Order Linear Partial Differential Equations Research Articles

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.

Boundary value problems for the Cimmino system via quaternionic analysis

In this paper, we study a class of boundary value problems for a first order linear partial differential equation (all of whose solutions are harmonic functions), which is called the Cimmino system. With the help of the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of Cimmino system’s solutions, necessary and sufficient conditions for the solvability of the non-homogeneous Cimmino system coupled by the boundary conditions are derived and its general solution is explicitly described.

An analysis on classifications of hyperbolic and elliptic PDEs

Abstract Purpose Our aim in this study is to generate some partial differential equations (PDEs) with variable coefficients by using the PDEs with non-constant coefficients. Methods Then by applying the single and double convolution products, we produce some new equations having polynomials coefficients. We then classify the new equations on using the classification method for the second order linear partial differential equations. Results Classification is invariant under single and double convolutions by applying some conditions, that is, we identify some conditions where a hyperbolic equation will be hyperbolic again after single and double convolutions. Conclusions It is shown that the classifications of the new PDEs are related to the coefficients of polynomials which are considered during the process of convolution product.

Two-point Taylor approximations of the solutions of two-dimensional boundary value problems

We consider two-dimensional elliptic boundary value problems of the form L(U)=F in Ω⊂R2,Ω bounded and open, with a Dirichlet boundary condition U|∂Ω=H, where L is a second order linear differential operator whose coefficients, as well as the functions F and H are differentiable up to a certain degree. In a recent paper [C. Kesan, Taylor polynomial solutions of second order linear partial differential equations, Appl. Math. Comput. 152 (2004) 29–41], a matrix algorithm is introduced to compute the standard Taylor polynomial of the solution U at a certain point in Ω. We propose an alternative formulation of the problem based on a redefinition of the unknown U and the use of the standard Frobenius method that simplifies the computation of the Taylor coefficients of U. We also consider the use of a two-point Taylor representation of the solution, instead of the classical one-point Taylor representation, which gives a more uniform approximation of the solution.

Weak-duality based adaptive finite element methods for PDE-constrained optimization with pointwise gradient state-constraints

Adaptive finite element methods for optimization problems for second order linear elliptic partial differential equations subject to pointwise constraints on the 2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.

An Lp‐Estimate for Weak Solutions of Elliptic Equations

We prove an Lp‐a priori bound, p > 2, for solutions of second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

Legendre Approximation for Solving Linear HPDEs and Comparison with Taylor and Bernoulli Matrix Methods

The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.

Explicit polynomial solutions of fourth order linear elliptic Partial Differential Equations for boundary based smooth surface generation

We present an explicit polynomial solution method for surface generation. In this case the surface in question is characterized by some boundary configuration whereby the resulting surface conforms to a fourth order linear elliptic Partial Differential Equation, the Euler–Lagrange equation of a quadratic functional defined by a norm. In particular, the paper deals with surfaces generated as explicit Bézier polynomial solutions for the chosen Partial Differential Equation. To present the explicit solution methodologies adopted here we divide the Partial Differential Equations into two groups namely the orthogonal and the non-orthogonal cases. In order to demonstrate our methodology we discuss a series of examples which utilize the explicit solutions to generate smooth surfaces that interpolate a given boundary configuration. We compare the speed of our explicit solution scheme with the solution arising from directly solving the associated linear system.

A group invariant solution for a pre-existing fluid-driven fracture in permeable rock

The propagation of a two-dimensional pre-existing fracture in permeable rock by the injection of a viscous, incompressible Newtonian fluid is considered. The fluid flow in the fracture is laminar. By the application of lubrication theory, a partial differential equation relating the half-width of the fracture to the fluid pressure and leak-off velocity is derived. The model is closed by the adoption of the PKN formulation in which the fluid pressure is proportional to the fracture half-width. The partial differential equation admits four Lie point symmetries provided the leak-off velocity satisfies a first order linear partial differential equation. The solution of this equation yields the leak-off velocity as a function of the distance along the fracture and time. The group invariant solution is derived by considering a linear combination of the Lie point symmetries. The boundary value problem is reformulated as a pair of initial value problems. The model in which the leak-off velocity is proportional to the fracture half-width is considered. The working condition of constant pressure at the fracture entry is analysed in detail.

Dynamical systems and Poisson structures

We first consider the Hamiltonian formulation of n=3 systems, in general, and show that all dynamical systems in R3 are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. The construction of the Poisson structures is based on solving an associated first order linear partial differential equations. We find the Poisson structures of a dynamical system recently given by Bender et al. [J. Phys. A: Math. Theor. 40, F793 (2007)]. Secondly, we show that all dynamical systems in Rn are locally (n−1)-Hamiltonian. We give also an algorithm, similar to the case in R3, to construct a rank two Poisson structure of dynamical systems in Rn. We give a classification of the dynamical systems with respect to the invariant functions of the vector field X⃗ and show that all autonomous dynamical systems in Rn are superintegrable.

Neumann problem for generalized n-Poisson equation

Using a hierarchy of integral operators having higher-order Neumann functions and their derivatives as kernels, the Neumann problem for a 2 n th order linear partial complex differential equation is discussed. The solvability of the problem is obtained.

Type-II hidden symmetries through weak symmetries for nonlinear partial differential equations

The Type-II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. In [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612–622] Abraham-Shrauner and Govinder have analyzed the provenance of this kind of symmetries and they developed two methods for determining the source of these hidden symmetries. The Lie point symmetries of a model equation and the two-dimensional Burgers' equation and their descendants were used to identify the hidden symmetries. In this paper we analyze the connection between one of their methods and the weak symmetries of the partial differential equation in order to determine the source of these hidden symmetries. We have considered the same models presented in [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612–622], as well as the WDVV equations of associativity in two-dimensional topological field theory which reduces, in the case of three fields, to a single third order equation of Monge–Ampère type. We have also studied a second order linear partial differential equation in which the number of independent variables cannot be reduced by using Lie symmetries, however when is reduced by using nonclassical symmetries the reduced partial differential equation gains Lie symmetries.

Three Dimensional Interface Problems for Elliptic Equations

The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular lines and some isolated singular points. It is proved that near a singular line or a singular point, each weak solution can be decomposed into two parts, a singular part and a regular part. The singular parts are some finite sum of particular solutions to some simpler equations, and the regular parts are bounded in some norms, which are slightly weaker than that in the Sobolev space H2.

On systems of first order linear partial differential equations with $L^p$ coefficients

We establish the existence, the uniqueness, and the stability of the solution to the Cauchy problem associated to a general class of systems of first order linear partial differential equations under minimal regularity assumptions on their coefficients.

Hyers-Ulam stability of first order linear partial differential equations with constant coefficients

Hyers-Ulam stability of first order linear partial differential equations with constant coefficients

On systems of first order linear partial differential equations with $L^p$ coefficients

We establish the existence, the uniqueness, and the stability of the solution to the Cauchy problem associated to a general class of systems of first-order linear partial differential equations under minimal regularity assumptions on their coefficients.

Computation of spiking activity for a stochastic spatial neuron model: Effects of spatial distribution of input on bimodality and CV of the ISI distribution

We obtain computational results for a new extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation and the synaptic input current is also a function of space and time, obeying a first order linear partial differential equation driven by a two-parameter random process. The model is first described explicitly with the inclusion of all biophysical parameters. Simplified equations are obtained with dimensionless space and time variables. A standard parameter set is described, based mainly on values appropriate for cortical pyramidal cells. When the noise is small and the mean voltage crosses threshold, a formula is derived for the expected time to spike. A simulation algorithm, involving one-dimensional random processes is given and used to obtain moments and distributions of the interspike interval (ISI). The parameters used are those for a near balanced state and there is great sensitivity of the firing rate around the balance point. This sensitivity may be related to genetically induced pathological brain properties (Rett’s syndrome). The simulation procedure is employed to find the ISI distribution for some simple patterns of synaptic input with various relative strengths for excitation and inhibition. With excitation only, the ISI distribution is unimodal of exponential type and with a large coefficient of variation. As inhibition near the soma grows, two striking effects emerge. The ISI distribution shifts first to bimodal and then to unimodal with an approximately Gaussian shape with a concentration at large intervals. At the same time the coefficient of variation of the ISI drops dramatically to less than 1/5 of its value without inhibition.

Spatial neuron model with two-parameter Ornstein–Uhlenbeck input current

We consider a new and extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation. The synaptic input current is also a function of space and time and satisfies a first order linear partial differential equation driven by a two-parameter random process. A natural choice for these random input processes is to make them two-parameter Poisson processes for both excitation and inhibition. For such inputs the mean subthreshold voltage is found in the case of an infinite cable and of finite cables. Assuming uniform amplitudes and rates exact expressions are obtained in the case of particular boundary conditions. We then consider a diffusion approximation and show that the membrane current is a two-parameter Ornstein–Uhlenbeck process, whose statistical properties are derived. Using representations for the voltage in terms of stochastic integrals in the plane we find, in the case of finite space intervals the mean, variance and covariance of the subthreshold voltage. For large times the voltage process is shown to be covariance stationary in time and the corresponding spectral density is found and compared with the result for a purely (two-parameter) white noise driven cable. The limiting white noise case is obtained from the extended model as the decay parameter becomes infinite. Finally, we develop useful simulation methods for the solution of the extended spatial model using properties of stochastic integrals involving eigenfunctions to obtain one-dimensional representations which are easily implemented.

Advertising for the introduction of an age‐sensitive product

AbstractAn age‐dependent market segmentation is often observed for real life products. We introduce a simple age‐structured model for the advertising process of a firm and the consequent goodwill evolution. The model formal structure is characterized by a first order linear partial differential equation. We formulate the advertising problem for a new product introduction as a distributed parameter optimal control problem and solve it using the suitable maximum principle conditions. Then, we discuss and solve the same problem under the assumption that a finite set of boundedly age‐selective advertising channels are available, instead of one infinitely age‐selective channel. Copyright © 2005 John Wiley & Sons, Ltd.

Uso de um esquema implícito e de splines para a simulação numérica do processo de resfriamento de frutas esféricas

Este trabalho teve como objetivo desenvolver um estudo teórico através do modelo matemático de Fourier, em sua forma unidimensional e em coordenadas esféricas, o que é descrito por uma equação diferencial parcial linear de segunda ordem, para a obtenção das curvas e do tempo de resfriamento de frutas esféricas. Utilizou-se o Método das Diferenças Finitas (MDF) segundo o esquema implícito no ambiente computacional Matlab 6.1, analisando-se e comparando os principais resultados através do método de ajuste spline. O modelo matemático foi validado com dados experimentais do resfriamento de laranja Valência num sistema com ar forçado (Q = 1933 m³ h-1, e velocidade em torno de 1 m s-1), com temperatura inicial de 26,57 ºC e resfriada até 1 ºC. Os resultados teóricos obtidos com a simulação foram comparados com os encontrados com a aplicação de um modelo bidimensional desenvolvido em coordenadas esferoidais prolato, com o emprego do Método dos Volumes Finitos (MVF) implementado em Fortran 77. Os dois modelos e métodos aplicados mostram-se aptos para caracterizar o processo de resfriamento das frutas, principalmente em termos de valores práticos específicos, uma vez que geraram resíduos desprezíveis. Apesar destes resultados, o valor do coeficiente convectivo de transferência de calor foi significativamente diferente; tal variação mostra como este coeficiente é sensível a vários fatores do processo de resfriamento e, ainda, que se deve aprofundar as pesquisas para a determinação efetiva do valor do mesmo.