Nonlinear Singular Problems Research Articles

Existence of Positive Solutions for Singular Difference Equations with Nonlinear Boundary Conditions

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by f(u)−auα with α>0, exhibits a singularity at u=0 and tends towards −∞ as u approaches 0+. By constructing some suitable auxiliary problems, the difficulty that arises from the singularity and semipositone of nonlinearity and the lack of a maximum principle is overcome. Subsequently, employing the Krasnosel’skii fixed-point theorem, we determine the parameter range that ensures the existence of at least one positive solution and the emergence of at least two positive solutions. Furthermore, based on our existence results, one can obtain the symmetry of the solutions after adding some symmetric conditions on the given functions by using a standard argument.

An efficient recursive technique with Padé approximation for a kind of Lane–Emden type equations emerging in various physical phenomena

The study numerically examined a class of nonlinear singular differential problems known as the Lane–Emden differential equation, which emerges in numerous real-world situations. The primary goal of this work is to formulate a computationally efficient iterative technique for solving the nonlinear Lane–Emden initial value problems. The proposed approach is a hybrid of the homotopy perturbation method and the Padé approximation. The nonlinear singular Lane–Emden initial value problem (SLEIVP) is transformed into an equivalent recursive integral employing the Picard’s approach. To resolve the singularity and nonlinearity, the recursive integral equation is transformed into a system of integral equations by using the homotopy notion. Furthermore, to enhance the convergence rate of the technique, Padé approximation is taken into account. The convergence analysis for the proposed approach is also conducted. The present technique is tested on SLEIVPs and numerical findings are compared with the existing techniques, to demonstrate the accuracy, effectiveness and ease of use.

Correction: Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets

Correction: Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets

Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets

Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets

Existence and regularity results for some nonlinear singular parabolic problems with absorption terms

Existence and regularity results for some nonlinear singular parabolic problems with absorption terms

Singular elliptic problem involving a Hardy potential and lower order term

We consider the following non-linear singular elliptic problem (1) − div ( M ( x ) | ∇ u | p − 2 ∇ u ) + b | u | r − 2 u = a u p − 1 | x | p + f u γ in Ω u > 0 in Ω u = 0 on ∂ Ω , where 1 < p < N; Ω ⊂ R N is a bounded regular domain containing the origin and 0 < γ < 1, a ⩾ 0 , b > 0 , 0 ⩽ f ∈ L m ( Ω ) and 1 < m < N p . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.

Solution of linear and nonlinear singular value problems using operational matrix of integration of Taylor wavelets

ABSTRACT In this paper, we present an effective method under Taylor wavelets and collocation technique to find an approximate solution of linear and non-linear second-order singular value differential equations. Application of this method on various problems confirms the efficiency, easy applicability, and rapid computation. The obtained solution is compared with some other existing numerical solutions. For many problems, this method gives a solution same as the exact solution of the problem which confirms the effectiveness and accuracy of this method. Additionally, we include graphs and figures to demonstrate that the Taylor wavelet method offers better accuracy for a number of problems. MATLAB software is used to process relative data and execute calculations.

Positive solutions for a class of singular (p, q)-equations

Abstract We consider a nonlinear singular Dirichlet problem driven by the ( p , q ) \left(p,q) -Laplacian and a reaction where the singular term u − η {u}^{-\eta } is multiplied by a strictly positive Carathéodory function f ( z , u ) f\left(z,u) . By using a topological approach, based on the Leray-Schauder alternative principle, we show the existence of a smooth positive 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.

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

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

Positive solutions for nonlinear singular problems with sign-changing nonlinearities

Positive solutions for nonlinear singular problems with sign-changing nonlinearities

Investigation of Nonlinear Vibrational Analysis of Circular Sector Oscillator by Using Cascade Learning

This paper analyzed the model of swinging oscillation of a solid circular sector arising in hydrodynamical machines, electrical engineering, heat transfer applications, and civil engineering. Nonlinear differential equations govern the mathematical model for frequency oscillation of the system. Furthermore, a computational strength of Cascade neural networks (CNNs) is utilized with backpropagated Levenberg–Marquardt (BLM) algorithm to study the oscillations in angular displacement θ , velocity θ ′ , and acceleration θ ″ . A data set for the supervised learning of the CNN-BLM algorithm for different angles α and radius R are generated by Runge–Kutta (RK-4) method. The BLM algorithm further interprets the dataset with log-sigmoid as an activation function for the solutions’ validation, testing, and training. The results obtained by the design scheme are compared with Akbari–Ganji’s (AG) method. The rapid convergence and quality of the solutions are validated through performance indicators such as mean absolute deviations (MAD), root means square error, and error in Nash–Sutcliffe efficiency (ENSE). The statistics demonstrate the design scheme’s applicability and efficiency to highly singular nonlinear problems.

Optimal Control of Investment in a Collective Pension Insurance Model: Study of Singular Nonlinear Problems for Integro-Differential Equations

Optimal Control of Investment in a Collective Pension Insurance Model: Study of Singular Nonlinear Problems for Integro-Differential Equations

Existence of solution for a singular fractional boundary value problem of Kirchhoff type

In this work, we investigate the existence of solution for some nonlinear singular problem of Kirchhoff type involving Riemann-Liouville Fractional Derivative and the p-Laplacian operator. The main tools are based on the variational method, precisely, we use the minimisation of the corresponding functional in a suitable fractional spaces. Our main result significantly complement and improves the previous ones due to [6] and [31].

Global multiplicity of solutions for a modified elliptic problem with singular terms * *C A Santos and J Zhou were supported by CAPES/Brazil Proc. No 2788/2015-02; Minbo Yang is the corresponding author who was partially supported by NSFC(11971436, 12011530199) and ZJNSF(LD19A010001).

We establish global multiplicity results of solutions for the singular nonlinear problem where is a smooth bounded domain; N ⩾ 3; a, b are bounded continuous functions, 0 < α < 1 < β ⩽ 22* − 1 and λ > 0 is a real parameter. We first prove a comparison principle to prove the existence of a minimal solution by the method of sub and super solutions and then we also obtain the second solution by critical point theory.

Singular Nonlinear Problems for Self-Similar Solutions of Boundary-Layer Equations with Zero Pressure Gradient: Analysis and Numerical Solution

Singular Nonlinear Problems for Self-Similar Solutions of Boundary-Layer Equations with Zero Pressure Gradient: Analysis and Numerical Solution

Nonlinear singular problems with convection

In this paper we study the existence of a positive solution for a nonlinear Dirichlet problem driven by the p -Laplacian and a reaction which has the competing effects of a singular term and of a convection (gradient dependent) term. Using the “frozen” variable method and eventually the Leray-Schauder Alternative Theorem, we show that the problem has a positive smooth solution. We mention that on the gradient dependent perturbation x ↦ f ( z , x , y ) , we do not impose any bilateral growth condition.

An anti-periodic singular fractional differential problem of Lane-Emden type

In this paper, we use both Riemann-Liouville integral and Caputo derivative to investigate a new nonlinear singular differential problem of Lane and Emden type. We note that for the studied singular problem, we will be concerned with some anti periodic conditions. So, we prove a first uniqueness result by application of Banach principle of contraction. Then, we prove a “second” existence result by application of Schaefer theorem. Two illustrative examples are discussed in details to show the applicability of the obtained results.

Semi-decoupling hybrid asymptotic and augmented finite volume method for nonlinear singular interface problems

An accurate semi-decoupling numerical method has been proposed for nonlinear singular differential equation with interface, which combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to decouple the singular interface problem and get high order accurate numerical solution. Key to our proposed new method is introducing three augmented variables involving the interface and the singularity, and reconstructing the representative of the solution as the Puiseux series expansions on the interface to decouple original problem as two standard nonlinear singular problems with the second jump condition. In this way, the augmented variables related with semi-analytic solutions near the interface and numerical solutions can be simultaneously solved in the remaining interval on both sides of the interface. It demonstrates that our method does not take more heavier works for handling jump conditions like other methods, and is independent of the interface and jump ratio. A rigorous error estimate for the solution of nonlinear singular differential equation with interface and augmented variables is obtained. Numerical experiments for those singular differential equations with interface confirm the theoretical analysis and accuracy of the new approach. In particular, an interesting example with blow-up coefficient at singular point shows that our approach can be extended to numerically solve strongly singular interface problem.

A hybrid asymptotic and augmented compact finite volume method for nonlinear singular two point boundary value problems

An accurate and efficient numerical method is proposed for nonlinear singular two point boundary value problems. The scheme combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to get high order accurate solution for nonlinear singular problems. The key of the new method is to express the solution as the Puiseux series expansion on a small subinterval involving the singular point. The expansion contains an undetermined parameter which is an augmented variable in the numerical method. In this way, a nonlinear system is constructed in other subinterval and the parameter related with the semi-analytic solution near the singular point and the numerical solution can be simultaneously obtained. A rigorous error estimate for the solution of the singular differential equation is conducted and fourth order accuracy is obtained. Numerical examples confirm the theoretical analysis and efficiency of the new method.