Singular Equations Research Articles

Toward uniform existence and convergence theorems for three-scale systems of hyperbolic PDEs with general initial data

Uniform existence of solutions to initial-value problems and convergence of appropriately filtered solutions are proven for a special class of three-scale singular limit equations, without any restriction on the initial data. The uniform existence is proven using a novel system of energy estimates. The convergence result is based on a detailed analysis of the fastest-scale oscillations, which unlike in two-scale systems have no explicit solution formula.

A convergence-preserving non-standard finite difference scheme for the solutions of singular Lane-Emden equations

The derivation of numerical schemes for the solution of Lane-Emden equations requires meticulous consideration because they are highly nonlinear in nature; have singularity behaviors at the origin and in some cases, their exact solutions are known for only a few parameter ranges. This explains the reason for the failure of some existing methods in approximating such problems. Thus, in this research, a convergence-preserving Non-Standard Finite Difference Scheme (NSFDS) shall be derived for the solution of Lane-Emden equations. The main idea in this work is the approximation of both the linear and nonlinear terms non-locally and the renormalization/reformulation of the numerator and denominator functions of the Lane-Emden equations. The need for this approach came up due to some setbacks of existing methods where the exact solutions’ qualitative properties are not routinely transferred to the approximate solutions. The analysis of some properties of the NSFDS like convergence, dynamical consistency, monotone dependence on the initial value, and monotonicity of solutions as well as perturbation solution analysis shall be carried out. The NSFDS was then employed in solving some classes of singular initial value problems of the Lane-Emden type and the results obtained show that the proposed method is efficient and computationally cheap.

An Approximation Method to Compute Highly Oscillatory Singular Fredholm Integro-Differential Equations

This paper appertains the presentation of a Clenshaw–Curtis rule to evaluate highly oscillatory Fredholm integro-differential equations (FIDEs) with Cauchy and weak singularities. To calculate the singular integral, the unknown function approximated by an interpolation polynomial is rewritten as a Taylor series expansion. A system of linear equations of FIDEs obtained by using equally spaced points as collocation points is solved to obtain the unknown function. The proposed method attains higher accuracy rates, which are proven by error analysis and some numerical examples as well.

Existence and smoothness of solutions of a singular differential equation of hyperbolic type

This paper investigates the question of the existence of solutions to the semiperiodic Dirichlet problem for a class of singular differential equations of hyperbolic type. The problem of smoothness of solutions is also considered, i.e., maximum regularity of solutions. Such a problem will be interesting when the coefficients are strongly growing functions at infinity. For the first time, a weighted coercive estimate was obtained for solutions to a differential equation of hyperbolic type with strongly growing coefficients.

Algorithms based on a 3-derivative method for singular differential equations including equations with blow-up solutions

We propose block algorithms based on a 3-derivative Nyström-type method (TDNM) for integrating singular second-order ordinary differential equations (ODEs), including parabolic and hyperbolic partial differential equations (PDEs) with blow-up solutions. The block algorithms are implemented in a block-by-block mode for initial value problems (IVPs) and block unification mode for boundary value problems (BVPs). In particular, the algorithms are applied to PDEs by first reducing them into ODEs via the method of lines, whereby the space variable is discretized to yield a system of initial value problems, or the time variable is discretized to yield a system of boundary value problems. It is shown that the numerical solution given by the block-by-block and block unification algorithms are superior to those available in the literature. It is also demonstrated that the implementation of the variable step version of the TDNM, which is generally recommended for IVPs, has no advantage over the block-by-block algorithm.

A Survey of the Earth Dams by Sturm-Liouville Equations - The Singular Case

In this work, we investigate the vibrations of embankments by the singular Sturm-Liouville equations. At first, we create the mathematical form of the vibrations by the shear beam (SB) model (see [21]) and transform this given form to the Sturm-Liouville form with a singularity. Finally, we discuss the numerical solution to the considered problem using the variational iteration method.

Chebyshev-quasilinearization method for solving fractional singular nonlinear Lane-Emden equations

In this paper, we propose a method for solving some classes of the singular fractional nonlinear Lane-Emden type equations. The method is proposed by utilizing the second-kind Chebyshev wavelets in conjunction with the quasilinearization technique. The operational matrices for the second-kind Chebyshev wavelets are used. The method is tested on the fractional standard Lane-Emden equation, the fractional isothermal gas spheres equation, and some other examples. We compare the results produced by the present method with some well-known results to show the accuracy and efficiency of the method.

Fast Multipole Boundary Element Method (FMM/BEM) for the solution of the Navier-Stokes in primitive variables based on the Burton and Miller formulation in two-dimensions

In the present work the fast multipole boundary element method (FMM/BEM) is presented for the solution of incompressible Navier-Stokes equations in 2 dimensions. This is a velocity-pressure primitive variable formulation abbreviated as uv-p. The Stokes-let fundamental solution is used as a weight function in the momentum equation eliminating the pressure from free variable. This way, the LBB conditions are bypassed removing any restriction in the type of grid, placement of nodes, or interpolation. Most of the numerical methods cannot use constant elements for all fields to solve the Navier-Stokes. BEM is the only known so far semi-analytical method capable of such an achievement. The continuity is enforced with the correction of velocity, using a scalar potential, the gradient of which is computed directly with hypersingular equations. The stress hypersingular equation is added to the boundary nodes rendering the well known Burton and Miller formulation and recommendation about the coupling constant is given. Accurate results for cavity Re = 1000 with constant elements and Re=10000 with linear quadrilaterals are obtained and presented. The reached degrees of freedom (dof) are beyond 1 million. Two more problems with mixed BCs are solved. The flow past cylinder is a kind of problem in which both singular and hypersingular equations must be used to ensure convergence. The main advantage of the method is the direct computation of the wall normal stresses or tractions, which is of cornerstone importance in many realistic applications. The tractions are obtained directly from the solution of the system of equations and can be integrated on the surface of a bluff body, to obtain forces and drag and lift coefficients. The pressure is computed in a post-processing phase with hypersingular equation derived in the framework of the present method. Smooth pressure field is obtained in all cases, even when the problem is solved with unstructured grids and constant elements. The method permits the use of non-conforming grids, and it can easily be combined with other numerical methods to obtain wall tractions and stresses via hypersingular equations. Moments for pressure and convective terms are derived.

On the Computational Study of a Fully Wetted Longitudinal Porous Heat Exchanger Using a Machine Learning Approach.

The present study concerns the modeling of the thermal behavior of a porous longitudinal fin under fully wetted conditions with linear, quadratic, and exponential thermal conductivities surrounded by environments that are convective, conductive, and radiative. Porous fins are widely used in various engineering and everyday life applications. The Darcy model was used to formulate the governing non-linear singular differential equation for the heat transfer phenomenon in the fin. The universal approximation power of multilayer perceptron artificial neural networks (ANN) was applied to establish a model of approximate solutions for the singular non-linear boundary value problem. The optimization strategy of a sports-inspired meta-heuristic paradigm, the Tiki-Taka algorithm (TTA) with sequential quadratic programming (SQP), was utilized to determine the thermal performance and the effective use of fins for diverse values of physical parameters, such as parameter for the moist porous medium, dimensionless ambient temperature, radiation coefficient, power index, in-homogeneity index, convection coefficient, and dimensionless temperature. The results of the designed ANN-TTA-SQP algorithm were validated by comparison with state-of-the-art techniques, including the whale optimization algorithm (WOA), cuckoo search algorithm (CSA), grey wolf optimization (GWO) algorithm, particle swarm optimization (PSO) algorithm, and machine learning algorithms. The percentage of absolute errors and the mean square error in the solutions of the proposed technique were found to lie between to and to , respectively. A comprehensive study of graphs, statistics of the solutions, and errors demonstrated that the proposed scheme’s results were accurate, stable, and reliable. It was concluded that the pace at which heat is transferred from the surface of the fin to the surrounding environment increases in proportion to the degree to which the wet porosity parameter is increased. At the same time, inverse behavior was observed for increase in the power index. The results obtained may support the structural design of thermally effective cooling methods for various electronic consumer devices.

Local Hölder continuity to a class of doubly singular nonlinear evolutionary equations

We establish the local regularity of the nonnegative, local, weak solutions to the class of doubly nonlinear parabolic equations∂t(uq)−div(|Du|p−2Du)=0,1<p<2andp−1<q<1. The proof relies on the study on an alternative argument together with an expansion of positivity procedure.

On the local behavior of local weak solutions to some singular anisotropic elliptic equations

Abstract We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations of the kind ∑ i = 1 s ∂ i i u + ∑ i = s + 1 N ∂ i ( A i ( x , u , ∇ u ) ) = 0 , x ∈ Ω ⊂ ⊂ R N for 1 ≤ s ≤ ( N − 1 ) , \mathop{\sum }\limits_{i=1}^{s}{\partial }_{ii}u+\mathop{\sum }\limits_{i=s+1}^{N}{\partial }_{i}({A}_{i}\left(x,u,\nabla u))=0,\hspace{1.0em}x\in \Omega \subset \hspace{-0.3em}\subset \hspace{0.33em}{{\mathbb{R}}}^{N}\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}1\le s\le \left(N-1), where each operator A i {A}_{i} behaves directionally as the singular p p -Laplacian, 1 &lt; p &lt; 2 1\lt p\lt 2 . Throughout a parabolic approach to expansion of positivity we obtain the interior Hölder continuity and some integral and pointwise Harnack inequalities.

Code verification for practically singular equations

The method-of-moments implementation of the electric-field integral equation (EFIE) yields many code-verification challenges due to the various sources of numerical error and their possible interactions. Matters are further complicated by singular integrals, which arise from the presence of a Green's function. To address these singular integrals, an approach was previously presented wherein both the solution and Green's function are manufactured. Because the arising equations are poorly conditioned, they are reformulated as a set of constraints for an optimization problem that selects the solution closest to the manufactured solution. In this paper, we demonstrate how, for such practically singular systems of equations, computing the truncation error by inserting the exact solution into the discretized equations cannot detect certain orders of coding errors. On the other hand, the discretization error from the optimal solution is a more sensitive metric that can detect orders less than those of the expected convergence rate.

A new existence result for some nonlocal problems involving Orlicz spaces and its applications

This paper studies some quasilinear elliptic nonlocal equations involving Orlicz–Sobolev spaces. On the one hand, a new sub-supersolution theorem is proved via the pseudomonotone operator theory; on the other hand, using the obtained theorem, we present an existence result on the positive solutions of a singular elliptic nonlocal equation. Our work improves the results of some previous researches.

Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula

We presented an interpolation method for solving weakly singular Volterra integral equations of the second kind (SVK2). The method based on the barycentric Lagrange interpolation.. For the chosen nodes of the two singular kernel variables, we created two rules that confirm that the denominator of the kernel will never become zero or have an imaginary value. The product of four matrices, one of which is the square coefficients matrix of the barycentric functions, expresses both the unknown and data functions. The singular kernel is interpolated twice with to its two variables. Furthermore, the kernel is represented by five matrices: two of which are the monomial basis of the kernel's variables, and one matrix expresses the kernel's values at the two sets of nodes for the kernel's variables. Without using the collocation approach, we get the matrix of the unknown coefficients by substituting the interpolated solution into the integral equation to obtain an equivalent algebraic system. By solving the algebraic system, we get the interpolated solution. We solved five examples, two for non-singular equations and three for weakly singular equations. By adopting lower interpolation degrees, the interpolated solutions for non-singular equations are shown to be equal to the exact ones, whereas, for singular equations, they are found to strongly converge to the exact ones and are superior to those obtained by other cited methods. This ensures the proposed method's uniqueness and high accuracy.

Collocation Methods for a Class of Singular Integro-Differential Equations

Collocation Methods for a Class of Singular Integro-Differential Equations

Modified decomposition solution of convex fins of different profile index with all power law dependent thermal properties

The present work investigates the non-linear mathematical model of one regular and two irregular convex fins where all thermo physical parameters are power law dependent. The extended length of regular convex fins varies parabolically along its length with a power exponent of ½ and two irregular convex fins with different power exponent ¼ and 9/10 respectively. The energy equations of one regular and two irregular convex fins are three different singular value equation but with a common non-linearity. The three different singular value equations are solved separately by the theory of modified Adomian decomposition method (MADM). The effects of power dependent thermal conductivity parameter, effect of conduction-convection parameter, effect of power exponent of heat transfer co-efficient, effect of power exponent of surface emissivity parameter, and effect of power exponent of heat generation number on the temperature distribution of three different convex fins are analyzed and compared separately.

Exact Solution to Axisymmetric Interface Crack Problem in Magneto-Piezo-Electric Transversely Isotropic Materials

Summary This seems to be the first exact closed-form solution to the problem of a penny-shaped interface crack, subjected to an axisymmetric normal and tangential loading. The crack is located at the boundary between two bonded magneto-piezo-electric transversely isotropic half-spaces, made of different materials. We use the combination of Green’s functions for two different half-spaces and Fourier transform. We derive first the governing equations, which are valid for a crack of arbitrary shape. The usual approach leads to five hypersingular integral equations, we arrive at four integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications or not noticed at all. In the particular case of axial symmetry, the problem is reduced to just one singular equation, for which an exact closed-form solution is known. We are not aware of any other publication with which our results can be compared.

Numerical approach for weakly singular equation with nonlinear delay

Numerical approach for weakly singular equation with nonlinear delay

Strong comparison principle for a p-Laplace equation involving singularity and its applications

We prove a strong comparison principle for radially decreasing solutions u,v∈C01,α(BR¯) of the singular equations −Δpu−λuδ=f(x) and −Δpv−λvδ=g(x) in BR, where 1<p≤2,δ∈(0,1) and λ>0. We assume that f and g are continuous radial functions with 0≤f≤g and f⁄≡g in BR. Also, a counterexample is provided where the strong comparison principle is violated when p>2. In addition, we prove a three solution theorem for p-Laplace equation as an application of strong comparison principle. This is illustrated with an example.

Scaling properties of a class of interfacial singular equations

This paper can be considered as an introductory review of scale invariance theories illustrated by the study of the equation ∂th=−∂x∂xh1−2ν+∂xxxh, where ν>1/2. The d−dimensionals version of this equation is proposed for ν≥1 to discuss the coarsening of growing interfaces that induce a mound-type structure without slope selection (Golubović, 1997). Firstly, the above equation is investigated in detail by using a dynamic scaling approach, thus allowing for obtaining a wide range of dynamic scaling functions (or pseudosimilarity solutions) which lend themselves to similarity properties. In addition, it is shown that these similarity solutions are spatial periodic solutions for any ν>1/2, confirming that the interfacial equation undergoes a perpetual coarsening process. The exponents β and α describing, respectively, the growth laws of the interfacial width and the mound lateral size are found to be exactly β=(1+ν)/4ν and α=1/4, for any ν>12. Our analytical contribution examines the scaling analysis in detail and exhibits the geometrical properties of the profile or scaling functions. Our finding coincides with the result previously presented by Golubović for 0<ν≤3/2.