Order Time Derivatives Research Articles

Second law and Liu relations: the no-reversible-direction axiom—revisited

AbstractA thermodynamic process is governed by balance equations in field-formulated thermodynamics. Especially the balance equation of entropy takes a prominent role: it introduces the Second Law in the form of a dissipation inequality via the non-negative entropy production. Balance equations and dissipation inequality are independent of the considered material which is described by additional constitutive equations which need the introduction of a state space which is spanned by the state space variables. Inserting these constitutive equations into the balance equations results in the balance equations on state space which include the first order time and position derivatives of the state space variables, called “higher derivatives” wich are directional derivatives in a mathematicle sense. Why do not appear the latter in the Liu Relations which pretend to describe material as well as the equations on state space do ? The answer is that the Liu Relations describe materials whose entropy production does not depend on the higher derivatives. Consequently, the Liu Relations are more specific than the balance equations on state space. A toy example concerning heat conduction in compressible fluids is in two different versions added for elucidation.

Innovative approache for the nonlinear atangana conformable Klein-Gordon equation unveiling traveling wave patterns

The aim of current work is to establish novel traveling wave solutions of the nonlinear Atangana conformable Klein - Gordon equation using a new extended direct algebraic technique. The Klein - Gordon equation is the relativistic state of the Schrödinger equation with a second - order time derivative and zero spin. Complex wave variable transformation is used to convert Atangana conformable nonlinear differential equation into an ordinary differential equation. Using the proposed technique based on Maple software structure, various types of solutions, such as, generalized trigonometric, generalized hyperbolic, and exponential functions, are established. When special parameteric values are considered for this method, solitary wave solutions can be obtained through other methods, such as the (G′G)-expansion method, the modified Kudryashov method, the sub-equation method, and so forth. A physical explanation is provided for the solutions under consideration to enhance comprehension of the physical phenomena resulting from the obtained solutions, provided that the physical parameters are set appropriately using 3D, 2D, and contour simulations. The results demonstrated that the new extended direct algebraic method provides a more potent mathematical tool for solving numerous more nonlinear partial differential equations with the aid of symbolic computation.

A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α∈(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x∈(0,L),0<x1<x2≤L. The problem is challenging since it is severely ill-posed for x∉[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.

T-phPINN: Physics-informed neural networks for solving 2D non-Fourier heat conduction equations

Non-Fourier heat conduction plays a dominant role in many extreme transient heat conduction processes, such as laser pulses and heat transfer in biological systems, but the heat wave effect makes it difficult to solve the temperature field accurately and quickly. In order to solve this problem, the first order time derivative enhanced parallel hard constraints physics-informed neural networks (T-phPINN) is proposed. T-phPINN comprises two subnetworks and incorporates a first order time derivative to capture sharp temperature changes. Two numerical cases show that the minimum relative error of T-phPINN is 0.001 % and 0.015 %, which is 1.04 % and 12.30 % of the error of conventional PINN respectively, proving the accuracy of our architecture. A transfer learning framework is established for scenarios of different parameters, the training only requires 1/6 iterations of the basic model, and close accuracy is obtained. The computational cost of T-phPINN is evaluated using the finite element method as the baseline. For the two cases, the single calculation time is 33.43 % and 51.50 % of the baseline, while the multiple calculation time under the acceleration of transfer learning is 11.59 % and 17.75 % of the baseline. This study will be helpful for solving large-scale non-Fourier heat conduction equations precisely and expeditiously.

Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations

A semilinear initial–boundary value problem with a Caputo time derivative of fractional order α∈(0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For an L2-type discretization of order 3−α, we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.

Numerical discretization of initial–boundary value problems for PDEs with integer and fractional order time derivatives

This paper is mainly concerned with introducing a numerical method for solving initial–boundary value problems with integer and fractional order time derivatives. The method is based on discretizing the considered problems with respect to spatial and temporal domains. With the help of finite difference methods, we transformed the studied problem into a set of fractional differential equations. Then, we implemented the fractional Adams method to solve this set in order to provide approximate solutions to the main problem. This combination results in an algorithm that can efficiently and accurately solve a general class of integer and fractional order initial–boundary value problems, such that it does not need to solve large systems of linear equations. In addition, we discussed the stability of the proposed scheme. Three illustrative examples are numerically solved to reveal the effectiveness and validity of the proposed technique.

Approximation of one and two dimensional nonlinear generalized Benjamin-Bona-Mahony Burgers' equation with local fractional derivative

This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with l∞ and l2 error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems.

New exact solutions of the conformable space-time two-mode foam drainage equation by two effective methods

In this article, the two-mode foam drainage equation in terms of time and space conformable sense has been investigated. Two effective methods, the generalized exponential rational function method (GERFM) and the improved version of the Bernoulli sub-equation function method (IBSEFM), are used to get new solutions of underlying equation. The fractional travelling wave transformation is applied to convert nonlinear partial differential equations to nonlinear ordinary differential equations. Proposed methods successfully extract trigonometric, hyperbolic and exponential solutions. Some of the obtained solutions are visualized to understand the effect of fractional orders of time and space derivatives on the wave profile and the dynamic behavior of the solutions.

Tracking Control for $(x,u)$-Flat Systems by Quasi-Static Feedback of Classical States

It is well known that for flat systems the tracking control problem can be solved by utilizing a linearizing quasi-static feedback of generalized states. If measurements (or estimates) of a so-called generalized Brunovský state are available, a linear, decoupled and asymptotically stable tracking error dynamics can be achieved. However, from a practical point of view, it is often desirable to achieve the same tracking error dynamics by feedback of a classical state instead of a generalized one. This is due to the fact that the components of a classical state typically correspond to measurable physical quantities, whereas a generalized Brunovský state often contains higher order time derivatives of the (fictitious) flat output which are not directly accessible by measurements. In this paper, a systematic solution for the tracking control problem based on quasi-static feedback and measurements of classical states only is derived for the subclass of $(x,u)$-flat systems.

On the Solution Existence for Collocation Discretizations of Time-Fractional Subdiffusion Equations

Time-fractional parabolic equations with a Caputo time derivative of order α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1)$$\\end{document} are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax–Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain m×m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\ imes m$$\\end{document} matrices (where m is the order of the collocation scheme), are verified both analytically, for all m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\\ge 1$$\\end{document} and all sets of collocation points, and computationally, for all m≤20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ m\\le 20$$\\end{document}. The semilinear case is also addressed.

Examining basic theorems and plane waves in the context of thermoelastic diffusion using a multi-phase-lag model with temperature dependence

This study employs a novel mathematical formulation of temperature-dependent thermoelastic diffusion with multi-phase delays (TDMT). The TDMT model, which is assumed, is obtained by converting the fundamental rules of Fourier’s and Fick’s laws into higher order time derivatives of the heat flow vector, the temperature gradient, diffusing mass flux, and the chemical potential gradient. Fundamental theorems such as energy, uniqueness, reciprocity theorems, and variational criterion are established using the fundamental equations for the assumed model TDMT. The reciprocity theorem is applied to a particular scenario by using instantaneous concentrated body forces, heat sources, chemical potential sources, and moving heat sources as examples. It is noted that both the variational criterion and these theorems depend on the field variables’ susceptibility as well as the fluctuations in the higher order temporal derivatives’ parameters. Additionally, the assumed model’s two-dimensional (2D) plane wave propagation is described. Three longitudinal waves are identified: the longitudinal (P) wave, the thermal (T) wave, the chemical potential (CP) wave, and the linked transverse (SV) wave. Wave properties (phase velocity and attenuation coefficient) are calculated numerically and shown visually. A few special examples are also investigated and compared with the established findings. The framework for additional research into basic issues in thermoelastic continua under various physical field variables is provided by this work. Numerous applications in material science, geomechanics, soil dynamics, and the electronic sector can be made of the current findings.

Effects of Brownian motion on solitary wave structures for 1D stochastic Poisson–Nernst–Planck system in electrobiochemical

This study deals with the time fractional 1D stochastic Poisson–Nernst–Planck (TFSPNP) system under the effect of multiplicative time noise. The M-truncated derivative (MTD) takes into consideration the fractional order time derivative. This is a steady-state Poisson–Nernst–Planck (PNP) equations that have applications in bioelectric dressings and bandages. To obtain the soliton solutions of TFSPNP, we use the generalized exponential rational functional method. These findings are presented in the form of trigonometric, exponential, and hyperbolic functions. Moreover, to show the effect of multiplicative time noise and MTD, we construct the plot of some solutions in the form of three-dimensional, two-dimensional, and their corresponding contours. These plots clearly show the effect of randomness in the wave structures for the exact solitary wave solutions that are attained. In general, the solutions become more stable when a noise term disrupts their symmetry.

Gravitational waves effects in a Lorentz–violating scenario

This paper focuses on how the production and polarization of gravitational waves are affected by spontaneous Lorentz symmetry breaking, which is driven by a self–interacting vector field. Specifically, we examine the impact of a smooth quadratic potential and a non–minimal coupling, discussing the constraints and causality features of the linearized Einstein equation. To analyze the polarization states of a plane wave, we consider a fixed vacuum expectation value (VEV) of the vector field. Remarkably, we verify that a space–like background vector field modifies the polarization plane and introduces a longitudinal degree of freedom. In order to investigate the Lorentz violation effect on the quadrupole formula, we use the modified Green function. Finally, we show that the space–like component of the background field leads to a third–order time derivative of the quadrupole moment, and the bounds for the Lorentz–breaking coefficients are estimated as well.

Modified nonlocal couple stress problem of magneto thermoelasticity in a multilayered cylinder with hall current, higher order time derivatives and two phase lags

Modified nonlocal couple stress problem of magneto thermoelasticity in a multilayered cylinder with hall current, higher order time derivatives and two phase lags

Analysis on reflected waves through semiconductor nanostructure medium with temperature dependent properties

The article is about the study of reflected waves through the surface of an elastic solid. The medium considered for the propagation of waves is homogeneous isotropic with semiconductor properties. The thermoelastic properties of the medium are a function of temperature. The governing equations are formulated by using non-local elastic theory. The conduction process of heat is studied by using the concept of three-phase lag along with fractional order time derivative. After reflecting through the surface one transverse and three longitudinal waves travel back into the media. The reflection coefficients and their energy ratios are computed analytically. Numerically simulated results are obtained for the intrinsic semiconductor material Silicon to depict the effect of temperature dependency parameters, nonlocal parameters, and time derivative fractional order on the different reflection coefficients. Some published results are also discussed as special cases.

Rayleigh wave through half space semiconductor solid with temperature dependent properties

The article focuses on Rayleigh wave propagation in a homogeneous isotropic semi-conductor thermoelastic medium rotating with fixed angular frequency with temperature-dependent properties. The elastic constants depend upon the temperature function. The effects of temperature dependency parameter, time derivative, and fractional order are illustrated. By the theory of thermo-elasticity, waves result in the generation of thermal signals that propagate through the medium. A heat conduction model of three-phase lag (3PL) along with fractional order time derivative is used to analyze the thermal signals. The secular equations of Rayleigh waves are derived mathematically at the stress-free, carrier density and thermally insulated boundaries. Some specific properties like velocity, attenuation coefficient, specific heat loss and penetration depth for Rayleigh waves have been evaluated and presented graphically. The secular equations are computed numerically and depicted graphically using Matlab.

Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations

This paper proposes a temporal second-order fully discrete two-grid finite element method (FEM) for solving nonlinear time-fractional variable coefficient diffusion-wave equations. The method involves introducing an auxiliary variable and utilizing a reducing order technique for the fractional derivative. This reduces the original fractional wave equations to a coupled system consisting of two equations with lower order derivative in time. The time-fractional derivative is discretized using the L2-1σ formula, while a second-order scheme is used for discretizing the first-order time derivative. The spatial direction is discretized using an efficient two-grid Galerkin FEM. The paper demonstrates that the optimal error estimates in L2-norm and H1-norm of the two-grid methods can achieve optimal convergence order when the mesh size satisfies H=h12 and H=hr2r+2, respectively. To handle the initial singularity of the solution and the historical memory of the fractional derivative term, the paper presents the nonuniform and fast two-grid algorithms. The numerical results validate the theoretical analysis and demonstrate the effectiveness of the proposed two-grid methods.

On Local Unique Solvability for a Class of Nonlinear Identification Problems

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders.

Uniqueness of solution with zero boundary condition for time-fractional wave equations

We consider a solution u to an initial boundary value problem in a bounded domain Ω over a time interval (0,T) for a time-fractional wave equation where the order of time derivative is between 1 and 2. We prove that if u|ω×(0,T)=0 for arbitrarily chosen subdomain ω⊂Ω, then u=0 in Ω×(0,T).

New results for convergence problem of fractional diffusion equations when order approach to $1^-$

This work studies the convergence problem for a class of fractional diffusion equations in which the time-derivative order approaches $1^-$. Up to now, few works have investigated this topic. The purpose of the article consists of three main contents. The first result is related to the convergence of the Caputo derivative and the Mittag-Leffler operators when $\alpha \to 1^-$. The second is to investigate the convergence problem for a linear fractional diffusion equation on $L^p$ spaces. And last result is concerned with the convergence problem for nonlinear fractional diffusion equations. The main analysis and techniques of the paper involve the evaluation related to Riemann-Liouville integration, Caputo derivative and Sobolev embeddings. Our analysis provides a complete and detailed answer to the convergence problem as fractional order tends to $1^-$.