation of the combined Korteweg-de Vries and modified Korteweg-de Vries equation (combined KdV-mKdV). The efficiency and effectiveness of the proposed method has been tested by computing the maximum error norm L_\infinity and discrete root mean square error L_2 . The newly found numerical approximations have been compared to available numerical approximations and this comparison has shown that the proposed method is an efficient one for solving