Order Linear Partial Differential Equations Research Articles

Formulation of physical fields from systems of first order linear partial differential equations

We present a heuristic approach to formulate physical fields from systems of first order linear partial differential equations by considering the symmetric and antisymmetric properties of the matrices that represent the systems of equations. Particularly, we show that Maxwell and Dirac fields can be established by imposing different types of symmetric and antisymmetric conditions on their associated representing matrices.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

The special solutions of two-dimensional drift-flux equations for the two-phase flow

This paper considers the special solutions of two-dimensional isentropic drift-flux equations for the two-phase flow. The nonlinear system is transformed into a second order linear partial differential equation by the hodograph and the polar coordinate transformations. Specific solutions (circulatory flow, radial flow, and spiral flow) are obtained for steady and irrotational systems. Moreover, the special solution of radial flow for the pseudo-steady two-phase flow system is obtained.

On the hyperbolic type differential equation with time involution

In the present paper, the initial value problem for the hyperbolic type involutory in t second order linear partial differential equation is studied. The initial value problem for the fourth order partial differential equations equivalent to this problem is obtained. The stability estimates for the solution and its first and second order derivatives of this problem are established.

Several families of new exact solutions for second order partial differential equations with variable coefficients

Several families of new exact solutions for a general second order linear partial differential equation with variable coefficients are derived in this paper. All the possible polynomial and polynomial-like solutions of this equation are derived. It is shown that there exist exactly two sets of such families of exact solutions. These solutions are extended to construct different families of exact solutions in terms of hypergeometric functions, which include polynomial solutions as particular cases. A total of eight families of exact solutions are derived using a novel method of balancing powers of the variables simultaneously. Several well known linear partial differential equations in applied mathematics and mechanics are special cases of the general equation considered in this paper and all the polynomial and polynomial-like solutions of these partial differential equations are also explicitly derived as special cases.

Effect of the Forcing Function values on the Numerical Simulation of General Second Order Linear Partial Differential Equation

The second-order PDEs are used to model a wide variety of application problems and due to their indispensable usage the computational analysis of various aspects have worked out by researchers in the field. In this study, a computational analysis of the finite difference numerical solution of the general linear second-order PDE with constant coefficients is presented. The study’s objective was to examine the simultaneous effect of the varied size of the computational domain and the forcing function values on the numerical simulation. From the results, it is revealed that the forcing function affects the simulation patterns as long as the size of the domain is increased from 1 × 1 square unit to 2 × 2, 3 × 3, and 4 × 4. Also, the local node-wise solution values change considerably with the varied values of the forcing function. The effect of forcing function G values on the numerical solution is observed higher when G < 0 and is lower when G > 0. The outcomes of this research study are expected to provide ways to predict the simulations obtained by the general second-order PDE based on the varied domain size and the forcing function with constant coefficients of the PDE.

Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

A high order discontinuous Galerkin method for the symmetric form of the anisotropic viscoelastic wave equation

We introduce a new symmetric treatment of anisotropic viscous terms in the viscoelastic wave equation. An appropriate memory variable treatment of stress-strain convolution terms, result into a symmetric system of first order linear hyperbolic partial differential equations, which we discretize using a high-order discontinuous Galerkin finite element method. The accuracy of the resulting numerical scheme is verified using convergence studies against analytical plane wave solutions and analytical solutions of the viscoelastic wave equation. Computational experiments are shown for various combinations of homogeneous and heterogeneous viscoelastic media in two and three dimensions.

Value functions in a regime switching jump diffusion with delay market model

<abstract><p>In this paper, we consider a market model where the risky asset is a jump diffusion whose drift, volatility and jump coefficients are influenced by market regimes and history of the asset itself. Since the trajectory of the risky asset is discontinuous, we modify the delay variable so that it remains defined in this discontinuous setting. Instead of the actual path history of the risky asset, we consider the continuous approximation of its trajectory. With this modification, the delay variable, which is a sliding average of past values of the risky asset, no longer breaks down. We then use the resulting stochastic process in formulating the state variable of a portfolio optimization problem. In this formulation, we obtain the dynamic programming principle and Hamilton Jacobi Bellman equation. We also provide a verification theorem to guarantee the optimal solution of the corresponding stochastic optimization problem. We solve the resulting finite time horizon control problem and show that close form solutions of the stochastic optimization problem exist for the cases of power and logarithmic utility functions. In particular, we show that the HJB equation for the power utility function is a first order linear partial differential equation while that of the logarithmic utility function is a linear ordinary differential equation.</p></abstract>

Streamlines in the Two-Dimensional Spreading of a Thin Fluid Film: Blowing and Suction Velocity Proportional to the Height

The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, vn satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which vn is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.

Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition

There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. In this paper we consider the convergence analysis of the explicit schemes for solving one dimensional, time-dependent wave equation with Drichlet and Neumann boundary condition. Taylor's series expansion is used to expand the finite difference approximations in the explicit scheme. We present the derivation of the schemes and develop a computer program to implement it We use spectral radius of Matrix obtained from discretization and Von Neumann stability condition to determine stability, and consistence of the method from truncated error from discretized method. Using Lax Equivalence Theorem, convergence of the methods was described by testing consistency and stability of the methods. And it is found out that the scheme is stable with the Drichlet boundary and conditionally stable with Derivative boundary condition.

Numerical solution of time fractional order partial differential equations

In this work, we have presented analysis of time fractional order linear and non-linear partial differential equations with initial value and boundary conditions by applying Riemann - Leivoulli fractional integral. As the explained differential equations are related to natural phenomenon, it may be observed under various circumstances for which the possible outcome may vary. The properties and nature of physical states of these equations have been emphasised more precisely by taking fractional order. Fractional order homotopy perturbation method has tackled the approximate solutions in the series form of well known time fractional order linear and non linear differential equations. Numerical simulations are demonstrated prominently in graphical format by using Matlab.

Existence and uniqueness results in weighted spaces for Dirichlet problem in unbounded domains

We study the Dirichlet problem for a second order linear elliptic partial differential equation with discontinuous coefficients in unbounded domains. We establish an existence and uniqueness result.

On the Riemann Function

Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in two variables. The first review of Riemann’s method was published by E.T. Copson in 1958. This study extends that work. Firstly, three solution methods were overlooked in Copson’s original paper. Secondly, several new approaches for finding Riemann functions have been developed since 1958. Those techniques are included here and placed in the context of Copson’s original study. There are also numerous equivalences between Riemann functions that have not previously been identified in the literature. Those links are clarified here by showing that many known Riemann functions are often equivalent due to the governing equation admitting a symmetry algebra isomorphic to S L ( 2 , R ) . Alternatively, the equation admits a Lie-Bäcklund symmetry algebra. Combining the results from several methods, a new class of Riemann functions is then derived which admits no symmetries whatsoever.

SOLUSI LEMAH MASALAH DIRICHLET PERSAMAAN DIFERENSIAL PARSIAL LINEAR ELIPTIK ORDER DUA

The weak solution is one of solutions of the partial differential equations, that is generated from derivative of the distribution. In particular, the definition of a weak solution of the Dirichlet problem for second order linear elliptic partial differential equations is constructed by the definition and the characteristics of Sobolev spaces on Lipschitz domain in R^n. By using the Lax Milgram Theorem, Alternative Fredholm Theorem and Maximum Principle Theorem, we derived the sufficient conditions to ensure the uniqueness of the weak solution of Dirichlet problem for second order linear elliptic partial differential equations. Furthermore, we discussed the eigenvalue of Dirichlet problem for second order linear elliptic partial differential equations with respect to the weak solution.

Calabi–Yau threefolds with small Hodge numbers associated with a one-parameter family of polynomials

We construct several quintic Calabi–Yau threefolds over the rationals with small Hodge numbers, by using certain members of a family of polynomial solutions of a second order linear partial differential equation.

English

English

The Mathematical Modelling of a Fixed Source of Dust

A mathematical model for the diffusion of dust particles emitted from a fixed source is investigated using the atmospheric diffusion equation. This model poses an initial boundary value problem with a second order linear partial differential equation. The steady state case of this problem when the uniform source is situated at ground level was examined by Sharan et al. [1]. The solution of the unsteady case in closed form for a time dependent source is derived. Two special cases, in which the source function of time is explicitly given and special values of the diffusion parameters are taken, are examined in detail. In the case when diffusion is present only in the vertical direction, it is shown that for small times, the particles spread with a front that travels with the speed of the wind. When diffusion is present only in the direction of the wind, there is no discontinuity front and the particles diffuse slowly into the direction of the wind. The solutions for the special cases considered are examined for large values of time. It is found that the solution approaches that of the corresponding steady state solution of the equation.

Spectra of certain large tridiagonal matrices

We characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal n by n matrices subject to arbitrary boundary conditions, i.e. with arbitrary elements on the first and last rows of the matrix. For large n, we show there are up to 4 eigenvalues, the so-called special eigenvalues, whose behavior depends sensitively on the boundary conditions. The other eigenvalues, the so-called regular eigenvalues vary very little as function of the boundary conditions. For large n, we determine the regular eigenvalues up to O(n−2), and the special eigenvalues up to O(κn), for some κ∈(0,1). The components of the eigenvectors are determined up to O(n−1).The matrices we study have important applications throughout the sciences. Among the most common ones are arrays of linear dynamical systems with nearest neighbor coupling, and discretizations of second order linear partial differential equations. In both cases, we give examples where specific choices of boundary conditions substantially influence leading eigenvalues, and therefore the global dynamics of the system.

Finit element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for linear convex quadrilaterals of cubic order Serendipity and Lagrange families

This paper presents an explicit integration scheme to compute the stiffness matrix of twelve node and sixteen node linear convex quadrilateral finite elements of Serendipity and Lagrange families using an explicit integration scheme and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique, In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties matrices and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+) in the bivariates over a 2-square (-1 ) with the nodes on the boundary and in the interior of this simple domain. The finite elements up to cubic order have nodes only on the boundary for Serendipity family and the finite elements with boundary as well as some interior nodes belong to the Lagrange family. The first order element is the bilinear convex quadrilateral finite element which is an exception and it belongs to both the families. We have for the present ,the cubic order finite elements which havee 12 boundary nodes at the nodal coordinates {(-1,-1),(1,-1),(1,1),(-1,1),(-1/3,-1), (1/3,-1),(1,-1/3),(1,1/3),(1/3,1),(-1/3,1),(-1,1/3),(-1,-1/3)} and the four interoior nodal coordinates at the points (-1/3,-1/3),(1/3,-1/3),(1/3,1/3),(-1/3,1/3)} in the local parametric space ( In this paper, we have computed the integrals of local derivative products with linear denominator (4+) in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be then applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all 12-node and 16-node linear(straight edge) convex quadrilaterals for a polygonal domain which provides the nodal coordinates and the element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is an unknown physical variable and in with Dirichlet boundary conditions over the convex polygonal domain.