Order Partial Differential Equations Research Articles

Numerical Simulation of Flood Routing, Model of Dynamic Equations

Dynamic equations of flow in a rectangular inclined channel are solved numerically for the case where the friction force of the channel wall is neglected by the gravity force. The flow discharge and the cross-section area along the channel are physical quantity that is calculated. In non-dimensional variables, the equations indicate solution in traveling wave occurring for critical flow. Near that type of flow, the perturbation method is applied to get second order equations that are first order partial differential equation with external force from the lower order equations. Numerical solution is obtained by predictor-corrector method, and the effect of these second order equations can be observed to the traveling wave, depending on the type of the flow, su-percritical or subcritical.

Some Fundamental Properties of Hunaiber Transform and its Applications to Partial Differential Equations

In this paper, we study some basic properties of a new integral transform '' Hunaiber transform''. Moreover, we apply Hunaiber transform to solve linear partial differential equations with initial and boundary conditions. We solve first order partial differential equations and Second order partial differential equations which are essential equations in mathematical physics and applied mathematics.

Boundary Integral Technique of 2nd Order Partial Differential Equation by Using Radial Basis

The solution of second order partial differential equation, with continuous change in coefficients by the formation of integral equation and then using radial basis function approximation (RBSA), has been developed in this paper. Use of boundary element method (BEM), which gives the solution of heat or mass diffusion in non-homogenous medium with function varying smoothly in space is also the part of this article. Discretization of boundary of integral domain instead of entire domain of the problem concerned is also the distinction of the recent work. The numerical solution of some problems with known value of the variable has also been included at the end.

Analysis of 2-D Steady State Heat Conduction In A Slab Subjected With Different Types of Boundary Conditions Using Method of Lines

Many disciplines in science and engineering, heat transfer plays a vital role and the problems associated with heat transfer are of great importance. Generally, two-dimensional steady state conduction is governed and transformed into a second order partial differential equation (PDE). Satisfying the differential equation along with four boundary conditions is essential for a solution to be valid. There are analytical solutions available, but only for simple boundary conditions and these are not suitable for complex boundary conditions. A technique for solving partial differential equations is Method of Lines (MOL), in which one dimension is discretized. In this study an analytical approach to a two-dimensional slab with steady state heat conduction under different types of boundary conditions is considered. The complete description of heat flow through slab using MOL is presented and the obtained PDEs are solved. Temperature distribution profiles in the slab by using MOL were plotted and compared with the profiles of analytical solutions by using MATLAB. The steady state analysis of temperature distribution in a slab with specified Dirichlet boundary conditions and Neumann boundary conditions was developed by using Method of Lines. From the profiles it is observed that as the number of lines increases, the error between analytical and MOL (semi analytical) solutions decreases, and the profiles of analytical and MOL converged which indicates the preference of higher number of lines in order to obtain accurate values.

Stability Analysis and Computational Interpretation of an Effective Semi Analytical Scheme for Fractional Order Non-Linear Partial Differential Equations

In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solve some non-linear fractional order partial differential equations. Since the Laplace transform cannot be applied to non-linear problems, the combination of the variational iteration technique with it will give a better and rapidly convergent sequence. Exact solutions may also exist, but we will show that the coupled technique is much better to approximate the exact solutions. The Caputo–Fabrizio fractional derivative will be used throughout the study. In addition, some possible implications of the results given here are connected with fixed point theory.

Holographic β Function in de Sitter Space

The scale invariance of the universe is slightly broken by slow roll parameters. It is likely the slow roll is dual to the random walk. We investigate the distribution function of the conformal zeromode. We identify de Sitter entropy $S_{dS}$ with the distribution entropy of the conformal zeromode $\rho(\omega)$. We have collected convincing support on our postulate. The semiclassical evidence is that the both are given by the gravitational coupling $1/g=\log N/2$ where $g=G_NH^2/\pi$ and $N$ is the e-folding number. We show the renormalized distribution function obeys gravitational Fokker-Planck equation (GFP) and Langevin equations . Under the Gaussian approximation, they boil down to a simple first order partial differential equation. The identical equation is derived by the thermodynamic arguments in the inflationary space-time. GFP determines the evolution of de Sitter entropy of the universe. It coincides with $\beta$ function of $g$. We find two types of the solutions of GFP:(1) UV complete spacetime and (2) inflationary spacetime with power potentials. The maximum entropy principle favors the scenario: (a) born small $\epsilon$ and (b) grow large by inflation. We like to convey the emerging notion of de Sitter duality. The inflationary universe: (bulk/geometrical) is dual to the stochastic space-time on the boundary (cosmological horizon ) as the both are the solutions of GFP.

On theoretical analysis of nonlinear fractional order partial Benney equations under nonsingular kernel

Abstract In the present article, the first step is devoted to develop some results about existence and uniqueness of solution to a general problem of fractional order partial differential equations (FPDEs) via classical fixed point theory. In the second step, a novel technique is used to handle the semi-analytical approximate solution for the considered general problem. Then, we extend the said result to fractional order partial Benney equations (FOPBEs) of the second and third order, which are special cases of the general problem we considered. We study the proposed problem under the Caputo-Febrizo fractional derivative (CFFD). With the help of the proposed method, we derive a series type approximate (semi-analytical) solution. Some numerical interpretations and visualizations are also given.

Numerical Modeling of a Propagation Process Using (Mpdx) Matrix Associated to Second Order Partial Diferential Equations with Two Independent Variables

Abstract This paper presents a preliminary version of case study for propagation process using Matrix of Partial Derivatives of the State Vector (Mpdx) Method associated to second order Partial Differential Equations with two Independent Variables and Approximating Solutions. Considering a thermally insulated layer of a specific thickness exposed on the upper part to a constant overtemperature with a uniform distribution over the entire heat transfer surface, using numerical modeling are emphasized some interpretation variants for evolution of processed signals for possible applications.

An improved PDE-constrained optimization fluid registration for image multi-frame super resolution

The main idea of multi-frame super resolution (SR) algorithms is to recover a single high-resolution image from a sequence of low resolution ones of the same object. The success of the SR approaches is often related to a well registration and restoration steps. Therefore, we propose a new approach based on a partial differential equation (PDE)-constrained optimization fluid image registration and we use a fourth order PDE to treat both the registration and restoration steps that guarantee the success of SR algorithms. Since the registration step is usually a variational ill-posed model, a mathematical study is needed to check the existence of the solution to the regularized problem. Thus, we prove the existence and of the well posed fluid image registration and assure also the existence of the used second order PDE in the restoration step. The results show that the proposed method is competitive with the existing methods.

Static and dynamic deformation response of smart laminated composite plates induced by inclined piezoelectric actuators

A Levi-type analytical solution procedure is developed to characterize static and dynamic deformation response of smart laminated simply-supported composite rectangular plates induced by inclined piezoelectric actuators under (1) constant electrical voltage and (2) time-dependent electrical voltage with excitation frequency. The key to development of this analytical solution is to employ higher order finite integral transform and discretized higher order partial differential unit step function equations. Unlike earlier studies, this research aims to investigate the effect of inclination angle of piezoelectric actuators on static and dynamic deformation response of laminated composite plates under both static and dynamic conditions. The developed analytical solution procedure is implemented computationally through Matlab-based computer code. Its accuracy is initially investigated through convergence study and results comparison with the published literature for a particular case when inclination angle is θ = 0°, which is only limited to bending deformation response. Since there is no published benchmark data for twisting deformation response analysis caused by inclination angle of piezoelectric actuators (θ ≠ 0°), a set of robust and realistic numerical analysis using Abaqus finite element analysis (FEA) is conducted. Good agreement between the analytical and numerical results is observed. Unlike applied electrical voltage, inclination angle of a piezoelectric actuator does not have a significant impact on twisting deformation response during static mode; whereas, both the excitation frequency and inclination angle can significantly influence maximum amplitude of vibration.

Analysis of the partially ionized kerosene oil-based ternary nanofluid flow over a convectively heated rotating surface

Abstract The main goal of this inspection is to explore the heat and mass transport phenomena of a three-dimensional magnetohydrodynamic (MHD) flow of ternary hybrid nanoliquid through a porous media toward a stretching surface. Nowadays, the low thermal conductivity is the key problem for scientist and researchers in the transmission of heat processes. Therefore, in order to improve the thermal conductivity of different base liquids, the scientist and researchers are mixing numerous types of solid particles in the base fluids. That is why the authors have mixed three different types of nanoparticles such as graphene oxide, silver, and copper in a kerosene oil base liquid. The influences of Hall current and ion-slip are also considered. Furthermore, the flow behavior is analyzed under the appliance of Darcy–Forchheimer, activation energy, and chemical reaction. By using the concept of boundary layer theory, the flow equations are modeled in the form of higher order nonlinear partial differential equations (PDEs) along with convective boundary conditions. Suitable similarity transformations are used for the transformation of higher order PDEs into the higher order nonlinear ordinary differential equations (ODEs). Analytical scheme known as the homotopic scheme is utilized for the simulation of the current problem. The impacts of discrete flow parameters on the velocities, temperature, and concentration profiles of the ternary hybrid nanoliquid are inspected. The skin friction coefficients, Nusselt number, and Sherwood number of the ternary hybrid nanofluid are investigated against various flow parameters. The outcomes of the current analysis showed that primary velocity of the ternary hybrid nanoliquid is augmented via Hall current and ion-slip number, while the reverse trend is observed via porosity parameter, Darcy–Forchheimer parameter, and magnetic field parameter. On the other hand, the higher values of Hall current and magnetic parameter enhanced the secondary velocity of the ternary hybrid nanoliquid, while the secondary velocity was reduced due to the increasing ion-slip number and rotation parameter. It is found that the heat transfer rate of the ternary hybrid nanofluid is 46% greater than the silver nanofluid.

Co-Evolution of the General Factor of Personality and the Big Five Factors: A Mathematical Model and Its Application to Methylphenidate

This study presents a generalization of a mathematical model, called the “bridge model” in Literature, to study the co-evolution or dynamical relationship between the General Factor of Personality (GFP) and the Big Five Factors (B5F) as a consequence of a stimulus. This generalization permits its application here to study such co-evolution after a dose intake of methylphenidate. On the one hand, a stimulus-response model, an integro-differential equation called the “response model,” predicts the short-term effects of a stimulus on the GFP and on the B5F. The fact that both the GFP and the B5F dynamics can be described by the response model is here referred as the “invariance principle.” On the other hand, the invariance principle leads to the derivation of the bridge model: a second order partial differential equation that relates every B5F to the GFP and time. The outcomes of both experimental designs permit the validation of the response model and of the new bridge model.

The Cauchy Problem for One Second Order Partial Differential Equation with Carleman Shift

The Cauchy Problem for One Second Order Partial Differential Equation with Carleman Shift

A Model for Cell Proliferation in a Developing Organism

In mathematical biology, there is a great deal of interest in producing continuum models by scaling discrete agent-based models governed by local stochastic rules. We discuss a particular example of this approach: a model for the proliferation of neural crest cells that can help us understand the development of Hirschprung’s disease, a potentially-fatal condition in which the enteric nervous system of a new-born child does not extend all the way through the intestine and colon. Our starting point is a discrete-state, continuous-time Markov chain model proposed by Hywood et al. (2013a) for the location of the neural crest cells that make up the enteric nervous system. Hywood et al. (2013a) scaled their model to derive an approximate second order partial differential equation describing how the limiting expected number of neural crest cells evolve in space and time. In contrast, we exploit the relationship between the above-mentioned Markov chain model and the well-known Yule-Furry process to derive the exact form of the scaled version of the process. Furthermore, we provide expressions for other features of the domain agent occupancy process, such as the variance of the marginal occupancy at a particular site, the distribution of the number of agents that are yet to reach a given site and a stochastic description of the process itself.

A Local Deep Learning Method for Solving High Order Partial Differential Equations

At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly, and high order derivatives lack robustness for training purposes. We propose a novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variables to rewrite the PDEs into a system of low order differential equations as what is done in the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neural network. By taking the residual of the system as a loss function, we can optimize the network parameters to approximate the solution. The whole process relies on low order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularly well-suited for high-dimensional PDEs with high order derivatives.

Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations

Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k≥2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction τ=O(hk) for stability. The implicit and implicit-explicit (IMEX) time-marching methods can overcome this constraint. However, for the equations with nonlinear high derivative terms, the IMEX methods are not good choices either, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The explicit-implicit-null (EIN) time-marching method is designed to cope with the above mentioned shortcomings. The basic idea of the EIN method discussed in this paper is to add and subtract a sufficiently large linear highest derivative term on one side of the considered equation, and then apply the IMEX time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we will discuss the high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations, respectively. By the aid of the Fourier method, we perform stability analysis for the schemes on the simplified equations with periodic boundary conditions, which demonstrates the stability criteria for the resulting schemes. Even though the analysis is only performed on the simplified equations, numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy for both one-dimensional and two-dimensional linear and nonlinear equations.

Hyers Ulam Rassias Stability Analysis of First Order Partial Differential Equations

Hyers Ulam Rassias Stability Analysis of First Order Partial Differential Equations

New soliton solutions for the space-time fractional modified third order Korteweg–de Vries equation

In present research work, the soliton solutions of the space-time fractional modified third order Korteweg–de Vries (KdV) equation are explored by Sardar-subequation method (SSM). The (KdV) equation has an established version based on the shallow water waves in the oceans, oceanography theory and the ion-acoustic waves in plasma. The fractional order partial differential equation (PDE) is transform to a non-linear ordinary differential equation (ODE) by using traveling wave transformation. It is noted that in this work the fractional differential equation is solved in the sense of conformable derivative. Then soliton solutions of reduced equation are established by SSM. The present technique provides bright, dark, singular, periodic singular, combined bright-dark and combined dark-singular soliton solutions.

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

The local meshless method based on Pascal polynomial basis functions for solving fourth-order PDEs

In this paper, we present a localized method based on Pascal polynomial basis functions to solve fourth-order partial differential equations (PDEs) even with variable coefficients. The proposed algorithm is simple and effective, since applying Pascal polynomial basis functions can avoid the derivation of the closed-form particular solutions for higher order PDEs. Also, the localized formulation can alleviate the ill-conditioned problem of the resulting coefficient matrix. Five numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed method in both regular and irregular domains.