On exact solutions of fractional differential-difference equations with Ψ-Riemann–Liouville derivative

The aim of this work is to modify the invariant subspace method (ISM) in order to obtain closed form solutions of fractional differential-difference equations with Ψ-Riemann–Liouville (Ψ-RL) fractional derivative for first time. We have investigated the cases of two-dimensional and the three-dimensional invariant subspaces (ISs) in the suggested scheme. Using the modified ISM, new exact generalized solutions for the general fractional mKdV Lattice equation and the fractional Volterra lattice system are obtained. Compared with similar solution techniques in literature, the presented solution scheme is highly efficient and is capable to find new general exact solutions which cannot be attained by other methods.

Impact-Angle and Terminal-Maneuvering-Acceleration Constrained Guidance against Maneuvering Target

A new, highly constrained guidance law is proposed against a maneuvering target while satisfying both impact angle and terminal acceleration constraints. Here, the impact angle constraint is addressed by solving an optimal guidance problem in which the target’s maneuvering acceleration is time-varying. To deal with the terminal acceleration constraint, the closed-form solutions of the new guidance are needed. Thus, a novel engagement system based on the guidance considering the target maneuvers is put forward by choosing two angles associated with the relative velocity vector and line of sight (LOS) as the state variables, and then the system is linearized using small angle assumptions, which yields a special linear time-varying (LTV) system that can be solved analytically by the spectral-decomposition-based method. For the general case where the closing speed, which is the speed of approach of the missile and target, is allowed to change with time arbitrarily, the solutions obtained are semi-analytical. In particular, when the closing speed changes linearly with time, the completely closed-form solutions are derived successfully. By analyzing the generalized solutions, the stability domain of the guidance coefficients is obtained, in which the maneuvering acceleration of the missile can converge to zero finally. Here, the key to investigating the stability domain is to find the limits of some complicated integral terms of the generalized solutions by skillfully using the squeeze theorem. The advantages of the proposed guidance are demonstrated by conducting trajectory simulations.

Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.

Existence and uniqueness of generalized solutions to hyperbolic systems with linear fluxes and stiff sources

Motivated by the modeling of boiling two-phase flows, we study systems of balance laws with a source term defined as a discontinuous function of the unknown. Due to this discontinuous source term, the classical theory of partial differential equations (PDEs) is not sufficient here. Restricting to a simpler system with linear fluxes, a notion of generalized solution is developed. An important point in the construction of a solution is that the curve along which the source jumps, which we call the boiling curve, must never be tangent to the characteristics. This leads to exhibit sufficient conditions which ensure the existence and uniqueness of a solution in two different situations: first when the initial data is smooth and such that the boiling curve is either overcharacteristic or subcharacteristic; then with discontinuous initial data in the case of Riemann problems. A numerical illustration is given in this last case.

A State-of-the-Art Review on Integral Transform Technique in Laser–Material Interaction: Fourier and Non-Fourier Heat Equations

Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties. Such wonderful features have enabled heat equations in various fields, including laser and electron beam processing. The integral transform technique (ITT) is a powerful general-purpose semi-analytical/numerical method that transforms partial differential equations into a coupled system of ordinary differential equations. Under this category, Fourier and non-Fourier heat equations can be implemented on both equilibrium and non-equilibrium thermo-dynamical processes, including a wide range of processes such as the Two-Temperature Model, ultra-fast laser irradiation, and biological processes. This review article focuses on heat equation models, including Fourier and non-Fourier heat equations. A comparison between Fourier and non-Fourier heat equations and their generalized solutions have been discussed. Various components of heat equations and their implementation in multiple processes have been illustrated. Besides, literature has been collected based on ITT implementation in various materials. Furthermore, a future outlook has been provided for Fourier and non-Fourier heat equations. It was found that the Fourier heat equation is simple to use but involves infinite speed heat propagation in comparison to the non-Fourier heat equation and can be linked with the Two-Temperature Model in a natural way. On the other hand, the non-Fourier heat equation is complex and involves various unknowns compared to the Fourier heat equation. Fourier and Non-Fourier heat equations have proved their reliability in the case of laser–metallic materials, electron beam–biological and –inorganic materials, laser–semiconducting materials, and laser–graphene material interactions. It has been identified that the material properties, electron–phonon relaxation time, and Eigen Values play an essential role in defining the precise results of Fourier and non-Fourier heat equations. In the case of laser–graphene interaction, a restriction has been identified from ITT. When computations are carried out for attosecond pulse durations, the laser wavelength approaches the nucleus-first electron separation distance, resulting in meaningless results.

METHOD OF GENERALIZED FUNCTIONS IN PLANE BOUNDARY VALUE PROBLEMS OF UNCOUPLED THERMOELASTODYNAMICS

Nonstationary boundary value problems of uncoupled thermoelasticity are considered. A method of boundary integral equations in the initial space-time has been developed for solving boundary value problems of thermoelasticity by plane deformation. According to generalized functions method the generalized solutions of boundary value problems are constructed and their regular integral representations are obtained. These solutions allow, using known boundary values and initial conditions (displacements, temperature, stresses and heat flux), to determine the thermally stressed state of the medium under the influence of various forces and thermal loads. Resolving singular boundary integral equations are constructed to determine the unknown boundary functions.

Elastodynamic Solutions of a Finite Soil Layer under Interior Distributed Actions

Through a method of displacement potentials, Fourier series, and Hankel integral transformation, the generalized solutions of an elastic layer resting on a rigid base under arbitrary, distributed, buried, and time-harmonic loads are developed in this study. With the proposed solution, the specific results for two kinds of uniform distributions as a kind of fundamental solutions in the interaction analysis of media and inclusions by the method of boundary integral equations are included as illustrations. Finally, numerical examples involving surface and buried patch loads are presented to validate the solutions and examine the effects of the thickness of the elastic layer. The results show that the proposed solution can cover the classical half-space solution by taking enough large thickness of the elastic layer (e.g., the ratio of the layer thickness beneath the load to the load radius ≥ 50 ) and the surface load solution by setting the load depth to zero; the underlying rigid base has significant and complex influence on the dynamic response of the thin layer due to wave reflections, which needs to be considered in the design and practice of related engineering.

APPLICATION OF THE FICTITIOUS DOMAIN METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS

The article shows the ways of applying the method of fictitious domains in solving problems for ordinary differential equations. In the introduction, a small review of the literature on this method, as well as methods for the numerical solution of these problems, is made. The problem statement for the method of fictitious domains for ordinary differential equations is considered. Further, the inequality of estimates was shown. The solution of the auxiliary problem approximates the solution of the original problem with a certain accuracy. The inequality of estimates is obtained in the class of generalized solutions. For the purpose of visual application of the fictitious domain method in problems, a boundary value problem for a one-dimensional nonlinear ordinary differential equation is considered. The problem was written in the form of a difference scheme and led to a solution using the sweep method. In the numerical solution of the problem, numerical calculations were carried out for various values of the parameter included in the auxiliary problem, based on the method of fictitious domains. The numbers of iterations, execution time, and graphs of these calculations are presented and analyzed.