Order Partial Differential Equations Research Articles

Mathematical analysis of a SIPC age-structured model of cervical cancer.

Human Papillomavirus (HPV), which is the main causal factor of cervical cancer, infects normal cervical cells on the specific cell's age interval, i.e., between the G1 to S phase of cell cycle. Hence, the spread of the viruses in cervical tissue not only depends on the time, but also the cell age. By this fact, we introduce a new model that shows the spread of HPV infections on the cervical tissue by considering the age of cells and the time. The model is a four dimensional system of the first order partial differential equations with time and age independent variables, where the cells population is divided into four sub-populations, i.e., susceptible cells, infected cells by HPV, precancerous cells, and cancer cells. There are two types of the steady state solution of the system, i.e., disease-free and cancerous steady state solutions, where the stability is determined by using Fatou's lemma and solving some integral equations. In this case, we use a non-standard method to calculate the basic reproduction number of the system. Lastly, we use numerical simulations to show the dynamics of the age-structured system.

COMPARTMENTAL MODELING AND DYNAMIC SIMULATION OF GINGER DRYING

The concept of Fourier Transformation and Laplace Transformation play an important role in diverse areas of Science, Engineering and Technology. Fourier Transform and Laplace Transform is also play an important role in the analysis of all kinds of physical phenomena. As a link between the various applications of these transforms the authors use the theory of signal and systems, as well as the theory of ordinary and partial differential equations. In this paper we established a Two Dimensional Fourier-Laplace Transform and investigated the Linearity property, Parseval’s theorem and Modulation theorem of Two Dimensional Fourier-Laplace Transform. The work may be useful for solving higher order ordinary and partial differential equations as well as integral equations.

Application of Partial Differential Equations in Multi Focused Image Fusion

Image Fusion is a process used to combine two or more images to form more informative image. More often, machine vision cameras are affected by limited depth of field and capture the clear view of the objects which are in focus. Other objects in the scene will be blurred. So, it is necessary to combine set of images to have the clear view of all objects in the scene. This is called Multi focused image fusion. This paper compares and presents the performance of second order and fourth order partial differential equation in multi focused image fusion.

Dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line

In this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schr\odinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G′/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

Meshfree numerical integration for some challenging multi-term fractional order PDEs

<abstract><p>Fractional partial differential equations (PDEs) have key role in many physical, chemical, biological and economic problems. Different numerical techniques have been adopted to deal the multi-term FPDEs. In this article, the meshfree numerical scheme, Radial basis function (RBF) is discussed for some time-space fractional PDEs. The meshfree RBF method base on the Gaussian function and is used to test the numerical results of the time-space fractional PDE problems. Riesz fractional derivative and Grünwald-Letnikov fractional derivative techniques are used to deal the space fractional derivative terms while the time-fractional derivatives are iterated by Caputo derivative method. The accuracy of the suggested scheme is analyzed by using $ L_\infty $-norm. Stability and convergence analysis are also discussed.</p></abstract>

Analytical solution for rotating cylindrical FGM vessel subjected to thermomechanical loadings

In the present work, an exact closed-form analytical solution is derived from the governing equations in order to compute the thermomechanical stress field for a rotating hollow cylinder made of nonhomogeneous materials (functionally graded materials: FGMs) subjected to internal pressure and uniform temperature. We assume a radial variation along the cylinder wall of the thermomechanical properties namely: the Young's modulus and the thermal conductivity. The Poisson’s ratio is assumed constant. In order to provide an exact solution of the displacement and stress fields, the well-known Navier’s equation which is a second order partial differential equation derived from the equilibrium equation was solved. The results obtained reveal the influence of thermal loading, power-law constant and internal pressure on the thermal and mechanical stresses of the nonhomogeneous cylindrical vessel.

A Posteriori Error Estimates of Spectral Approximations for Second Order Partial Differential Equations in Spherical Geometries

A Posteriori Error Estimates of Spectral Approximations for Second Order Partial Differential Equations in Spherical Geometries

Time recursive control of stochastic dynamical systems using forward dynamics and applications

The solution of a stochastic optimal control problem may be associated with that of the Hamilton–Jacobi–Bellman (HJB) equation, which is a second order partial differential equation subject to a terminal condition. When this equation is semilinear and satisfies certain other constraints, it can be solved via a nonlinear version of the Feynman–Kac formula. According to this approach, the solution to the HJB equation can be obtained by simulating an associated pair of partly coupled forward–backward stochastic differential equations. Although an elegant way to interpret and solve a partial differential equation, simulating the system of forward–backward equations can be computationally inefficient. In this work, the HJB equation pertaining to the optimal control problem is reformulated such that instead of the given terminal condition, it is now subject to an appropriate initial condition. In the process, while the total cost associated with the control problem remains unchanged, pathwise solutions may not. Associated with the new partial differential equation, we then derive a set of stochastic differential equations whose solutions move only forward in time. This approach has a significant computational advantage over the original formulation. Moreover, since the forward–backward approach generally requires simulating stochastic differential equations over the current to the terminal time at every step, the integration errors may accumulate and carry forward in estimating the control. This error is particularly high initially since the time of integration is the longest there. The proposed method, numerically implemented for the control of stochastically excited oscillators, is free of such errors and hence more robust. Unsurprisingly, it also exhibits lower sampling variance.

Electro-osmosis-modulated biologically inspired flow of solid–liquid suspension in a channel with complex progressive wave: application of targeted drugging

Electrokinetic microperistaltic pumps are important biomechanical devices that help target drug delivery to specific body parts. The current study focuses on the mathematical modelling and analysis of some important aspects of such flows in a channel with complex waves. It is considered that solid particles are uniformly distributed in the flow, and that these particles are non-conducting. Parameters such as the particle volume fraction coefficient, electro-osmotic parameter, and Helmholtz–Smoluchowski parameter are specifically focused on in this study. Equally sized spherical particles were uniformly floated in a non-Newtonian Powell–Eyring base fluid. The defined flow problem was modelled and analyzed analytically for the transport of a solid–liquid suspension. It is accepted that the flow is steady, non-turbulent, and propagating waves have a considerably longer wavelength compared to the amplitude. The conditions and assumptions lead to a model of the coupled partial differential equations of order two. The exact results of the homotopy perturbation method expansion method are obtained and shown accordingly. Predictions of the behavior of important parameters are displayed in the figures. The impact of several parameters was analyzed. The current study involved transporting or targeted drug delivery systems using peristaltic micropumps and magnetic fields in the pharmacological engineering of biofluids, such as blood.

A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations

The virtual element method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order [Formula: see text], for any integer [Formula: see text]. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of [Formula: see text], [Formula: see text] being the computational domain and [Formula: see text] another suitable integer number. In this review, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation spaces with a global high-order regularity on [Formula: see text]. Then, we illustrate specific examples in the case of second- and fourth-order partial differential equations, that correspond to the cases [Formula: see text] and [Formula: see text], respectively. Finally, we investigate numerically the effect on the approximation properties of the conforming highly-regular method that results from different choices of the degree of continuity of the underlying virtual element spaces and how different stabilization strategies may impact on convergence.

A symplectic analytical approach for free vibration of orthotropic cylindrical shells with stepped thickness under arbitrary boundary conditions

Exact analytical solutions for free vibration of isotropic and orthotropic cylindrical shells with uniform and stepped thickness subject to general boundary conditions are presented by means of a symplectic analytical approach. The Reissner shell theory is adopted to formulate a theoretical model. By introducing a Hamiltonian system, the governing higher order partial differential equation is reduced to a set of ordinary differential equations which can be analytically solved by separating the variables. Applying the end boundary and interface continuous conditions, a set of analytical characteristic frequency equations are obtained, and exact solutions can be determined. To ensure accuracy and validity of the symplectic method, the analytical solutions for uniform and stepped cylindrical shells with isotropic or orthotropic material properties and with arbitrary boundary conditions are compared with available published data and FEM solution. A set of comprehensive new result for orthotropic stepped cylindrical shells under classical and elastic restraints are presented. Some typical mode shapes for various examples are illustrated. In addition, the effects of boundary conditions and orthotropic properties on vibration frequency are analyzed. The result shows that the fundamental frequency is higher for a boundary with more constraints.

Multivector Fields of Noether Symmetries in the Lagrangian Formalism and Belinfante Tensor

Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar\'e-Cartan form.\\ The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.

Fractional Chebyshev deep neural network (FCDNN) for solving differential models

Differential and integral equations have been used vastly in modeling engineering and science problems. Solving these equations has been always an active and important area of research. In this paper, we propose the Fractional Chebyshev Deep Neural Network (FCDNN) for solving fractional differential models. Chebyshev orthogonal polynomials are basic functions in spectral methods. These functions are used as activation functions in FCDNN. The marching in time technique and the Gaussian method are applied in the fractional operations to simplify the calculations. We show how FCDNN can be used to solve fractional Fredholm integral equations (FFIEs). We also propose a solution to the extension of fractional time order partial differential equations (FPDEs). In this approach, fractional PDEs are first discretized by the finite difference and the marching in time methods and then are solved using FCDNN. Fractional Fredholm integral equations are also first approximated by the numerically Gaussian quadrature method and then are solved using FCDNN. A comparison between the results from FCDNN and some other methods is presented to validate the effectiveness and advance of the proposed method.

Modulated periodic wavetrains in the spherical Gardner equation

The spherical Gardner (sG) equation is derived by reducing the (3+1)-dimensional Gardner–Kadomtsev–Petviashvili (Gardner–KP) equation with a similarity reduction. As a special case, the step-like initial condition is considered along a paraboloid front. By applying a multiple-scale expansion, a system of first order partial differential equations for the slowly varying parameters of a periodic wavetrain is obtained. The corresponding modulation system is transformed into a simpler form with the help of Riemann type variables. This basic form is important to investigate the dispersive shock wave (DSW) phenomena in the sG equation. DSW solution is also compared with the numerical solution of the sG equation and good agreement is found between these solutions.

Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.

Interpolating Stabilized Element Free Galerkin Method for Neutral Delay Fractional Damped Diffusion-Wave Equation

A numerical solution for neutral delay fractional order partial differential equations involving the Caputo fractional derivative is constructed. In line with this goal, the drift term and the time Caputo fractional derivative are discretized by a finite difference approximation. The energy method is used to investigate the rate of convergence and unconditional stability of the temporal discretization. The interpolation of moving Kriging technique is then used to approximate the space derivative, yielding a meshless numerical formulation. We conclude with some numerical experiments that validate the theoretical findings.

Photovoltaic Energy All-Day and Intra-Day Forecasting Using Node by Node Developed Polynomial Networks Forming PDE Models Based on the L-Transformation

Forecasting Photovoltaic (PV) energy production, based on the last weather and power data only, can obtain acceptable prediction accuracy in short-time horizons. Numerical Weather Prediction (NWP) systems usually produce free forecasts of the local cloud amount each 6 h. These are considerably delayed by several hours and do not provide sufficient quality. A Differential Polynomial Neural Network (D-PNN) is a recent unconventional soft-computing technique that can model complex weather patterns. D-PNN expands the n-variable kth order Partial Differential Equation (PDE) into selected two-variable node PDEs of the first or second order. Their derivatives are easy to convert into the Laplace transforms and substitute using Operator Calculus (OC). D-PNN proves two-input nodes to insert their PDE components into its gradually expanded sum model. Its PDE representation allows for the variability and uncertainty of specific patterns in the surface layer. The proposed all-day single-model and intra-day several-step PV prediction schemes are compared and interpreted with differential and stochastic machine learning. The statistical models are evolved for the specific data time delay to predict the PV output in complete day sequences or specific hours. Spatial data from a larger territory and the initially recognized daily periods enable models to compute accurate predictions each day and compensate for unexpected pattern variations and different initial conditions. The optimal data samples, determined by the particular time shifts between the model inputs and output, are trained to predict the Clear Sky Index in the defined horizon.

New exact solutions of time conformable fractional Klein Kramer equation

The Klein Kramer equation (KKE) stands for the probability distribution function (PDF) that describes the diffusion of particles subjected an external force in the presence of friction. It is applicable in statistical and stochastic treatments of chemical dynamics, in particular the diffusive description of chemical reactions. Here, we are concerned with finding the exact solutions of the conformable fractional derivative (CFD) KKE. An approach is presented to transform linear partial differential equations (PDEs) to a set of first order PDEs. On the other hand the CFD is shown to be reduced to the classical one’s by using similarity transformations. Here, the objective of this work is to find the exact solutions of CFD-KKE. To this issue, the approach presented is applied. The solutions are found by implementing extended unified method. It is found that, the integrability condition is that the external force is constant. The numerical results of the solutions are calculated and the are shown graphically. Calculations are carried by MATHEMATICA 12.

Solution of Fractional Partial Differential Equations Using Fractional Power Series Method

In this paper, we are presenting our work where the noninteger order partial differential equation is studied analytically and numerically using the noninteger power series technique, proposed to solve a noninteger differential equation. We are familiar with a coupled system of the nonlinear partial differential equation (NLPDE). Noninteger derivatives are considered in the Caputo operator. The fractional-order power series technique for finding the nonlinear fractional-order partial differential equation is found to be relatively simple in implementation with an application of the direct power series method. We obtained the solution of nonlinear dispersive equations which are used in electromagnetic and optics signal transformation. The proposed approach of using the noninteger power series technique appears to have a good chance of lowering the computational cost of solving such problems significantly. How to paradigm an initial representation plays an important role in the subsequent process, and a few examples are provided to clarify the initial solution collection.