Space Derivatives Research Articles

An analysis on time optimal feedback control for Caputo fractional neutral systems with delay

ABSTRACT This paper aims to provide a systematic approach to investigate the optimal feedback control systems governed by the fractional neutral evolution system with fixed delay in the sense of Caputo fractional derivatives in separable reflexive Banach spaces. Initially, we study the existence result of mild solutions for the system by applying fixed point theorem for a condensing map due to Martelli. The existence of feasible pairs is obtained by using the Filippov theorem and the Cesari property. Furthermore, the existence of optimal feedback control and time optimal control are studied for the corresponding control problem. Finally, an example is given to validate our findings.

Frequency space derivation of linear and nonlinear memory gravitational wave signals from eccentric binary orbits

Frequency space derivation of linear and nonlinear memory gravitational wave signals from eccentric binary orbits

Application of an Extended Cubic B-Spline to Find the Numerical Solution of the Generalized Nonlinear Time-Fractional Klein–Gordon Equation in Mathematical Physics

A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The flexibility, symmetry and high-order accuracy of the B-spline functions make it possible to tackle the best solutions. In this study, extended cubic B-spline (ECBS) functions are utilized for the numerical solutions of the generalized nonlinear time-fractional Klein–Gordon Equation (TFKGE). Initially, the Caputo time-fractional derivative (CTFD) is approximated using standard finite difference techniques, and the space derivatives are discretized by utilizing ECBS functions. The stability and convergence analysis are discussed for the given numerical scheme. The presented technique is tested on a variety of problems, and the approximate results are compared with the existing computational schemes.

Numerical Studies on the Classical Long Wave System Describing Shallow Water Waves With Dispersion

In this study, a reliable and efficient local discontinuous Galerkin scheme for numerically solving the classical long wave system is proposed. This scheme approximates the temporal derivatives using the explicit strong-stability-preserving higher-order Runge-Kutta method, while the space derivatives are approximated using the local discontinuous Galerkin method, yielding an ordinary differential equation system. Numerical examples for various test problems are presented to offer a comprehensive understanding of the accuracy and reliability of the proposed method. The results obtained, which confirm the sub-optimal order of accuracy, are displayed in multiple tables. Additionally, several two-dimensional and three-dimensional graphical depictions of the problem have been provided to illustrate the behavior of the solution.

An FFT based adaptive grid framework to represent non-singular dislocations

Non-singular theories aim at regularizing the unrealistic stress singularity around a dislocation line predicted by elasticity by spreading the Burgers vector around the core. The use of these approaches in discrete dislocation models or field dislocation mechanics requires very fine discretization around the dislocation lines. FFT solvers commonly used in these problems rely on a regular grid which enforces the use of very fine discretizations or very small domains. In this work a methodology is proposed to solve the boundary value problems associated with dislocation modeling using FFT with an adaptive grid which is refined around dislocation cores. The framework introduces a regular domain to compute Fourier derivatives mapped to a physical domain with more points concentrated in the areas of interest. The linear differential operators involved are obtained by the product of the Fourier derivatives in the regular space and the Jacobian of the map defining the transformation between computational and physical domains. The periodic boundary value problem involved is transformed into a linear system that can be efficiently solved using Krylov solvers. The method for adaptive FFT grids is fully general and can be applied to any other micromechanical problem. It is demonstrated that the method allows to use several grid points within the core region of 2D and 3D dislocations even using coarse discretizations, allowing to resolve the non-singular stress fields within this region and also strongly reducing the numerical noise. Moreover, the method allows to preserve the accuracy of the results in fine meshes by reducing the grid spacing up to four times.

Explicit P1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space.

An adaptive low-rank splitting approach for the extended Fisher–Kolmogorov equation

The extended Fisher–Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biological systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence of the proposed scheme is proved rigorously. Based on the frame of the full-rank splitting scheme, we design a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve the energy dissipation.

Stress wave in the mesoscopic discontinuous medium by fractional approach

A fractional approach based on the generalized Taylor's formula is proposed to describe the dynamic responses in the meso-discontinuous medium (e.g., oyster shell). The governing equations with the Caputo-defined fractional derivatives in space are derived. In order to describe the dynamic effect of geometric structures in the meso- discontinuous medium, the finite element method (FEM) is employed together with the fractional governing equations to establish the relation of the equivalent fractional order (α¯) with the fractal dimension (D), and the linear relation is obtained as: α¯=0.99D−0.97. The results also show that the equivalent effects of porous structure on wave propagation could be determined by fractal dimension and fractional order. The CO2 pulse laser together with VISAR measurement is then employed to investigate the dynamic responses in the oyster shell specimens. The evaluated equivalent fractional order decreases with increasing of the density, and the equivalent fractional orders (α¯) of pure nacre and pure chalk are estimated to be respectively 0.61 and 0.69, while the estimated α¯ of pure nacre and pure chalk by FEM are 0.63 and 0.79. The difference of estimated α¯(0.79) with actual α¯(0.69) of chalk is due to the irregular interface, which would cause greater wave attenuation. Due to the stronger and much more meso-discontinuous interfaces (i.e., the brick-mud structures) in the nacre, wave amplitude decays more seriously in the nacre (denser with higher density) than those in the chalk (looser with lower density), and measured velocities decrease with increasing of the density. This work provided a new approach to investigate the dynamic mechanical properties of meso-discontinuous mediums.

Double upscaling procedure for the Sine–Gordon equation with highly-oscillating coefficients: Homogenization and modulation equations

We study the sine-Gordon equation with h-periodic in space coefficients. Leading-order homogenization yields an effective sine-Gordon equation for which traveling wave periodic solutions of wavelength λ≫h can be determined. The periodic solutions are then modulated on a scale Λ≫λ. As we know, the corresponding Whitham equations are elliptic, which ensures that the periodic solution is unstable. However, the instability scenarios are not universal. In this paper, such scenarios are described both in the low and high energy regimes and for supersonic compared to the averaged sound speed case. In the low energy case the space derivatives of the solutions “explode” in finite time (a caustic appears), while in the high energy case the solutions grow at most linearly in time.

Temporal Regularity of Symmetric Stochastic p-Stokes Systems

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An improved axisymmetric interfacial lattice Boltzmann flux solver for large-density-ratio multiphase flows

In this paper, an improved axisymmetric interfacial lattice Boltzmann flux solver abandoning the previous predictor-corrector scheme is proposed. Unlike the previous model starting from the two-dimensional standard lattice Boltzmann method (LBM), the present method is developed using the axisymmetric LBM, which directly incorporates the axisymmetric effects into the distribution functions. As a result, the proposed solver does not need the corrector step involving complicated space derivatives. It makes this method simpler and more computationally efficient. In the present solver, the resultant governing equation is globally resolved by the finite volume method, while the fluxes are reconstructed by local application of the axisymmetric LBM. Therefore, the inconsistency between the local reconstruction and the global governing equation can be eliminated because the global equation can be strictly derived from the axisymmetric LBM, which holds stronger physical basis than the previous method. Numerical experiments about the interface capturing and the multiphase flows are conducted to test the proposed model. Results show that the present method is superior to the fractional step solver in terms of the accuracy, stability, and computational efficiency. In addition, this solver has the capacity of simulating large-density-ratio and complex interfacial change.

Impact of fractional magnetohydrodynamic and hall current on ree-eyring fluid flow by using radial basis function method

In the present work, an analysis has been arranged to study the Magnetohydrodynamic of Ree-Eyring fluid with the effect of Hall current between two parallel plates. Initially, both plates are at rest then suddenly plates are bought to steady velocities v0 and v1 at y=0 and y=d respectively. The governing problem is converted to the fractional partial differential equation and then numerically solved by using the radial basis functions collocation scheme. The Crank-Nicolson scheme is applied for the discretization of space derivatives. The numerical computation is performed for 0≤α≤1,0≤m≤1,0≤β≤1. The performance of the method is investigated with the help of tabulated results. The obtained results show the efficiency of the method for different parameters. The numerical results are compared for different fractional orders, and it is concluded that decreasing the fractional order increases the velocity which enhances the flow. Results are tabulated for different time levels, and it is concluded that method produces smooth solution for increasing values of T. Time fractional derivatives play a significant role in understanding non-Newtonian fluids and complex flow behaviors. They accurately model resistance forces in fluid flow, particularly in systems with non-standard behavior. This allows for a more detailed analysis of transient phenomena, providing valuable insights into fluid behavior over time. Ultimately, the use of time fractional derivatives enhances our capacity to describe and predict fluid flow in various practical applications such as physiology, biomedicine, and engineering.

THE NEWTON-RAPHSON METHOD OF REAL-VALUED FUNCTIONS IN DISCRETE METRIC SPACE

This paper studies the Newton-Raphson method to approximate a root of a real-valued function in one-dimensional real discrete metric space. The method involves a derivative and is considered to be convergent very fast. However, the derivative is derived from the limit definition with respect to the Euclidean distance, different from that of the discrete metric space. This research investigates the Newton-Raphson method with respect to derivatives defined in discrete metric spaces by deriving the derivative first. The results show that the constructed Newton-Raphson method can be an alternative root-finding method exemplified by some examples.
 Keywords: Newton-Raphson method, discrete metric space, metric space derivative

A new MIB-based time integration method for transient heat conduction analysis of discrete and continuous systems

By incorporating the matched interface and boundary (MIB) method into time integration computation, a new MIB-based time integration (MIBTI) approach is proposed for solving transient heat conduction problems in this work. Indeed, this is a truly effective time-domain MIB method that can directly discretize space and time derivatives, and the transient heat conduction problem (an initial-value problem) is transformed into a pseudo-space boundary-value problem. The initial and final time points are regarded as boundary points in the transformed space domain. Two algorithms are developed to deal with the final time point. Unlike conventional recursive schemes for time integration, all unknown variables can be determined by solving the transformed boundary-value problem in the present approach. Various discrete and continuous systems without and with internal heat sources under different boundary conditions are considered to examine the working performance of the proposed method. The applicability of the time stability criterion in the conventional recursive schemes to the proposed method is also discussed. It is found that excellent numerical stability and convergence can be achieved by using the present MIBTI method for solving transient heat conduction problems.

Extension of Haar wavelet technique for numerical solution of three-dimensional linear and nonlinear telegraph equations

In this manuscript, a hybrid numerical technique is presented for solving three-dimensional hyperbolic telegraph equations. The proposed technique is based on the Haar wavelet collocation method and the finite difference method. In this technique, the space derivatives are estimated by truncated Haar wavelet series, while the time derivatives are approximated using finite difference method. The method is applied to some linear and non-linear three-dimensional hyperbolic telegraph equations. In the linear case, the resulting algebraic system is solved by the Gauss elimination technique. Whereas in the nonlinear case, the quasi-Newton linearization technique is utilized to remove the nonlinearity of the problem. The maximum absolute errors, root mean square errors and computational rate of convergence are calculated for a different number of collocation points at different time steps of the final time t=1. The stability analysis for the proposed method is also discussed. Overall, the recently devised numerical method boasts straightforward implementation, rapid convergence, and enhanced accuracy in addressing both linear and nonlinear problems.

Magnetotactic bacteria and Fe3O4–water in a wavy walled cavity

PurposeThe purpose of this study is to investigate the interaction between magnetotactic bacteria and Fe3O4–water nanofluid (NF) in a wavy enclosure in the presence of 2D natural convection flow.Design/methodology/approachUniform magnetic field (MF), Brownian and thermophoresis effects are also contemplated. The dimensionless, time-dependent equations are governed by stream function, vorticity, energy, nanoparticle concentration and number of bacteria. Radial basis function-based finite difference method for the space derivatives and the second-order backward differentiation formula for the time derivatives are performed. Numerical outputs in view of isolines as well as average Nusselt number, average Sherwood number and flux density of microorganisms are presented.FindingsConvective mass transfer rises if any of Lewis number, Peclet number, Rayleigh number, bioconvection Rayleigh number and Brownian motion parameter increases, and the flux density of microorganisms is an increasing function of Rayleigh number, bioconvection Rayleigh number, Peclet number, Brownian and thermophoresis parameters. The rise in buoyancy ratio parameter between 0.1 and 1 and the rise in Hartmann number between 0 and 50 reduce all outputs average Nusselt, average Sherwood numbers and flux density of microorganisms.Research limitations/implicationsThis study implies the importance of the presence of magnetotactic bacteria and magnetite nanoparticles inside a host fluid in view of heat transfer and fluid flow. The limitation is to check the efficiency on numerical aspect. Experimental observations would be more effective.Practical implicationsIn practical point of view, in a heat transfer and fluid flow system involving magnetite nanoparticles, the inclusion of magnetotactic bacteria and MF effect provide control over fluid flow and heat transfer.Social implicationsThis is a scientific study. However, this idea may be extended to sustainable energy or biofuel studies, too. This means that a better world may create better social environment between people.Originality/valueThe presence of magnetotactic bacteria inside a Fe3O4–water NF under the effect of a MF is a good controller on fluid flow and heat transfer. Since the magnetotactic bacteria is fed by nanoparticles Fe3O4 which has strong magnetic property, varying nanoparticle concentration and Brownian and thermophoresis effects are first considered.

Fractional advection diffusion asymmetry equation, derivation, solution and application

The non-Markovian continuous-time random walk model, featuring fat-tailed waiting times and narrow distributed displacements with a non-zero mean, is a well studied model for anomalous diffusion. Using an analytical approach, we recently demonstrated how a fractional space advection diffusion asymmetry equation, usually associated with Markovian Lévy flights, describes the spreading of a packet of particles. Since we use Gaussian statistics for jump lengths though fat-tailed distribution of waiting times, the appearance of fractional space derivatives in the kinetic equation demands explanations provided in this manuscript. As applications we analyse the spreading of tracers in two dimensions, breakthrough curves investigated in the field of contamination spreading in hydrology and first passage time statistics. We present a subordination scheme valid for the case when the mean waiting time is finite and the variance diverges, which is related to Lévy statistics for the number of renewals in the process.

A higher order compact numerical approach for singularly perturbed parabolic problem with retarded term

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper.

Numerical approximation of the space-time fractional diffusion problem

Fractional differential equations have become central tools for the accurate modeling of real-world phenomena in various fields. This work focuses on the discretization of the space-time fractional diffusion problem with Caputo derivative in time and Riesz-Caputo derivative in space. We introduce a collocation method based on a B-spline representation of the solution. This approach strategically exploits the symmetry properties of both the spline basis functions and the Riesz-Caputo operator, resulting in an efficient method for solving the given fractional differential problem. Preliminary numerical tests are presented to validate the proposed collocation method.

On the controllability results of semilinear delayed evolution systems involving fractional derivatives in Banach spaces

<abstract><p>The paper investigated the exact controllability of delayed fractional evolution systems of order $ \alpha\in (1, 2) $ in abstract spaces. At first, the exact controllability result is obtained when the nonlinear term $ f $ is locally Lipschitz continuous. Then, the certain compactness conditions and the measure of noncompactness conditions were applied to demonstrate the exact controllability of the concerned problem. The discussion was based on the fixed point theorems and the cosine family theory.</p></abstract>