Singular Initial Value Problems Research Articles

Strong convergence of the tamed Euler-Maruyama method for stochastic singular initial value problems with non-globally Lipschitz continuous coefficients

In our previous works [1] and [2], we delved into numerical methods for solving stochastic singular initial value problems (SSIVPs) that involve coefficients satisfying the global Lipschitz condition. The paper addresses the limitations of our previous work by introducing an explicit method, called the tamed Euler-Maruyama method, for numerically solving SSIVPs with non-globally Lipschitz continuous coefficients, which is both easy-to-implement and exceptionally efficient. We prove the existence and uniqueness theorem and the boundedness of the moments of the solution to SSIVPs under the non-globally Lipschitz condition. Moreover, we provide a sharp analysis of the strong convergence of the proposed method, along with the uniform boundedness of numerical solutions. We also apply our results to the stochastic singular Ginzburg-Landau system and provide numerical simulations to illustrate our theoretical findings.

A new pair of block techniques for direct integration of third-order singular IVPs

This paper proposes a new pair of block techniques (NPBT) for the direct solution of third-order singular initial-value problems (IVPs). The proposed method uses a polynomial and two intermediate points to approximate the theoretical solution of third-order singular IVPs, resulting in a reasonable approximation within the integration interval. The method's essential features, including stability and convergence order, are analyzed. The proposed NPBT method is improved by using an embedding-like strategy that allows it to be executed in a variable step size mode in order to gain better efficiency. The effectiveness of the proposed method is assessed using various model problems. The approximate solution provided by the proposed NPBT method is more accurate than that of the existing methods utilized for comparison. This efficient solution positions NPBT as a good numerical method for integrating third-order singular IVP models in the fields of applied sciences and engineering.

Polytropic Dynamical Systems with Time Singularity

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.

Explicit numerical methods for solving singular initial value problems for systems of second-order nonlinear ODEs

Explicit numerical methods for solving singular initial value problems for systems of second-order nonlinear ODEs

Multi-derivative hybrid block methods for singular initial value problems with application

This study presents a novel class of multi-derivative integrating techniques designed to effectively address singular initial value problems. This study not only delves into the application of these methods but also explores their potential in solving parabolic partial differential equations. We introduce the underlying mathematical framework of the multi-derivative method, derived from power series polynomials. This approach encompasses the strategic incorporation of collocation and interpolation techniques, utilizing a combination of hybrid and non-hybrid points within the problem domain. This multi-derivatives method combines the advantages of derivative-based methods, which offer high accuracy, and finite difference methods, which provide simplicity and stability. By utilizing multiple derivative approximations at various points within the problem domain, the multi-derivative method effectively handles the singularities and captures the discountinuity of the solutions. Furthermore, our investigation includes comprehensive qualitative analysis, demonstrating the method's inherent stability, consistency, and convergence when applied to practical problems. The numerical results obtained through the application of the multi-derivative method attest to its reliability and efficiency, particularly in tackling singular and stiff problems.

On strong convergence of two numerical methods for singular initial value problems with multiplicative white noise

Mean-square convergence of the Milstein method and the Euler–Maruyama (EM) method are investigated for stochastic singular initial value problems (SIVPs) driven by multiplicative white noise. For the first-order model equation, the existence, uniqueness, and moment boundedness of the exact solution are developed under some appropriate assumptions. On this basis, it is proved that the Milstein method is of 1/2−ϵ order convergence in the mean-square sense, and the convergence order can be increased to 1−ϵ when the diffusion coefficient vanishes at the origin, where ϵ is an arbitrarily small positive number. It is significantly different from the first-order mean-square convergence of the Milstein method to solve the classical stochastic ordinary differential equations. Meanwhile, whether the diffusion coefficient vanishes at the origin or not, the mean-square convergence order of the EM method is always 1/2−ϵ. Furthermore, a second-order stochastic SIVP with multiplicative noise is considered by converting it to a first-order system. When solving the system, it is shown that the EM method has the same form as the Milstein method, and thus enjoys the convergence order of the Milstein method. Numerical examples are provided to verify our theoretical prediction.

The Solution of some Kinds of Higher Order Differential Equations by Adomian Decomposition Method

In this paper, we presented a new differential operator to solve singular initial value problems of higher order differential equations by utilizing Adomian decomposition method (ADM). The suggested approach is applicable to linear as well as non linear problems. Some exampls are used in testing and the obtained results demonstrate that the suggested strategy is effective.Keywords: Adomian decomposition method; singular initial value problems ; higher order diferential equations

On the numerical integration of singular initial and boundary value problems for generalised Lane–Emden and Thomas–Fermi equations

We propose a geometric approach for the numerical integration of singular initial and boundary value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane–Emden equations and the Thomas–Fermi equation.

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

Asymptotically Kasner-like singularities

abstract: We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form $$ {}^{(4)}g = -dt^2 + \sum_{i,j=1}^3 a_{ij}t^{2 p_{\max\{i,j\}}}\,{\rm d} x^i\,{\rm d} x^j $$ on $(0,T]_t\times\Bbb{T}^3_x$, where $a_{ij}(t,x)$ and $p_i(x)$ are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as $t\to 0^+$. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the ``singular hypersurface'' $\{t=0\}$. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-$t$ hypersurfaces.

A new adaptive nonlinear numerical method for singular and stiff differential problems

In this work, a new adaptive numerical method is proposed for solving nonlinear, singular, and stiff initial value problems often encountered in real life. Starting with a fixed step size, the new method’s performance can be significantly enhanced by introducing an adaptive step-size approach. The qualitative properties of the proposed method have been investigated to determine the efficiency and reliability of the method. The proposed method is of fifth-order accuracy, zero stable, L-stable, and consistent. In addition, the proposed method is convergent, and its stability properties are also shown through its Order Stars. Finally, numerical experiments are conducted to illustrate the performance of the method. The results obtained show that the proposed method compares favourably with existing methods.

Solving SIVPs of Lane–Emden–Fowler Type Using a Pair of Optimized Nyström Methods with a Variable Step Size

This research article introduces an efficient method for integrating Lane–Emden–Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nyström technique with a set of formulas applied at the initial step to circumvent the singularity at the beginning of the interval. The variable step-size formulation is implemented using an embedded-type approach, resulting in an efficient technique that outperforms its counterpart methods that used fixed step-size implementation. The numerical simulations confirm the better performance of the variable step-size implementation.

A NEW DIFFERENTIAL OPERATOR FOR SOLVING MULTI SINGULAR INITIAL VALUE PROBLEMS OF SECOND ORDER

Modified Adomian Decomposition Method (MADM) was used in this research to solve singular initial value problems in the second-order ordinary differential equation. In the beginning, we studied the genral equation, and we gave several illustrations for this equation. In addition, we studied some different cases of second order ordinary differential equations which we got from genral equation. From these cases, we got different types of equations such as multi singular initial value problems, like Bessel’s equation from half order and other equations as well. Using the examples under investigation, we conclude that the solution is converging towards the exact solution.

Singular Nonlinear Problems for Phase Trajectories of Some Self-Similar Solutions of Boundary Layer Equations: Correct Formulation, Analysis, and Calculations

We study a singular initial value problem for a nonlinear non-autonomous ordinary differential equation of the second order, defined on a semi-infinite interval and degenerating in the initial data for the phase variable. The problem arises in the dynamics of a viscous incompressible fluid as an auxiliary problem in the study of self-similar solutions of the boundary layer equations for a stream function with a zero pressure gradient (plane-parallel laminar flow in a mixing layer). It is also of independent mathematical interest. Using the previously obtained results on singular nonlinear Cauchy problems and parametric exponential Lyapunov series, a correct formulation and a complete mathematical analysis of this singular initial value problem are given. Restrictions on the “self-similarity parameter” for the global existence of solutions are formulated, two-sided estimates of solutions, and results of calculations of the phase trajectories of solutions for different values of this parameter are given.

Renormalization Group Method for Singular Perturbation Initial Value Problems with Delays

Renormalization Group Method for Singular Perturbation Initial Value Problems with Delays

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.

Uniqueness of Local, Analytic Solutions to Singular ODEs

We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem to an equilibrium problem of a regular ODE. Then, we apply classical invariant manifold theory. We demonstrate that the class of ODEs under consideration captures models which describe the shape of axially symmetric surfaces which are closed on one side. Our main result guarantees smoothness at the tip of the surface.

On numerical methods to second-order singular initial value problems with additive white noise

In this work, we investigate the strong convergence of the Euler–Maruyama method for second-order stochastic singular initial value problems with additive white noise. The singularity at the origin brings a big challenge that the classical framework for stochastic differential equations and numerical schemes cannot work. By converting the problem to a first-order stochastic singular differential system, the existence and uniqueness of the exact solution is studied. Moreover, under some suitable assumptions, it is proved that the Euler–Maruyama method is of (1/2−ɛ) order convergence in mean-square sense, where ɛ is an arbitrarily small positive number, which is different from the consensus that the Euler–Maruyama method is convergent with first order in strong sense when solving stochastic differential equations with additive white noise. While, it is found that if the diffusion coefficient vanishes at the origin, the convergence order in mean-square sense will be increased to 1−ɛ. Our theoretical findings are well verified by numerical examples.

To solving second order ordinary differential equations with singular points by Adomian decomposition method

The Adomin Decomposition Method (ADM) is used to solve differential equations, so in this study we used (ADM) to solve second order ordinary differential equations with singular initial value problem then, the equation was given a generalization.

Crystalline flow starting from a general polygon

<p style='text-indent:20px;'>This paper solves a singular initial value problem for a system of ordinary differential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.</p>