Types Of Partial Differential Equations Research Articles

A nonlocal operator method for solving partial differential equations

A nonlocal operator method is proposed which is generally applicable for solving partial differential equations (PDEs) of mechanical problems. The nonlocal operator can be regarded as the integral form “equivalent” to the differential form in the sense of a nonlocal interaction model for solving the unknown field. The variation of a nonlocal operator plays an equivalent role as the derivatives of the shape functions in the meshless methods or those of the finite element method, thus it circumvents many difficulties in the calculation of shape functions and their derivatives. The nonlocal operator method can consistently applied with common procedure leading to the weak forms, i.e. the variational principle and the weighted residual method. Based on these, the residual and the tangent stiffness matrix can be obtained with ease. The nonlocal operator method is enhanced here also with an operator energy functional to satisfy the linear consistency of the field. Higher order nonlocal operators and higher order operator energy functional are hereby generalized. A highlighted of the present method is the functional derived based on the nonlocal operator can convert the construction of residual and stiffness matrix into a series of matrix multiplications using the predefined nonlocal operators. The nonlocal strong forms of different functionals can be obtained easily via the concept of support and dual-support, the two basic elements introduced in the paper. Several numerical examples of different types of PDEs are presented in the end to show the effectiveness of the present method and also serve for validation.

Collocation method with fractional powers exponential trial functions for singularly perturbed reaction-convection-diffusion equation

We develop a collocation method for solving the singularly perturbed convection-diffusion equation and reaction-diffusion equation of elliptic type partial differential equations. When the singularly perturbed solution is expressed in terms of a set of fractional powers 2D exponential trial functions, and collocate points to satisfy the boundary conditions and the governing equation, we can obtain a small scale normal linear system, which is solved by the Gaussian elimination method or the conjugate gradient method to determine the expansion coefficients. The numerical algorithm with complexity O(n) is very time saving, effective and accurate to find the solutions of highly singularly perturbed reaction-convection-diffusion problems, which are defined in a rectangular domain as well as in an arbitrary domain, and the numerical examples together with an experimental result confirm the assessments.

Exact solutions of initial boundary value problems for Sobolev type equations of elastic oscillations

The article is devoted to the problem of the unique solvability of initial boundary value problems for partial differential equations. These initial boundary value problems are dissipative dynamic mathematical models from the field of linear elasticity. The considered equations are not resolved with respect to the highest time derivative. Therefore, these equations are referred to the so-called Sobolev type partial differential equations. The research of initial boundary value problems is carried out from general position in the form of the complete second order differential equation in abstract spaces. The theory of the distributions with values in Banach spaces and the concept of a fundamental solution of differential operator are used. Theorems of existence and uniqueness of the solutions of considered initial boundary value problems in the class of functions of finite smoothness with respect to time are proved. These solutions are obtained as an functional series of the Kummer confluent hypergeometric function.

Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

Physics-informed neural networks (PINNs) have recently emerged as an alternative way of numerically solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In particular, in addition to the deep neural network (DNN) for the solution, an auxiliary DNN is considered that represents the residual of the PDE. The residual is then combined with the mismatch in the given data of the solution in order to formulate the loss function. This framework is effective but is lacking uncertainty quantification of the solution due to the inherent randomness in the data or due to the approximation limitations of the DNN architecture. Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty. We first account for the parametric uncertainty when the parameter in the differential equation is represented as a stochastic process. Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors. We can then make predictions from new sensor measurements very efficiently with the trained DNNs. Moreover, we employ dropout to quantify the uncertainty of DNNs in approximating the modal functions. We then design an active learning strategy based on the dropout uncertainty to place new sensors in the domain in order to improve the predictions of DNNs. Several numerical tests are conducted for both the forward and the inverse problems to demonstrate the effectiveness of PINNs combined with uncertainty quantification. This NN-aPC new paradigm of physics-informed deep learning with uncertainty quantification can be readily applied to other types of stochastic PDEs in multi-dimensions.

Stability Analysis of Linear Partial Differential Equations With Generalized Energy Functions

We present a method for the stability analysis of a large class of linear partial differential equations (PDEs) in one spatial dimension. We rely on Lyapunov analysis to establish the exponential stability of the systems under consideration. The proposed test for the verification of the underlying Lyapunov inequalities relies on the existence of solutions of a system of coupled differential equations. We illustrate the application of this method using a PDE actuated by a backstepping computed feedback law. Furthermore, for the case of PDEs defined by polynomial data, we formulate a numerical methodology in the form of a convex optimization problem which can be solved algorithmically. We show the effectiveness of the proposed numerical methodology using examples of different types of PDEs.

Effects of Straw and Biogas Co-firing on the Combustion Characteristics

In this study, effects of straw co-firing with a renewable fuel – low heating value biogas – on the combustion characteristics, heat output from the device and composition of flue gas, are studied and analyzed with the aim to achieve more efficient use of straw for energy production. The study combines experimental research and mathematical modelling of the processes developing at co-firing of straw pellets with a low heating value biogas, which consists of 60 % methane and 40 % CO2. The complex research of the effects of straw co-firing on the gasification and combustion characteristics suggests that the increase of the biogas supply from 0 up to 1.35 l/min (additional heat input up to 0.4 kW) stabilizes the after-flame stage of the combustion of straw pellets, thus completing the burnout of volatiles, increasing the average value of the axial flow velocity, the flame temperature and the heat output from the device by about 25 – 30 %. The mathematical modelling of the straw co-firing with biogas is provided with account of variations of combustible volatiles by increasing the biogas supply into the experimental setup. A mathematical model of the combustion dynamics has been developed using MATLAB. A system of dimensionless parabolic type partial differential equations was used to describe the formation of the 2D compressible reacting swirling flow, the variations of the mass fraction of volatiles, flame species and flame temperature at thermo-chemical conversion of volatiles. Results of the numerical simulation were analyzed considering the processes developing downstream the combustor.

Meromorphic solutions of Fermat type partial differential equations

The main purpose of this paper is to describe the pseudo-primeness of the meromorphic solutions of the Fermat type functional equation F(uz1,⋯,uzn)+uk=1, where F(z1,⋯,zn) is a nonconstant homogeneous polynomial with degree m in n variables z1,⋯,zn and k is a positive integer. In addition, we give a simple proof and strengthening of a theorem which is related to the Fermat type functional equation f3+g3=1 in C2 by improving the condition so that fz2 and gz1 have the same zeros (ignoring multiplicities).

Pressure-driven gas flow in viscously deformable porous media: application to lava domes

The behaviour of low-viscosity, pressure-driven compressible pore fluid flows in viscously deformable porous media is studied here with specific application to gas flow in lava domes. The combined flow of gas and lava is shown to be governed by a two-equation set of nonlinear mixed hyperbolic–parabolic type partial differential equations describing the evolution of gas pore pressure and lava porosity. Steady state solution of this system is achieved when the gas pore pressure is magmastatic and the porosity profile accommodates the magmastatic pressure condition by increased compaction of the medium with depth. A one-dimensional (vertical) numerical linear stability analysis (LSA) is presented here. As a consequence of the pore-fluid compressibility and the presence of gravitation compaction, the gradients present in the steady-state solution cause variable coefficients in the linearized equations which generate instability in the LSA despite the diffusion-like and dissipative terms in the original system. The onset of this instability is shown to be strongly controlled by the thickness of the flow and the maximum porosity, itself a function of the mass flow rate of gas. Numerical solutions of the fully nonlinear system are also presented and exhibit nonlinear wave propagation features such as shock formation. As applied to gas flow within lava domes, the details of this dynamics help explain observations of cyclic lava dome extrusion and explosion episodes. Because the instability is stronger in thicker flows, the continued extrusion and thickening of a lava dome constitutes an increasing likelihood of instability onset, pressure wave growth and ultimately explosion.

Symmetries of partial differential equations and stochastic processes in mathematical physics and in finance

In 1971, B. Kent Harrison and Frank B. Estabrook introduced a method to determine the symmetries of partial differential equations (PDEs). These last years, the determination of the symmetries of PDEs in Mathematical Physics, in Mathematical Biology and in Financial Mathematics has proved useful. The computations effected in all these cases let appear a remarkable degree of similarity between them. So with the same aim in mind, we develop a general framework for the computation of the symmetries with this method, we give properties of isovectors for a rather general type of PDE’s and some results on the Lie algebra itself. Finally we present three examples for which all the results we exposed hold.

Solving partial differential equations in computational mechanics via nonlocal numerical approaches

AbstractThe work addresses solving partial differential equations (PDE)s for a continuum solid body using a nonlocal formulation. Second order spatial derivatives are of concern as they enable to deal with many dynamic problems in physics. The authors introduce a nonlocal finite difference (FD) scheme to show its applicability to computational mechanics. In the theoretical part of the work exemplary solutions of the second order ordinary derivatives are discussed. The capability of numerical dispersion reduction, which is identified for the proposed approach, is demonstrated for the crude spatial domain discretization. Based on the provided nonlocal FD scheme, discrete forms for the two types of PDEs are derived to indicate potential areas of practical applications for the presented method. Elastic wave equation and thermal diffusion equation are taken into account. Moreover, for the former case, the peridynamics – which is also a nonlocal approach – is also used to derive an alternative discrete formulation. The stability conditions for the provided numerical schemes are determined using von Neumann stability analysis. The limits for the allowed simulation time steps are found. The second part of the paper is devoted to the practical applications of the elaborated nonlocal formulations. There are studied selected thermal and mechanical properties of a gas foil bearing (GFB) component. A rod‐like structural part of GFB made of superalloy Inconel 625 is analyzed. A one‐dimensional model is built to determine temperature and displacement distributions via transient simulations. Again, reduced numerical dispersion is confirmed for the model with a crude mesh.

An inverse boundary value problem for the p-Laplacian: a linearization approach

This work tackles an inverse boundary value problem for a p-Laplace type partial differential equation parametrized by a smoothening parameter . The aim is to numerically test reconstructing a conductivity type coefficient in the equation when Dirichlet boundary values of certain solutions to the corresponding Neumann problem serve as data. The numerical studies are based on a straightforward linearization of the forward map, and they demonstrate that the accuracy of such an approach depends nontrivially on and the chosen parametrization for the unknown coefficient. The numerical considerations are complemented by proving that the forward operator, which maps a Hölder continuous conductivity coefficient to the solution of the Neumann problem, is Fréchet differentiable, excluding the degenerate case that corresponds to the classical (weighted) -Laplace equation.

Initial boundary value problems for Sobolev type equations of hereditary continuum mechanics

The article is devoted to the problem of the unique solvability of initial boundary value problems for partial integro-differential equations. These problems are dynamic mathematical models fromthe fields of plasma physics, fluids dynamics and the theory of viscoelasticity. The differential parts of considered equations are not resolved with respect to the highest time derivative. Therefore, these equations are referred to the so-called Sobolev type partial differential equations. Volterra type integral terms have an Abel kernel, which admits a weak singularity. The research of initial boundary value problems is carried out from general position in the form of the equations in abstract spaces. The theory of the distributions with values in Banach spaces and the concept of a fundamental solution of integrodifferential operator are used. Theorems of existence and uniqueness of the solutions of considered initial boundary value problems in the class of functions of finite smoothness with respect to time are proved. Formulas of these solutions are obtained.

THE TREFFTZ TEST FUNCTIONS METHOD FOR SOLVING THE GENERALIZED INVERSE BOUNDARY VALUE PROBLEMS OF LAPLACE EQUATION

The issue of data completion is important for the elliptic type partial differential equation. In the inverse Cauchy problem, we need to complete the boundary data by over-specifying Dirichlet and Neumann data on a portion of the boundary. In this paper, we numerically solve the generalized inverse boundary value problems of Laplace equation in a rectangle with one boundary function and two boundary functions missing, which are more difficult than the inverse Cauchy problem. By using the technique of a boundary integral equation method together with a specially designed Trefftz test function, we can complete the boundary data by requiring minimal extra data. Then solving the Laplace equation with the given data and recovered data by the multiple-scale Trefftz method, we can find the numerical solution in the interior nodal points.

A meshless method for solving the nonlinear inverse Cauchy problem of elliptic type equation in a doubly-connected domain

In the paper a nonlinear inverse Cauchy problem of nonlinear elliptic type partial differential equation in an arbitrary doubly-connected plane domain is solved using a novel meshless numerical method. The unknown Dirichlet data on an inner boundary are recovered by over-specifying the Cauchy data on an outer boundary. A homogenization function is derived to annihilate the Cauchy data on the outer boundary, and then a homogenization technique generates a transformed equation in terms of a transformed variable, whose outer Cauchy boundary conditions are homogeneous. When the numerical solution is expanded by a sequence of boundary functions, which automatically satisfy the homogeneous Cauchy boundary conditions on the outer boundary, we can solve the transformed equation by the domain type meshless collocation method. For the nonlinear inverse Cauchy problems we require to iteratively solve the linear systems with the right-hand sides varying per iteration step. The accuracy and robustness of the homogenization boundary function method (HBFM) are examined through seven numerical examples, where we compare the exact Dirichlet data on the inner boundary to the ones recovered by the HBFM under a large noisy disturbance.

Equivalence and symmetries for variable coefficient linear heat type equations. II. Fundamental solutions

We present a comparative study of fundamental solutions (heat kernels) of variable coefficient heat type partial differential equations based on Lie symmetry group methods and equivalence transformations discussed in the work of Güngör [J. Math. Phys. 59, 051507 (2018)]. Applications will include both one- and two-dimensional equations.

Zarys metody szacowania trwałości elementów lub zespołów urządzeń z zachowaniem wymaganego poziomu niezawodności

The paper includes a probabilistic method for evaluating the durability of components and device assemblies which operate under the impact of destructive processes. As a result of these processes, wear that causes deterioration of their cooperation conditions occurs. It is assumed that a component operates reliably when the wear does not exceed the acceptable (limit) values. In mathematical terms, this method is based on a differential equation, after the transformation of which, it is possible to obtain the Fokker-Planck type partial differential equation. The specific solution of this equation allows for obtaining the density function of the probability wear in the normal distribution form. The paper presents two methods for determining the durability. The first one involves the application of the wear density function, and the second one consists in determining the probability density function of the time of reaching the acceptable state, and its use in order to determine the component or assembly durability. The paper presents a numerical example on the aircraft technology operation process

Solving singularly perturbed problems by a weak-form integral equation with exponential trial functions

The second-order singularly perturbed problem is transformed to a singularly perturbed parabolic type partial differential equation by using a fictitious time technique. Then we use Green’s second identity to derive a boundary integral equation in terms of the adjoint Trefftz test functions, namely a weak-form integral equation method (WFIEM). It accompanying with the exponential trial functions, which are designed to satisfy the boundary conditions automatically, can provide very accurate numerical solutions of linear and nonlinear singularly perturbed problems. For the latter problem the iterative procedure is convergent very fast.

Exact Traveling Wave Solutions to Benney- Luke Equation

By using the improved (G¢/G) -expansion method, we obtained some travelling wave solutions of well-known nonlinear Sobolev type partial differential equations, namely, the Benney-Luke equation. We show that the improved (G¢/G) -expansion method is a useful, reliable, and concise method to solve these types of equations.GANIT J. Bangladesh Math. Soc.Vol. 37 (2017) 1-14

Transient Vibrations of a Half-Space Under a Massive Line Loading

Dynamic response of a half-space subjected to loads is of considerable practical interest for engineers and scientists. Recent investigations demonstrated that the inertial influences of moving loads are of importance in some cases. Although there are several works on moving inertial loads (masses), very few works were performed on the interaction of a half-space and a stationary inertial foundation on its surface. In this study, a new analytical–numerical method has been used to investigate the vertical interaction between a massive strip foundation and a homogenous, isotropic elastic half-space. Navier’s equations of motion were transformed to a system of wave-type partial differential equations using the Helmholtz resolution. The interactive tractions between the strip foundation and the half-space were imposed to the problem as boundary conditions. A concurrent two-sided and one-sided Laplace integral transform was used for the wave type partial differential equations subjected to the specific boundary conditions and eventually the solution in transformed form was obtained. In order to inverse the transformed solution, the Cagniard–De Hoop method accompanying a numerical procedure was implemented. Final results revealed the influence of the inertia of the massive strip foundation on the dynamic response of the half-space.

A Semi-Automatic Numerical Algorithm for Turing Patterns Formation in a Reaction-Diffusion Model

The Turing pattern formation is modeled by reaction - diffusion (RD) type partial differential equations , and it plays a crucial role in ecological studies. Big data analytics and suitable frameworks to manage and predict structures and configurations are mandatory. The processing and resolution procedures of mathematical models relies upon numerical schemes, and concurrently upon the related automated algorithms. Starting from a RD model for vegetation patterns, we propose a semi-automatic algorithm based on a smart numerical criterion for observing ecological reliable results. Numerical experiments are carried out in the case of spot’s formations.