Singular Differential Equations Research Articles

Singular dissipative third-order operator and its characteristic function

In this work, we describe well-defined dissipative boundary conditions related with a singular third-order differential equation in lim-3 case at singular point. Using the characteristic function of the corresponding dissipative operator we introduce a completeness theorem.

Crank–Nicolson/Galerkin spectral method for solving two-dimensional time-space distributed-order weakly singular integro-partial differential equation

The fractional PDEs based upon the distributed-order fractional derivative have several applications in physics. The two-dimensional time-space distributed-order weakly singular integro-partial differential model is investigated by a combination of finite difference and Galerkin spectral methods. A second-order finite difference formula is employed to approximate the temporal variable. In this stage, the stability and convergence of the semi-discrete scheme are proved. Then, the Galerkin spectral method based on the modified Jacobi polynomials is applied to discrete the space variable. Also, in this step, the error estimate of the full-discrete scheme is studied. Finally, two test problems have been presented to confirm the theoretical results.

A finite-difference procedure to solve weakly singular integro partial differential equation with space-time fractional derivatives

The main aim of the current paper is to propose an efficient numerical technique for solving space-time fractional partial weakly singular integro-differential equation. The temporal variable is based on the Riemann–Liouville fractional derivative and the spatial direction is based on the Riesz fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order $${{\mathcal {O}}}(\tau ^{\frac{3}{2}})$$. Also, the space variable is discretized using a finite difference scheme with second-order accuracy. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency, applicability and simplicity of the proposed technique.

Optimal successive complementary expansion for singular differential equations

In this article, we consider a singular ordinary differential equation of a two‐point boundary value problem in an asymptotic sense. We discuss the method of successive complementary expansion (SCEM) with asymptotics beyond all orders approach thinking and also scrutinise the solutions that already exist in the literature along with their advantages and drawbacks. In particular, we present an approach using techniques in exponential asymptotics that extends the SCEM for singular differential equations well beyond the leading‐order terms and consider the interaction of small parameter in the solutions. We optimally truncate the divergent outer solution and do not eliminate the growing exponentials. Through this analysis, we show that the extended method exhibits more information, such as the effects of exponential smallness and Stokes lines, than the regular SCEM that cannot reveal the information from such singular differential equations.

Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform $L_{\infty}$ bound, and $W_2^{1,1}$-energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient.

Radon measure solutions for steady compressible Euler equations of hypersonic-limit conical flows and Newton's sine-squared law

We formulate a mathematical problem on hypersonic-limit of three-dimensional steady uniform non-isentropic compressible Euler flows of polytropic gases passing a straight cone with arbitrary cross-section and attacking angle, which is to study Radon measure solutions of a nonlinear hyperbolic system of conservation laws on the unit 2-sphere. The construction of a measure solution with density containing Dirac measures supported on the surface of the cone is reduced to find a regular periodic solution of highly nonlinear and singular ordinary differential equations (ODE). For a circular cone with zero attacking angle, we then proved the Newton's sine-squared law by obtaining such a measure solution. This provides a mathematical foundation for the Newton's theory of pressure distribution on three-dimensional bodies in hypersonic flows.

Langevin Equation Involving Three Fractional Orders

In this paper, the existence and uniqueness of initial value problems for nonlinear Langevin equation involving three fractional orders are discussed. We use a new norm that is convenient for the fractional and singular differential equations. This norm and the contraction mapping principles are the main tools for investigating the existence and uniqueness of the desired issue. The fractional derivatives are described in Caputo sense.

Multiple positive solutions to singular fractional differential equations with integral boundary conditions involving p-q-order derivatives

In this paper, we investigate the existence for a class of higher-order fractional differential equations with integral boundary value conditions involving p-q-order derivatives. As an application of the height functions on some special bounded sets, we obtain the existence of two positive solutions by means of the Leray–Schauder nonlinear alternative and cone expansion and cone compression fixed point theory. The nonlinearity may take negative infinity, and there may appear a singular phenomenon on both time and space variables.

Статическая термоустойчивость пологой геометрически нерегулярной оболочки из ортотропного термочувствительного материала

Рассматривается пологая ортотропная геометрически нерегулярная оболочка (ГНО) постоянного кручения, термомеханические параметры которой линейно зависят от температуры. При достижении температуры определенного значения происходит скачкообразно смена формы равновесия, что вызывает изменение первоначальной геометрии оболочки. Эти значения температур называют критическими. Для практики значительный интерес представляют соотношения, связывающие критические температуры с геометрическими и термомеханическими параметрами ГНО. Решение задач статической термоустойчивости ГНО, как правило, начинается с анализа их исходного безмоментного состояния. Тангенциальные усилия, вызванные нагревом оболочки, определяются как решения системы сингулярных дифференциальных уравнений безмоментной термоупругости. Эти усилия содержатся в формах Брайена или Рейсснера в уравнениях статической термоустойчивости, и от их структуры существенно зависит успех в дальнейшем решении задачи. В работе решение сингулярной безмоментной термоупругости найдено в элементарных функциях. Уравнения моментной термоупругости, записанные в компонентах поля перемещений, методом функции перемещений сведены к одному сингулярному дифференциальному уравнению в частных производных восьмого порядка. Решение записывается в виде двойного тригонометрического ряда, коэффициенты которого на основании процедуры Галeркина определяются как решения линейной однородной алгебраической системы уравнений. Из равенства нулю определителя этой системы (в первом приближении) получено алгебраическое уравнение пятой степени для относительной критической температуры, наименьший положительный действительный корень которого и есть искомая температура. Проводится количественный анализ влияния геометрических и термомеханических параметров ГНО на величины критических температур.

Discrete fractional solutions to the effective mass Schrödinger equation by mean of Nabla operator

In the current article, we investigate the second order singular differential equation namely the effective mass Schrödinger equation by means of the fractional nabla operator. We apply some classical transformations in order to reduce the governing equation, and also restrict the difference parameters involved in order to find them values. In order to achieve these important results, certain tools such as the Leibniz rule, the index law, the shift operator, and the power rule are provided in view of the discrete fractional calculus. We use all these mentioned data for two representations of the given model for homogeneous and non-homogeneous instances. The main advantage of the fractional nabla operator is to apply the singular differential equations and transform them into a fractional order model. As a result, we produce some new exact fractional solutions to the present model for a given potential.

Positive periodic solution for third-order singular neutral differential equation with time-dependent delay

In this paper, we investigate a class of third-order singular neutral differential equations with time-dependent delay. Applying Krasnoselskiios fixed point theorem, we prove the existence results of a positive periodic solution for this neutral equation.

Consistent Investment of Sophisticated Rank-Dependent Utility Agents in Continuous Time

We study portfolio selection in a complete continuous-time market where the preference is dictated by the rank-dependent utility. As such a model is inherently time inconsistent due to the underlying probability weighting, we study the investment behavior of sophisticated consistent planners who seek (subgame perfect) intra-personal equilibrium strategies. We provide sufficient conditions under which an equilibrium strategy is a replicating portfolio of a final wealth. We derive this final wealth profile explicitly, which turns out to be in the same form as in the classical Merton model with the market price of risk process properly scaled by a deterministic function in time. We present this scaling function explicitly through the solution to a highly nonlinear and singular ordinary differential equation, whose existence of solutions is established. Finally, we give a necessary and sufficient condition for the scaling function to be smaller than 1 corresponding to an effective reduction in risk premium due to probability weighting.

A Bismut–Elworthy–Li formula for singular SDEs driven by a fractional Brownian motion and applications to rough volatility modeling

In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multidimensional fractional Brownian motion with Hurst parameter H <1/2. Singular here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of the δ price sensitivity of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDE's. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed local time variational calculus.

Optimal Type-3 Fuzzy System for Solving Singular Multi-Pantograph Equations

In this study a new machine learning technique is presented to solve singular multi-pantograph differential equations (SMDEs). A new optimized type-3 fuzzy logic system (T3-FLS) by unscented Kalman filter (UKF) is proposed for solution estimation. The convergence and stability of presented algorithm are ensured by the suggested Lyapunov analysis. By two SMDEs the effectiveness and applicability of the suggested method is demonstrated. The statistical analysis show that the suggested method results in accurate and robust performance and the estimated solution is well converged to the exact solution. The proposed algorithm is simple and can be applied on various SMDEs with variable coefficients.

具非局部边界条件的奇异分数阶微分方程正解的存在性和唯一性

具非局部边界条件的奇异分数阶微分方程正解的存在性和唯一性

On high-order generalized neutral singular differential equations with time-dependent deviating argument

On high-order generalized neutral singular differential equations with time-dependent deviating argument

The general solution for singular equations of (n+1) order using Adomian decomposition method

This article proposes a general and efficient modification of Adomain decomposition method (ADM) to obtain the general solutions for (n+1)- order nonlinear singular differential equations with different boundary conditions. This technique is proposed to overcome the singular behaviour of this type of problems. We study several nonlinear problems which will illustrate the efficiency of using that developed technique of the given method to clearly confirm the effectiveness and accuracy of ADM. In addition, we compare the numerical results with the exact solution to explain the rapid convergence of the approximation series as the solution.

Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation

The bounded traveling wave solutions of the Zakharov-Rubenchik equation are investigated by using the method of dynamical system theorems in this paper. After suitable transformations we find that the traveling wave equations of the Zakharov-Rubenchik equation are fully determined by a second-order singular ordinary differential equation (ODE) with three real coefficients which can be arbitrary constants. We derive abundant exact bounded periodic and solitary wave solutions of the Zakharov-Rubenchik equation via studying the bifurcations and exact solutions of the derived ODE.

Some modified Adams‐Bashforth methods based upon the weighted Hermite quadrature rules

In this paper, we first introduce a modification of linear multistep methods, which contain, in particular, the modified Adams‐Bashforth methods for solving initial‐value problems. The improved method is achieved by applying the Hermite quadrature rule instead of the Newton‐Cotes quadrature formulas with equidistant nodes. The related coefficients of the method are then represented explicitly, the local error is given, and the order of the method is determined. If a numerical method is consistent and stable, then it is necessarily convergent. Moreover, a weighted type of the new method is introduced and proposed for solving a special case of the Cauchy problem for singular differential equations. Finally, several numerical examples and graphical representations are also given and compared.

Existence results for a semipositone singular fractional differential equation

This paper provides sufficient conditions to guarantee the existence and multiplicity of positive solutions for a singular semipositone fractional equation, subject to integral boundary value condition and parametric dependence $\mu $. Our approach relies on Krasnoselskii's fixed point theorem. Some examples are presented to illustrate our results.