Technique For Partial Differential Equations Research Articles

The relationship between two kinds of structural credit migration models

<abstract><p>In this paper, we presented an asymptotic relationship between the single threshold model and the model with different upgrade and downgrade thresholds for credit migration problems with fixed boundaries. By partial differential equation (PDE) techniques, we proved that the solution of the asymmetric threshold problem converges to that of a single threshold problem uniformly as one of the asymmetric thresholds approaches the other.</p></abstract>

Similarity solution of fractional viscoelastic magnetohydrodynamic fluid flow over a permeable stretching sheet with suction/injection

A novel easy‐to‐use similarity technique for spatial fractional partial differential equations is introduced to solve the flow and heat transfer of fractional viscoelastic magnetohydrodynamic (MHD) fluid over a permeable stretching sheet. The proposed method represents a new technique for deriving similarity transformations, which allows the conversion of the fractional partial differential equations to fractional ordinary differential equations. Moreover, using the Grünwald–Letnikov formula, the fourth‐order Runge–Kutta method is adopted here to solve the obtained similarity equations, using the shooting method. The transformed governing equations were found to be dependent on six physical parameters, namely, fractional parameter, magnetic field parameter, generalized local Reynolds number, generalized Prandtl number, the wall suction/injection parameter, and the wall stretching exponential. The results reveal that maximums of the skin friction coefficient and local Nusselt number are obtained for fractional‐order derivative of about 0.5. Moreover, application of a magnetic field on electrically conducting fractional fluid thins the velocity boundary layer and thickens the thermal boundary layer; therefore, the skin friction coefficient increases whereas the local Nusselt number decreases. Furthermore, an increase in any parameters of the local Reynolds number, suction parameter, or the wall stretching exponential leads to thinner boundary layers and higher values of skin friction coefficient and local Nusselt number, whereas imposing injection has opposite effects. Finally, increasing the generalized Prandtl number leads to a thinner thermal boundary layer and higher local Nusselt number for the fractional viscoelastic fluid. It would be of interest to extend the proposed method to other practical problems in the same field.

Similarity Transforming Techniques of Partial Differential Equations and its Applications: A review

Similarity Transforming Techniques of Partial Differential Equations and its Applications: A review

Numerical approximation with the splitting algorithm to a solution of the modified regularized long wave equation

In this article, a Lie-Totter splitting algorithm, which is highly reliable, flexible and convenient, is proposed along with the collocation finite element method to approximate solutions of the modified regular long wave equation. For this article, quintic B-spline approximation functions are used in the implementation of collocation methods. Four numerical examples including a single solitary wave, the interaction of two- three solitary waves, and a Maxwellian initial condition are presented to test the closeness of the solutions obtained by the proposed algorithm to the exact solutions. The solutions produced are compared with those in some studies with the same parameters that exist in the literature. The fact that the present algorithm produces results as intended is a proof of how useful, accurate and reliable it is. It can be stated that this fact will be very useful the application of the presented technique for other partial differential equations, with the thought that it may lead the reader to obtain superior results from this study.

Stochastic modeling for describing crystallization droplets in water emulsion

This paper introduces a new, stochastic mathematical model for the crystallization of emulsion in dispersed media. The mathematical model reads as a stochastic partial differential equation by combining the heat energy equation and the nucleation theory with specified drift and diffusion. We show the existence and uniqueness of the solution of the model by using techniques of stochastic partial differential equations. Numerical experiments are drawn to support the theoretical results. Moreover, comparison of numerical results to experimental ones is provided.

Bayesian inversion techniques for stochastic partial differential equations

Bayesian inversion techniques for stochastic partial differential equations

Performance of spatial derivatives using interpolation with radial basis functions

Initially applied in the context of the Boundary Element Method as an auxiliary tool, interpolating the domain core and allowing its transformation into boundary integrals, the radial basis functions have expanded their field of application. They are currently widely used as a solution technique for partial differential equations, generating the meshless formulations of the Finite Element Method. These functions also have recently become an important numerical resource as a more straightforward solver for spatial derivatives calculation. Of course, precision is lost compared to the performance as a direct interpolation tool. Still, even so, it can be advantageous because of the complexity of procedures related to the analytical derivation of primary variables and other more classical techniques. This work evaluates a series of characteristics of the derivation procedure, such as the influence of the dimensions of the problem on the results. In this work, the initial results for the research are performed through the cubic radial function, following the guidelines of the previous works. This function has no arbitrary parameter in its structures, which is a great advantage.

On a mixed singular/switching control problem with multiple regimes

AbstractThis paper studies a mixed singular/switching stochastic control problem for a multidimensional diffusion with multiple regimes on a bounded domain. Using probabilistic partial differential equation and penalization techniques, we show that the value function associated with this problem agrees with the solution to a Hamilton–Jacobi–Bellman equation. In this way, we see that the regularity of the value function is $ \textrm{C}^{0,1}\cap \textrm{W}^{2,\infty}_{\textrm{loc}}$ .

Error estimates for deep learning methods in fluid dynamics

In this study, we provide error estimates and stability analysis of deep learning techniques for certain partial differential equations including the incompressible Navier–Stokes equations. In particular, we obtain explicit error estimates (in suitable norms) for the solution computed by optimizing a loss function in a Deep Neural Network approximation of the solution, with a fixed complexity.

On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Methods

Dynamics of meandering rivers in finite-length channels: linear theory

Meandering channels are dynamic landforms that arise as a result of fluid mechanic and sedimentary processes. Their evolution has been described by the meander-morphodynamic equations, which dictate that channel curvature and bed topography give rise to local perturbations in streamwise fluid velocity, prompting the preferential erosion and sediment deposition that constitute meander behaviour. Novel mathematical conditions are presented to guarantee unique solutions for the linearized equations in non-periodic domains with finite boundaries. With the boundary condition specification sufficient for the uniqueness proof one finds a well-posed initial-boundary-value problem amenable to standard numerical techniques for partial differential equations. This provides a pathway for improved numerical algorithms for simulations of meandering river dynamics. Previous theoretical analysis for linear stability theory in meandering dynamics has been restricted to spatially periodic systems. The present effort develops new results for linear stability theory in non-periodic systems with temporal driving at system boundaries as well as non-homogeneous initial conditions. Predictions for temporal driving at the inlets for non-periodic finite domains provide clarification for observed behaviour in laboratory flumes where driven conditions at the inlet avoids the long-term decay of all meanders observed in flumes with fixed entry conditions. Linear stability theory for finite domains confirms that a continuous perturbation is required for sustained meandering. Original scaling arguments are presented for the dependence of the meander migration rate on geological parameters, showing that the rate of channel migration increases with increased width, downreach slope and bank erodibility, and decreases with increased volumetric flow rate.

The attachment length in orificed hollow cathodes

A first-principles approach to obtain the attachment length within a hollow cathode with a constrictive orifice, and its scaling with internal cathode pressure, is developed. This parameter, defined herein as the plasma density decay length scale upstream of (away from) the cathode orifice, is critical because it controls the utilization of the hollow cathode insert and influences cathode life. A two-dimensional framework is developed from the ambipolar diffusion equation for the insert-region plasma. A closed-form solution for the plasma density is obtained using standard partial differential equation techniques by applying an approximate boundary condition at the cathode orifice plane. This approach also yields the attachment length and electron temperature without reliance on measured plasma property data or complex computational models. The predicted plasma density profile is validated against measurements from the NSTAR discharge cathode, and calculated electron temperatures and attachment lengths agree with published values. Nondimensionalization of the governing equations reveals that the solution depends almost exclusively on the neutral pressure-diameter product in the insert plasma region. Evaluation of analytical results over a wide range of input parameters yields scaling relations for the variation of the attachment length and electron temperature with the pressure-diameter product. For the range of orifice-to-insert diameter ratio studied, the influence of orifice size is shown to be small except through its effect on insert pressure, and the attachment length is shown to be proportional to the insert inner radius, suggesting high-pressure cathodes should be constructed with larger-diameter inserts.

Method of approximations for the convection-dominated anomalous diffusion equation in a rectangular plate using high-resolution compact discretization

The present method describes the high-resolution compact discretization method for the numerical solution of the nonlinear fractal convection-diffusion model on a rectangular plate by employing the Hausdorff distance metric. Estimation of anomalous diffusion is formulated by averaging forward and backward mesh stencils. The higher-order fractional derivatives are appropriately approximated on a minimum mesh stencil and subsequently considered for designing a numerical method that falls in the scope of expanded accuracy. Compact discretization is an efficient technique for partial differential equations; however, studies that apply high-resolution scheme for fractional-order systems are still uninvestigated. A second and fourth-order numerical method for the fractional-order convection-dominated anomalous diffusion equation in two dimensions is constructed for practical applications. Convergence of high-order method is obtained for the nonlinear partial differential equations employing Hausdorff fractal distance metric. The numerical simulations with fractal Graetz-Nusselt equation, fractal Poisson equation, fractal Schrödinger equation, and anomalous diffusion equations with variable and constant coefficients are considered to illustrate the utility of the numerical method in the context of local fractional partial differential equations.•The paper demonstrates a computational method for the fractal convection-diffusion model on a rectangular plate.•Two numerical methods of order two and four for the mildly nonlinear fractional-order convection-dominated anomalous diffusion equations are proposed.•The high-resolution scheme is computationally efficient and makes use of minimal data storage.Method name: High-order method for 2D convection-dominated anomalous diffusion equation, Graetz-Nusselt equation, Poisson equation, and Schrödinger equation in fractal media

Combination of Laplace transform and residual power series techniques of special fractional-order non-linear partial differential equations

<abstract><p>This paper investigates fractional-order partial differential equations analytically by applying a modified technique called the Laplace residual power series method. The analytical solution was utilized to test the accuracy and precision of the proposed methodologies and shown by tables and graphs. The solution is a convergent series established on Taylor's new form. When determining the series coefficients like RPSM, the fractional derivatives must be calculated every time. We only need to perform a few computations to obtain the coefficients because LRPSM only requires the concept of an infinite limit. The advantage of this method is that it does not require Adomian polynomials or he's polynomials to solve nonlinear problems. As a result, the method's reduced computation size is a strength. The outcome we got supports the idea that the suggested method is the best one for handling any non-linear models that appear in technology and science.</p></abstract>

A novel collocation technique for parabolic partial differential equations

The present study focuses on an accurate, convergent, stable and efficient method for solving the parabolic Fisher’s type equation with three different cases. The method makes use of the modified quintic trigonometric B-spline collocation technique over the Crank-Nicolson scheme for spatial discretization while the time derivative is discretized by the usual finite difference method. The nonlinear terms are linearized using the Rubin-Graves type linearization process. The efficiency and accuracy of the method are examined by taking three numerical examples which show that the method presents more accurate solutions than existing solutions. The convergence analysis and von Neumann stability of the method show that the method is fourth-order convergent in space and the discretized system is unconditionally stable. Computational efficiency of the method is investigated by small values of CPU-time. The proposed method can be implemented to solve the fourth-order partial differential equations in higher dimensional including physical, mechanical, or biophysical effects.

Matrix Equation Techniques for Certain Evolutionary Partial Differential Equations

We show that the discrete operator stemming from time-space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines projection techniques with the full exploitation of the entry-wise structure of the involved coefficient matrices is proposed. The resulting scheme is able to efficiently solve problems with a tremendous number of degrees of freedom while maintaining a low storage demand as illustrated in several numerical examples.

Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs

We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of bounded Riesz systems and generalizes random partial Fourier matrices. Our main result improves the currently available results for the null space and restricted isometry properties of such random matrices. The main novelty of our analysis is a new upper bound for the expectation of the supremum of a Bernoulli process associated with a restricted isometry constant. We apply our result to prove new performance guarantees for the CORSING method, a recently introduced numerical approximation technique for partial differential equations (PDEs) based on compressive sensing.

An Efficient Analytical Technique for Time-Fractional Parabolic Partial Differential Equations

In this work, we examine the time-fractional fourth-order parabolic partial differential equations with the aid of the optimal homotopy asymptotic method (OHAM). The 2nd order approximate results obtained by using the suggested scheme are compared with the exact solution. It has been noted that the results achieved via OHAM have a large convergence rate for the problems. The solutions are graphically analyzed, and the relative errors are presented in tabular form.

Numerical study of an influenza epidemic dynamical model with diffusion

In this paper, a deterministic model is formulated in the aim of performing a thorough investigation of the transmission dynamics of influenza. The main advantage of our model compared to existing models is that it takes into account the effects of hospitalization as well as the diffusion. The proposed model consisting of a dynamical system of partial differential equations with diffusion terms is numerically solved using fast and accurate numerical techniques for partial differential equations. Furthermore, the basic reproduction number that guarantees the local stability of disease-free steady state without diffusion term is calculated. Various numerical simulation for different values of the model input parameters are finally presented in order to show the effect of the effective contact rate on the steady state of the different population compartments.

Free boundaries of credit rating migration in switching macro regions

In this paper, under the structure framework, a valuation model for a corporate bond with credit rating migration risk and in macro regime switch is established. The model turns to a free boundary problem in a partial differential equation (PDE) system. By PDE techniques, the existence, uniqueness and regularity of the solution are obtained. Furthermore, numerical examples are also presented.