Pseudo-spectral Collocation Method Research Articles

Novel Numerical Investigation of Reaction Diffusion Equation Arising in Oil Price Modeling

Consideration is given to a reaction–diffusion free boundary value problem with one or two turning points arising in oil price modeling. First, an exact (analytical) solution to the reduced problem (i.e., no diffusion term) was obtained for some given parameters. The space–time Chebyshev pseudospectral and superconsistent Chebyshev collocation method is proposed for both reaction diffusion (RDFBP) and reduced free boundary value problem. Error bounds on the discrete L2–norm and Sobolev norm (Hp) are presented. Adaptively graded intervals were introduced and used according to the value of turning points to avoid the twin boundary layers phenomena. Excellent convergent (spectrally) and stable results for some special turning points were obtained for both reduced and RDFBP equations on an adaptively graded interval and this has been documented for the first time.

On the Numerical Solution of Boundary Value Problem (BVP) of the Ordinary Differential Equation (ODE) - The Case of Steady-State Bio-Heat Equation with Combined Heat Transfer Coefficient by Pseudo-Spectral Collocation Method

Spectral methods for the solution of a boundary value problem of an ordinary differential equation are reviewed with particular emphasis laid on pseudo-spectral collocation method. The pseudo-collocation method is then used to solve the one dimensional bio-heat equation with metabolic heat generation in cylindrical coordinates applied to human tissue. It was noticed that an increase in heat transfer coefficient (hA), enhanced the temperature but a decrease in the tissue thickness was observed when this coefficient was increased. The effects of the combined heat transfer coefficient are analyzed and the results indicate that the obtained solution can be used in the study of the thermal behaviour of a biological system with the potential to locate tumours in the living tissue.

Pseudospectral convex optimization for on-ramp merging control of connected vehicles

It can be a daunting task for human drivers to merge into highways because of the intricate vehicle negotiations and potential risk within limited time and space. Connected vehicle (CV) technologies could be a solution to this problem and offer many benefits to the road safety, traffic mobility, and energy efficiency. However, real-time optimal control of CVs is still an open challenge, due to the nonlinear vehicle dynamics, non-convex fuel consumption model, and highly dynamic uncertain inter-vehicle interactions. To tackle these issues, a novel real-time optimal control approach that balances the computational efficiency and solution optimality is proposed for the purpose of onboard application. To this end, the pseudospectral collocation method is integrated with a sequential convex programming approach to develop two new optimization algorithms, which are implemented within a model predictive control (MPC) framework to allow for real-time generation of optimal merging speed profiles. One algorithm leverages the line search technique to improve convergence, and the other benefits from the trust region method for better computational efficiency. The optimality and convergence process of both proposed algorithms are investigated by comparing their solutions with a popular non-linear solver. Simulation results show that the proposed methods outperform the benchmark in terms of computational cost, fuel consumption, and traffic efficiency. In particular, the proposed fuel-economy merging rule can save 57.1% fuel consumption on average on four different traffic volumes. Meanwhile, the proposed optimal control algorithms can reduce 2.2% travel time on average comparing to the “first-in-first-out” merging rule.

Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

We introduce new differentiation matrices based on the pseudospectral collocation method. Monic Chebyshev polynomials (MCPs) were used as trial functions in differentiation matrices (D-matrices). Those matrices have been used to approximate the solutions of higher-order ordinary differential equations (H-ODEs). Two techniques will be used in this work. The first technique is a direct approximation of the H-ODE. While the second technique depends on transforming the H-ODE into a system of lower order ODEs. We discuss the error analysis of these D-matrices in-depth. Also, the approximation and truncation error convergence have been presented to improve the error analysis. Some numerical test functions and examples are illustrated to show the constructed D-matrices’ efficiency and accuracy.

Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order

The classical pseudospectral collocation method based on the expansion of the solution in a basis of Chebyshev polynomials is considered. A new approach to constructing systems of linear algebraic equations for solving ordinary differential equations with variable coefficients and with initial (and/or boundary) conditions makes possible a significant simplification of the structure of matrices, reducing it to a diagonal form. The solution of the system is reduced to multiplying the matrix of values of the Chebyshev polynomials on the selected collocation grid by the vector of values of the function describing the given derivative at the collocation points. The subsequent multiplication of the obtained vector by the two-diagonal spectral matrix, inverse with respect to the Chebyshev differentiation matrix, yields all the expansion coefficients of the sought solution except for the first one. This first coefficient is determined at the second stage based on a given initial (and/or boundary) condition. The novelty of the approach is to first select a class (set) of functions that satisfy the differential equation, using a stable and computationally simple method of interpolation (collocation) of the derivative of the future solution. Then the coefficients (except for the first one) of the expansion of the future solution are determined in terms of the calculated expansion coefficients of the derivative using the integration matrix. Finally, from this set of solutions only those that correspond to the given initial conditions are selected.

PySCP: A Multiple-Phase Optimal Control Software Using Sequential Convex Programming

Optimal control problems are common in aerospace engineering. A Python software program called PySCP is described for solving multiple-phase optimal control problems using sequential convex programming methods. By constructing a series of approximated second-order cone programming subproblems, PySCP approaches to the solution of the original optimal control problem in an iterative way. The key components of the software are described in detail, including convexification, discretization, and the adaptive trust region method. The convexification of the first-order differential dynamic equation is implemented using successive linearization. Six discretization methods, including zero-order hold, first-order hold, Runge-Kutta, and three hp pseudospectral collocation methods, are implemented so that different types of optimal control problems can be tackled efficiently. Adaptive trust region method is employed, and robust convergence is achieved. Both free-final-time problem and fixed-final-time problem can be solved by the software. The application of the software is demonstrated on three optimal control problems with varying complexity. PySCP provides researchers a useful toolkit to solve a wide variety of optimal control problems using sequential convex programming.

Optimal control of the nonspherical oscillations of encapsulated microbubbles

Encapsulated microbubbles (EMBs) were originally developed as contrast agents for ultrasound imaging but are more recently emerging as vehicles for intravenous drug and gene delivery. Ultrasound can excite nonspherical oscillations, or shape modes, that can enhance the acoustic signature of an EMB and also incite rupture, which promotes drug and gene delivery at targeted sites. Therefore, the ability to control shape modes can improve the efficacy of both the diagnosis and treatment mediated by EMBs. This work uses optimal control theory to determine the ultrasound input that maximizes a desired nonspherical EMB response (e.g., to enhance scattering or rupture), while minimizing the total acoustic input in order to enhance patient safety and reduce unwanted side effects. The optimal control problem is applied to a model of an EMB that accounts for small amplitude shape deformations. This model is solved subject to a cost function that maximizes the incidence of rupture or acoustic echo while minimizing the acoustic energy input. The optimal control problem is solved numerically through pseudospectral collocation methods using commercial optimization software. Single frequency and broadband acoustic forcing schemes are explored and compared. The results show that broadband forcing significantly reduces the acoustic effort required to incite EMB rupture relative to single frequency schemes. Furthermore, the acoustic effort required depends strongly on the shape mode that is forced to become unstable.

Accelerate the Electrolyte Perturbed-Chain Statistical Associating Fluid Theory-Density Functional Theory Calculation With the Chebyshev Pseudo-Spectral Collocation Method. Part II. Spherical Geometry and Anderson Mixing.

To improve the efficiency of electrolyte perturbed-chain statistical associating fluid theory–density functional theory (ePC-SAFT-DFT) calculation of the confined system, in this work, first, the Chebyshev pseudo-spectral collocation method was extended to the spherical pores. Second, it was combined with the Anderson mixing algorithm to accelerate the iterative process. The results show that the Anderson mixing algorithm can reduce the computation time significantly. Finally, based on the accelerated ePC-SAFT-DFT program, a systematic study of the effects of the temperature, pressure, pore size, and pore shape on the CO2 solubilities in the ionic liquids (ILs) confined inside the silica nanopores was conducted. Based on the simulation results, to obtain high CO2 solubilities in the ILs confined in silica, a better option is to use the silica material with a narrow spherical pore, and the IL-anion should be selected specifically considering that it has a more significant impact on the absorption enhancement effect.

A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method.

Rotating Black Holes in Randall-Sundrum II Braneworlds.

We find rotating black hole solutions in the Randall-Sundrum II (RSII) model, by numerically solving a three-dimensional PDE problem using pseudospectral collocation methods. We compute the area and equatorial innermost stable orbits of these solutions. For large black holes compared with the AdS length scale ℓ the black hole exhibits four-dimensional behavior, approaching the Kerr metric on the brane, while for small black holes, the solution tends instead towards a five-dimensional Myers-Perry black hole with a single nonzero rotation parameter aligned with the brane. This departure from exact four-dimensional gravity may lead to different phenomenological predictions for rotating black holes in the RSII model to those in standard four-dimensional general relativity. This Letter provides a stepping stone for studying such modifications.

Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis

<abstract><p>In this article, we evaluated the approximate solutions of one-dimensional variable-order space-fractional diffusion equations (sFDEs) by using a collocation method. This method depends on operational matrices for fractional derivatives and the integration of generalized Fibonacci polynomials. In this method, a Caputo fractional derivative of variable order is applied. Some properties of these polynomials (using boundary conditions) are presented to simplify and transform sFDEs into a system of equations with the expansion coefficients of the solution. Also, we discuss the convergence and error analysis of the generalized Fibonacci expansion. Finally, we compare the obtained results with those obtained via the other methods.</p></abstract>

RETRACTED: Theoretical study of MHD electro-osmotically flow of third-grade fluid in micro channel

• MHD electro-osmotic flow of non-Newtonian fluid. • Perturbation and Pseudo-Spectral collocation methods. • Velocity of the fluid retards subject to strong magnetic fields. • Velocity and temperature diminishing via electro-kinetic parameter Γ 2 . The electro-osmatic flow of non-newtonian fluid in a micro-channel is investigated in this article. The governing equations are derived with the help of a stress tensor then later transformed into dimensionless form with non-dimensional quantities. The perturbation method is used to obtain the approximate analytical solution of the given problem. The viscosity of the fluid is taken as a variable in terms of temperature. The validity of the perturbation solution is provided. The Pseudo-spectral collocation method is used to calculate the error in the perturbation solution of velocity and temperature. The magnitude of error in velocity and temperature is 10 −4 and 10 −2 , respectively. The impact of emerging parameters on velocity and heat profiles is also presented. The computational results reveal that the velocity profile diminishing via non-Newtonian parameter Γ 1 , electro-kinetic parameter Γ 2 , magnetic field parameter M and viscosity parameter D while an opposite behavior is noted in the velocity versus pressure gradient parameter Γ 3 , viscosity parameter E and wall's temperature θ w , respectively. Further, the temperature profile increases against the enhancement of Brinkman number, Joule heating parameter, pressure gradient parameter, wall's temperature, viscosity parameter D while reverse behavior is observed via electro-kinetic and magnetic field parameters. This study will help to understand the basic idea of MHD electro osmotic flow of non-Newtonian fluid bounded within a microchannel under the influence of variable viscosity. Further, this research can be useful to design the different types of biomedical lab-on-a-chip and thermal microfluidic devices. The developed devices can be helpful in DNA analysis and biomedical diagnosis etc.

On the steady laminar boundary-layer flow over the family of rotating tori

We present the laminar boundary layer flow over the standard tori rotating in an otherwise quiescent incompressible fluid. The mean flow equations are derived by using a specific form of the toroidal–poloidal coordinate system. A pseudo-spectral collocation method mixed with the first-order Euler scheme is employed to solve the related steady mean flow equations. A fixed aspect ratio R is defined as the ratio of radii of the torus and its tube that distinguishes each fixed torus within the general family of standard tori. Tori of spindle type turn into a sphere when the aspect ratio R=0 and thus the mean flow results for the rotating sphere are consistently reproduced. The influence of aspect ratio R on the steady mean velocity components and the wall shear stresses are presented in detail. In a precise manner, the mean flow results for each rotating torus are explainable in a consistent way to the already established case of steady laminar boundary layer flow over the rotating sphere. The significance of the current work is discussed in terms of the hydrodynamic instabilities of boundary layer flows over the family of rotating tori.

Decentralized closed-loop optimization for 6-DOF self-assembly satellites

A decentralized closed-loop optimization based on the difference of convex programming is proposed to assemble large telescopes autonomously in-orbit. This study focuses on assembly problems of mirror satellites with six degrees of freedom. Firstly, the hp-Radau pseudospectral collocation method is adopted to increase computational efficiency. Subsequently, the nonconvex constraints are convexified by applying undominated decomposition and sequential convex programming. Additionally, the fixed trajectory assumption enables the translational optimization problem to be a decentralized problem with convergence-guaranteed proof. The six degrees of freedom optimization problem is decoupled by utilizing the proposed algorithm in translational and rotational planning, and coupled by thruster allocation method. To improve the robustness of the system, the algorithm is wrapped in the model predictive control framework, and the recursive feasible property is ensured using the outer-bounding tube. Therefore, the closed-loop optimal control is robust under additive uncertainties. In the numerical experiments, mirror satellites are released from a high-accuracy fitting carrier satellite to accomplish the assembly, and these satellites are utilized to verify the high-speed algorithm and the effectiveness of the closed-loop optimization.

Accelerate the ePC-SAFT-DFT Calculation with the Chebyshev Pseudospectral Collocation Method

ePC-SAFT-DFT is a powerful tool for studying the properties of confined ionic liquids for CO2 separation, in which efficient algorithms are required to obtain the calculation efficiently. In this work, the feasibility of accelerating the ePC-SAFT-DFT calculation with the Chebyshev pseudo-spectral collocation method was discussed for the confined ionic liquid (IL)–CO2 systems. In addition, a general scheme was proposed to search the electrical boundary potential. The new algorithm was further combined with the general scheme to model the confined IL–CO2 systems. It was found that the Chebyshev pseudo-spectral collocation method can improve the efficiency of the ePC-SAFT-DFT calculation significantly. Moreover, the new algorithm can be further combined with the general scheme to efficiently describe the density profile of the IL–CO2 system inside the electroneutral nanopores.

High precision simulation of thermal-mechanical problems in functionally graded materials by spectral element differential method

Our purpose is to establish a numerical method meeting the requirements of accuracy and easy-using for thermal-mechanical analysis of functionally graded structures. Faced with these demands, a new point collocation pseudo-spectral method named spectral element differential method (SEDM), is proposed in this paper. In this paper the Chebyshev polynomial is used to obtain the derivatives of the variables with respect to intrinsic coordinates. By using the analytical expressions of the differentiation for the shape functions which are used for geometry mapping, the first- and second- derivatives of the variables with respect to global coordinates can be obtained directly. Besides, the local equilibrium equation technique is proposed to connect the elements to form the final system of equations in this paper. Based on the spectral derivative matrix and the element mapping technique, an accurate and efficient strong-form numerical method without any variational principles or energy principles can be obtained lastly for irregular domains. Some examples with complex material properties or geometries are calculated by the proposed method to illustrate the accuracy and convergence. The thermoelastic performances of functionally graded structures with different material distribution and temperature-dependent properties are investigated.

Thermal radiation and double diffusive effects on bioconvection flow of a nanofluid past an inclined wavy surface

In this article, the influence of thermal radiation on double-diffusive natural convection of a nanofluid past an inclined wavy surface in the presence of gyrotactic microorganisms is analysed. A coordinate transformation is used to transform the complex wavy surface to a smooth surface. Using the pseudo-similarity variables, the governing non-linear differential equations and their associate boundary stipulations are non-dimensionalized. The resulting system of non-linear partial differential equations is linearized using a local linearization method. The obtained linear system of partial differential equations is solved using the bivariate Chebyshev pseudo-spectral collocation method. The effect of pertinent physical and geometrical parameters on the physical quantities of the flow are illustrated through graphs and analyzed.

Analytical Study on Couple Stress Fluid in an inclined Channel

Numerical and analytical solutions of Stokes theory of couple stress fluid under the effects of constant, space, and variable viscosity in the inclined channel are discussed here. The considered couple stress fluid is described mathematically with the definition of the stress tensor. The dimensional form of the boundary value problem is transformed into dimensionless form by defining dimensionless quantities and then solved with help of the perturbation technique. The analytical expressions of velocity and temperature of all cases based on the viscosity of the couple stress fluid are presented. For the validity of the perturbation solution, the Pseudo-Spectral collocation method is employed for each case of the viscosity model including constant, space, and Vogel’s models, respectively. The solution of the perturbation method and Pseudo-Spectral methods are shown together in the graphs. The effects of couple stress parameters on velocity and temperature distributions are also elaborated with physical reasoning in the results and discussion part. It is observed that velocity and temperature of fluid escalate via the pressure gradient parameter and Brinkman number while decelerating via couple stress parameter.

Numerical and scale analysis of non-Newtonian fluid (Eyring-Powell) through pseudo-spectral collocation method (PSCM) towards a magnetized stretchable Riga surface

A numerical simulation of a two-dimensional Eyring-Powell nanofluid flow is carried out through a Riga plate with radiative walls. Simultaneous contributions of chemical reaction and activation energy have also been taken into account on the mixed convection flow. Lubrication effects are applied by considering second order slip boundary condition in order to reduce the role of skin friction. A set of highly nonlinear and coupled differential equations are tackled with the help of “Pseudo-spectral collocation method” by developing numerical code on MATLAB. A detailed parametric study is performed that reveals that more energy adds to the system in response to slip parameter and Brownian motion of the nano-particles. Numerical results have also been compiled and tabulated. This is worth noticing that skin friction coefficient reduces due to modified Hartmann factor while heat flux is an increasing function of Prandtl number. However, there is a prominent increase in mass flux against all parameter, except activation energy.

Numerical simulation of advection–diffusion equation with caputo-fabrizio time fractional derivative in cylindrical domains: Applications of pseudo-spectral collocation method

The study of the unsteady fractional advection–diffusion equation (ADE) is carried out in cylindrical geometry along with time-exponential concentration on a cylindrical surface. We have used the Caputo-Fabrizio time-fractional derivative for the fractional model of the advection–diffusion. The analytical solutions for the solute concentration are determined by using integral transformations. For comparison, we also present a numerical scheme to get a numerical solution for different parameters' values. For time ordinary derivative approximation, we use the finite difference method. For space derivatives, we use a pseudo-spectral collocation method for higher-order accuracies. The advection-diffusion’ classical model is obtained by taking the fractional parameter to be unit. The influence of the memory, namely, the Caputo-Fabrizio time-fractional derivative on the solute concentration is studied and compared with the ordinary case. Also, the impact of the drift velocity is analyzed by employing the Peclet number. It has been found that the concentration is decreasing with the Peclet number and is increasing with the radial coordinate. The present investigation will be helpful in future research to use a higher-order approximation for ordinary derivatives. As the derivatives in space are ordinary derivatives, we use highly accurate pseudo-spectral collocation approximation for them.