Self-adjoint Differential Equations Research Articles

Solving the nonlinear Riccati equation for the 1D plane-wave reflection response from a velocity gradient interface by a Heun to hypergeometric reduction

For 1D acoustic wave propagation in the frequency domain, the complex nonlinear Riccati differential equation describes the dependence of the plane-wave reflection response in depth, defined as the upgoing part of the wavefield divided by the downgoing part of the wavefield at that depth. In the case that the model is a velocity gradient interface expressed in terms of a smooth Heaviside function defined by the Fermi-Dirac function, the Riccati equation is shown to have an analytical solution for the reflection response. The solution accounts for all wave phenomena in the gradient zone. The solution of the Riccati equation for the reflection response is obtained by introducing the Riccati equation for the reciprocal of the reflection response which can be related to a second-order self-adjoint linear differential equation. The two latter equations obey the same radiation condition. Furthermore, a connection between the second-order linear equation and the Heun equation is obtained, both being characterized by having four regular singular points. By a change of variables, the Heun equation can be transformed into Gauss’ hypergeometric equation, whereby the solution is expressed as the sum of two linearly independent functions containing hypergeometric functions. The advantage of applying the Heun-to-hypergeometric reduction to the nonlinear Riccati equation is that solutions to the hypergeometric equation are well known. The two functions allow us to analytically determine the Riccati solution because the ratio of the linearly independent functions at infinity is constant. In contrast to the classic frequency-independent reflection coefficient from two layers in welded contact, the reflection coefficient at the surface associated with the gradient interface depends on frequency. In the limit of zero frequency, the reflection coefficient approaches the classic one. We have implemented the iterative solution of the integral form of the Riccati equation and found that the solution builds up nonlinearly to the correct solution. The model is verified by a numerical example valid for 1D wave propagation.

Adaptive two-layer ReLU neural network: II. Ritz approximation to elliptic PDEs

In the companion paper [1], we introduce adaptive network enhancement (ANE) method for the best least-squares approximation to a target function by using two-layer ReLU neural networks (NNs). In this paper, we apply the ANE method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The underlying PDE is discretized by the Ritz method using a two-layer spline neural network based on either the primal or dual formulations that minimize the respective energy or complimentary functionals. Essential boundary conditions are imposed weakly through the functionals with proper norms. It is proved that the Ritz approximation is the best approximation in the energy norm; moreover, effect of numerical integration for the Ritz approximation is analyzed as well. Two estimators for adaptive neuron enhancement method are introduced, one is the so-called recovery estimator and the other is the least-squares estimator. Finally, numerical results for diffusion problems with either corner or intersecting interface singularities are presented.

Hille-Nehari type non-oscillation criteria for half-linear dynamic equations with mixed derivatives on a time scale

This article deals with half-linear dynamic equations that have two types of derivatives, and obtains sufficient conditions for all solutions to be non-oscillatory. The obtained results extend a previous Hille-Nehari type theorems for problems of dynamic equations. To prove our main result, we use a generalized Riccati inequality. As an application, we apply the main result to self-adjoint Euler type linear differential and difference equations with a changing sign coefficient. The equation selected for this application is of Mathieu type. For more information see https://ejde.math.txstate.edu/Volumes/2021/78/abstr.html

On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.

Solution of the Neumann problem for a linear self-adjoint differential equation of the second order with variable coefficients by the Fourier transform method

Solution of the Neumann problem for a linear self-adjoint differential equation of the second order with variable coefficients by the Fourier transform method

On the analytic representation of Newtonian systems

We show that the theory of self-adjoint differential equations can be used to provide a satisfactory solution of the inverse variational problem in classical mechanics. A Newtonian equation when transformed to the self-adjoint form allows one to find an appropriate Lagrangian representation (direct analytic representation) for it. On the other hand, the same Newtonian equation in conjunction with its adjoint provides a basis to construct a different Lagrangian representation (indirect analytic representation) for the system. We obtain the time-dependent Lagrangian of the damped Harmonic oscillator from the self-adjoint form of the equation of motion and at the same time identify the adjoint of the equation with the so called Bateman image equation with a view to construct a time-independent indirect Lagrangian representation. We provide a number of case studies to demonstrate the usefulness of the approach derived by us. We also present similar results for a number of nonlinear differential equations by using an integral representation of the Lagrangian function and make some useful comments.

Adaptive linear solution process for single-phase Darcy flow

This article presents an adaptive approach for solving linear systems arising from self-adjoint Partial Differential Equations (PDE) problems discretized by cell-centered finite volume method and stemming from single-phase flow simulations. This approach aims at reducing the algebraic error in targeted parts of the domain usinga posteriorierror estimates. Numerical results of a reservoir simulation example for heterogeneous porous media in two dimensions are discussed. Using the adaptive solve procedure, we obtain a significant gain in terms of the number of time steps and iterations compared to a standard solve.

Interval criteria for oscillation of second-order self-adjoint impulsive differential equations

In this paper, oscillation theorems are given for second-order self-adjoint impulsive differential equations. The obtained results extend the well-known Kamenev-type and Philos-type oscillation theorems. A generalized Riccati transformation is used to prove these results. There are two advantages of using the generalized Riccati transformation rather than the standard Riccati transformation. One is that Kamenev-type and Philos-type oscillation theorems cannot be applied to conditionally oscillatory differential equations such as Euler’s equations, but the obtained results can be applied even to such equations. The other advantage is the ability to prove that the impulsive differential equation may become oscillatory even if the total impulse is small. A specific example is included to demonstrate the merits of the results obtained.

Explicit group AOR method for solving elliptic partial differential equations

The explicit group overrelaxation methods for the numerical solution of the sparse linear systems derived from the discretisation of self-adjoint elliptic partial differential equations have been presented (Yousif & Evans, 1986) and has been shown to be computationally superior in comparison with the implicit 1-line and 2-line block succsessive overrelaxation (SOR) iterative methods. In this work a new explicit 4-point group accelerated overrelaxation (Group AOR) iterative method is presented. A comparison with the point AOR method for the model problem shows the computational advantages of the new technique.

Minimal length Schrödinger equation via factorisation approach

The fourth-order modified Schrodinger equation due to the generalised uncertainty principle is considered in one dimension with a box problem. The factorisation of fourth-order self-adjoint differential equations is then discussed and thereby the wave functions and energy spectra of the modified Schrodinger equation are derived.

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as generalized polynomial chaos expansions. A low-rank variant of the inverse subspace iteration algorithm is presented for computing one or several minimal eigenvalues and corresponding eigenvectors of parameter-dependent matrices. In the algorithm, the iterates are approximated by low-rank matrices, which leads to significant cost savings. The algorithm is tested on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant. Numerical experiments show that the low-rank algorithm produces accurate solutions compared to the Monte Carlo method, and it uses much less computational time than the original algorithm without low-rank approximation.

On new traveling wave solutions and conserved densities for the 2D Ricci flow model

In this paper some travelling wave solutions and conservation laws for the 2D Ricci flow model in conformal gauge are investigated. A guideline able to classify the types of solutions according to the values of some parameters is provided by making use of two versions of the auxiliary equation method. The key feature of these approaches is to take a second order linear ordinary differential equation (ODE), respectively a first order nonlinear ODE with at most an eighth-degree nonlinear term as auxiliary equations. Conserved forms of the travelling wave equation for the Ricci flow are derived through three specific approaches, namely the variational approach, the Ibragimov method for nonlinear self-adjoint differential equations and the one based upon a relationship between conserved forms and their associated symmetries. The former two methods generated similar results, while the latter one has revealed new conserved densities.

Group classification and conservation laws of anisotropic wave equations with a source

Linear and nonlinear waves in anisotropic media are useful in investigating complex materials in physics, biomechanics, biomedical acoustics, etc. The present paper is devoted to investigation of symmetries and conservation laws for nonlinear anisotropic wave equations with specific external sources when the equations in question are nonlinearly self-adjoint. These equations involve two arbitrary functions. Construction of conservation laws associated with symmetries is based on the generalized conservation theorem for nonlinearly self-adjoint partial differential equations. First we calculate the conservation laws for the basic equation without any restrictions on the arbitrary functions. Then we make the group classification of the basic equation in order to specify all possible values of the arbitrary functions when the equation has additional symmetries and construct the additional conservation laws.

Solution of Boundary-Value Problems using Kantorovich Method

We propose a computational scheme for solving the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated as solutions of the parametric eigenvalue problem for an ordinary second-order differential equation. As a result, the initial problem is reduced to a boundary-value problem for a set of self-adjoint second-order differential equations for functions of the second independent variable. The discrete formulation of the problem is implemented using the finite element method with Hermite interpolation polynomials. The effciency of the calculation scheme is shown by benchmark calculations for a square membrane with a degenerate spectrum.

Nonlinear Self-Adjoint Classification of a Burgers-KdV Family of Equations

The concepts of strictly, quasi, weak, and nonlinearly self-adjoint differential equations are revisited. A nonlinear self-adjoint classification of a class of equations with second and third order is carried out.

Comments on the paper “Conservation laws of the (2 + 1)–dimensional KP equation and Burgers equation with variable coefficients and cross terms” by Li-Hua Zhang

The recent paper mentioned in the title contains a confusing statement on computing conservation laws corresponding to symmetries of nonlinearly self-adjoint differential equations. The present brief article contains clarifying comments.

Distribution of zeros of solutions of self-adjoint fourth order differential equations

In this paper, for self-adjoint fourth order differential equations, we establish some lower bounds on the distance between zeros of a nontrivial solution and also lower bounds on the distance between zeros of a solution and/or its derivatives. We also give new results related to boundary value problems which arise in the bending of rods. The main results will be proved by making use of some generalizations of Hardy, Opial and Wirtinger type inequalities.

Eigenvalue singular problem of factorized fourth-order self-adjoint differential equations

This paper deals with the eigenvalue singular problem of the spectral type differential equations of the fourth-order self-adjoint differential operators1(1-x)p(1+x)qd2dx2(1-x)p+2(1+x)q+2d2ydx2-μy=0,y(-1),y(1)finite where, p⩾1,q⩾1. It has been reported in the literature that spectral type differential equations above have general solutions in terms of hypergeometric functions. In this work it is showed that the general solution in terms of hypergeometric functions reduces to Jacobi orthogonal polynomials (as eigenfunctions) in the case of eigenvalue singular problem. The corresponding eigenvalues are found. As application, the natural frequencies and mode shapes of mechanical transverse vibrations of a nonuniform singular structure are reported.

Solving a Class of Self-adjoint Differential Equations of the Fourth Order and its Algorithms in MATLAB

In many of applied sciences, various Self-adjoint differential equations are generated, where, methods for their solution is very complex. Usually, numerical methods used to solve them. Leighton et al were investigated oscillation properties of solutions of self-adjoint differential equations of the fourth order, with specific conditions. In this paper, we use a new method for the solving a class of Self-adjoint differential equations of the fourth order. We use a variable change in the equation, and then obtain an analytical solution for the equation with a specific condition. Because in this method, an analytical solution is obtained, therefore, it is not necessary to use numerical methods to solve the problem.

Factorization of self-adjoint ordinary differential equations

This paper deals with the factorization of self-adjoint differential operators L(2n)=1ρdndxnρβndndxn, and their spectral type differential equations. Sufficient conditions of factorization are reported. A large class of differential operators and equations that can be factorized is obtained. The factorizations of fourth- and sixth-order operators and equations are explicitly given. A particular fourth-order spectral type differential equation in which ρ(x)=(1-x)p(1+x)q, p⩾1,q⩾1, is considered. Its general solution is obtained in terms of hypergeometric functions. As application, the natural frequencies and mode shapes of mechanical transverse vibrations of a nonuniform structure are found.