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In this paper we present the inverse problem of determining a time dependent heat source in a two-dimensional heat equation accompanied with Dirichlet–Neumann–Wentzell boundary conditions. The model is of significant practical importance in applications where the time dependent internal source is to be controlled from total energy measurements in the case when the boundary is defined with partially absorbing (Dirichlet), partially reflecting (Neumann), and partially having the capacity for storing heat (Wentzell) conditions. The considered spatial domain is a rectangle which allows the application of a Fourier series analysis to the solution of the problem whose auxiliary spectral problem is a novel eigenproblem. This eigenvalue problem is defined using the two-dimensional Laplace operator where the spectral parameter resides in one of the accompanying boundary conditions. Under this setting we study the well-posedness of both the direct and inverse problems. In addition, we use the results obtained on the well-posedness to establish the numerical solvability of all the problems considered in the paper. In order to achieve this, we make use of a Chebyshev spectral collocation method for the spatial discretisation and a backward differentiation formula for the temporal discretisation. All the numerical results are shown to be consistent with the theory and in good agreement with the analytical solutions. • Inverse problem of finding a transient heat source in a 2D heat equation is analysed. • Partially absorbing, reflecting, and heat storing capacity boundaries are considered. • Fourier series analysis is employed for the well-posedness of the inverse problem. • The adjunct eigenproblem is defined by the Laplace operator with a Wentzell condition. • A numerical method is derived to solve all the direct, inverse, and eigenproblems. [ABSTRACT FROM AUTHOR]

Scientists are aware of the importance of studying interdependent networks, as they represent real-world scenarios better than isolated networks, such as the interdependent communication and power networks. However, in real scenarios, higher-order interactions exist in these networks. Recent research has shown that higher-order interactions that cannot be reflected in simple networks (i.e., a network only has pairwise interaction) can be reflected through hypergraph. This paper proposes an interdependent hypergraph model that considers the dependencies of interlayer nodes. We studied the cascading failures in the hypergraph with different interlayer dependencies. Through theoretical analysis, we have determined the maximum attack intensity the network can withstand and how its robustness changes under different attack intensities. In completely interdependent hypergraph, the network becomes increasingly fragile as the attack intensity increases, and the final network size suddenly jumps to zero, indicating a first-order phase transition in the network. We conducted experiments on multiple artificially synthesized hypergraph. The experimental results indicate that our analytical solution is consistent with the simulated solution. These findings will help us better understand the impact of different topologies on network robustness in interdependent hypergraph, enabling us to take more effective measures to address potential network failures and attacks. • Two interdependent hypergraph models are proposed based on layer dependencies. • Consider the impact of topology on the robustness of interdependent hypergraphs. • The critical value of hypergraph collapse is determined through theoretical analysis. • There is a first-order phase transition for completely interdependent hypergraphs. [ABSTRACT FROM AUTHOR]

In this paper, we deal with numerical approximations for solving the Black–Scholes Partial Differential Equation (PDE) for European and American options pricing with local volatility. This PDE is well-known to be degenerated. Local volatility model is a model where the volatility depends locally of both stock price and time. In contrast to constant volatility or time-dependent volatility models for which analytical representations of the exact solution is known for European Call options, there is no analytical solution for local volatility. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called Fitted Two-Point Flux Approximation (FTPFA). The Fitted Two-Point Flux Approximation (FTPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. This combination yields our fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support theoretical results are provided. [ABSTRACT FROM AUTHOR]

Using the flow switchability theory of the discontinuous dynamical systems, the present paper is to develop mechanical complexity in a periodic-excited horizontal impact pair with dry friction. The impact pair studied models the motions of a single bolted connection which is vibrated in the plane perpendicular to the bolt axis. According to motion character, the phase space can be partitioned into several domains and boundaries, in which a continuous dynamical system is defined in each domain, and it possesses dynamical properties different from its adjacent subsystem, the boundaries have different properties and can fall into two kinds – displacement boundaries and velocity boundaries. In this paper, using G–functions defined on separation boundaries to study flow switching on corresponding boundaries, the analytical switching conditions on each boundary are developed: the sufficient and necessary conditions of occurrence and disappearance of sliding-stick motion and side-stick motion are obtained, the sufficient and necessary conditions of grazing motion appearing on velocity boundaries are also obtained, and the analytical conditions of appearance for grazing motion on displacement boundaries are preliminarily discussed. Thus it can be seen that dynamical behaviors of the horizontal impact pair with or without dry friction are essentially different, in particular flow switchability on displacement boundaries depend on whether the conditions of passable flows on velocity boundaries are satisfied. The numerical simulations are given to demonstrate the analytical results of two stick motions and grazing motions in such pair. More details of the motions for the object reaching the intersection point of displacement boundary and velocity boundary need to be considered further in the future. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel and simple Yee Finite-Difference Time-Domain (FDTD) scheme to solve numerically the nonlinear second-order thermoviscous Navier–Stokes and the Continuity equations. In their original form, these equations cannot be discretized by using the Yee's mesh, at least, easily. As it is known, the use of the Yee's mesh is recommended because it is optimized in order to obtain higher computational performance and remains at the core of many current acoustic FDTD softwares. In order to use the Yee's mesh, we propose to rewrite the aforementioned equations in a novel form. To achieve this, we will use the substitution corollary. This procedure is novel in the literature. Although the scheme can be extended to more than one dimension, in this paper, we will focus only on the one-dimensional solution because it can be validated with two analytical solutions to the Burgers equation: the Mendousse mono-frequency solution and the Lardner bi-frequency solution. Numerical solutions are excellently consistent with the analytical solution, which demonstrates the effectiveness of our formulation. • A novel FDTD scheme to solve the nonlinearsecond-order thermoviscous hydrodynamic model. • Use substitution corollary to rewrite the hydrodynamic equations. • The numerical results obtained show good agreement with the analytical solutions. • The scheme is competitive for its simplicity, accuracy and speed. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a solution strategy for an overdetermined ODE system that uses elliptic functions. In particular, we show how an apparently ill-posed ODE system can be made amenable to treatment by Jacobian elliptic functions by introducing additional free parameters and still obtain a solution that corresponds well to the physically expected behaviour. [Display omitted] [ABSTRACT FROM AUTHOR]

Characterization of nonlocality is an open problem in physics and engineering. This paper conducts a detailed investigation on two nonlocal models, namely, the fractional derivative model and the peridynamic model for anomalous diffusion. A generalized nonlocal model combining the advantages of the fractional derivative model and the peridynamic model, is introduced. In this paper, analytical solutions and the mean squared displacements of the two models are provided and discussed. In addition, their intrinsic relations and notable differences are investigated. Preliminary applications indicate that the peridynamic model can well capture an unremarkable transition from normal-diffusion to super-diffusion, while the fractional derivative model presents super-diffusion behaviors in the whole process. At last, a generalized nonlocal operator is proposed as a more general strategy to solve nonlocal problems. • The analytical solutions and mean squared displacement of the two models are obtained. • The relation and difference between fractional derivative model and peridynamic model are investigated. • The peridynamic model captures an unremarkable transition from normal-diffusion to super-diffusion. • A generalized nonlocal operator is presented to solve nonlocal problems. [ABSTRACT FROM AUTHOR]

In this paper, control engineering methods are presented with regard to modelling and simulations of signal propagation in time-fractional (TF) electrodynamics. That is, signal propagation is simulated in electromagnetic media described by Maxwell's equations with fractional-order constitutive relations in the time domain. We demonstrate that such equations in TF electrodynamics can be considered as a continuous-time system of state–space equations in control engineering. In particular, we derive continuous-time analytical solutions based on state-transition matrices for electromagnetic-wave propagation in the TF media. Then, discrete-time zero-order-hold equivalent models are developed and their analytical solutions are derived. It is demonstrated that the proposed models give the same results as other reference methods presented in the literature. However, due to the application of finite-difference scheme, they remain more flexible in terms of the number of simulation scenarios which can be tackled. • Control engineering methods are applied to time-fractional electrodynamics. • Fractional Maxwell's equations are a continuous-time system of state-space equations. • Continuous-time analytical solutions based on state-transition matrices are found. • Discrete-time zero-order-hold equivalent models are developed. • Proposed models give the same results as other reference methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider the analytical solutions of multi-term time-space coupling fractional delay partial differential equations for general mixed Robin boundary conditions on a finite domain. Firstly, integral transforms and method of reduction to integral equations are used to obtain the analytical solutions of multi-term time coupling delay fractional differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time-space coupling fractional delay partial differential equations to the multi-term time coupling fractional delay differential equations. By applying the obtained analytical solutions to the resulting multi-term time coupling fractional delay differential equations, the desired analytical solutions of the multi-term time-space coupling fractional delay partial differential equations are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability. [ABSTRACT FROM AUTHOR]

• An optional decoupling condition approach is proposed for deriving the lump-stripe solutions and lump-soliton solutions to the KPI equation. • New and more general solutions are derived, and there exists a link between the two kinds of interaction solutions. • The optional decoupling condition approach can be applied to a wide range of nonlinear evolution equations. As a kind of analytical exact solutions to the nonlinear evolution equations, the interaction solutions are of great value in the study of the interacting mechanism in nonlinear science. In this paper, an optional decoupling condition approach is proposed for deriving the lump-stripe solutions and lump-soliton solutions to the KPI equation. We derive new and more general solutions to the KPI equation and discuss the link between the two kinds of interaction solutions, which has not been reported before. The interaction solutions to the KPI equation are analyzed and simulated numerically, which show that all the interaction phenomena are completely inelastic. Although we are concerned on the KPI equation in this paper, this approach can be applied to a wide class of nonlinear evolution equations and lays out the framework of deriving the lump-multi-stripe and/or lump-multi-soliton solutions. [ABSTRACT FROM AUTHOR]

In this paper, a PDE-constrained optimization problem corresponding to the bi-Laplacian operator is investigated. The optimization problem arises when we search for the maximum load potential in a uniform elastic plate hinged at its boundary with a given total amount of force. It is established that there is a bang–bang solution to the optimization problem, and an optimality condition is derived. Based on the optimality condition, we obtain some topological and geometrical properties of the maximizer. Although analytical solutions for PDE-constrained problems are very rare, an analytical solution is determined if the plate is circular and its thickness has a particular form. In those cases, it is proved that the solution is unique. For plates of general shape, an efficient numerical algorithm is developed to solve the PDE-constrained optimization problem, and its convergence is studied. According to the results, one can determine the profile of the regions in a plate where the largest and smallest external force should be exerted to have the maximum load potential. According to the research results, the thickness of the plate plays a crucial role in determining the shape of those regions. Therefore, disregarding the impact of plate thickness in our studies for the sake of simplicity would be impractical and might result in inaccurate outcomes. By utilizing our findings, engineers and designers can determine the optimal external force pattern that yields the maximum load potential for plates. This, in turn, enables them to create plates that are more efficient and effective, tailored to meet their specific needs and requirements. • A PDE-constrained optimization problem corresponding to the bi-Laplacian operator is investigated. • The problem arises when we search for the maximum load potential in a elastic plate hinged at its boundary with a given total amount of force. • The existence of a solution and its topological properties are investigated. • An efficient numerical algorithm is developed to solve the PDE-constrained optimization problem, and its convergence is studied. • We can determine the profile of the regions in a plate where the largest and smallest external force should be exerted to have the maximum load potential. [ABSTRACT FROM AUTHOR]

In this paper, the analytical solution for 2D quasi-Laplace equation with piecewise constant conductivities is provided. The analytical solution can be expressed as an infinite power series with a group of intrinsic non-integer power exponents around each singular point. Combined with the given boundary conditions, the coefficient of each term can be determined by numerical methods. [ABSTRACT FROM AUTHOR]

This paper aims to reveal the nonlinear dynamic mechanism of asymmetric tristable energy harvesters and enhance the energy harvesting performance for different excitations. The general harmonic balance solutions of asymmetric tristable energy harvesters and the corresponding Jacobian matrix for estimating the stability of these analytical solutions are deduced. It is found that asymmetric tristable energy harvesters have seven analytical solutions (four stable solutions) under the appropriate excitation frequency and amplitude, which are verified by using basins of attraction. The influence mechanism of asymmetry of potential wells on energy harvesting performance is analyzed. The results show that the potential barrier is a key factor to determine the orbit height of high-energy interwell oscillations, which influences the amplitude of the output voltage. The influence essence of asymmetry on tristable energy harvesters is to change their potential wells and adjust the distribution of their potential energy. By changing the unstable equilibrium positions, a series of tristable energy harvesters with different dynamic characteristics can be obtained, which will benefit energy harvesting under various excitation conditions. [ABSTRACT FROM AUTHOR]

Using the Pontryagin’s maximum principle for a time-delayed optimal control problem results in a system of coupled two-point boundary-value problems (BVPs) involving both time-advance and time-delay arguments. The analytical solution of this advance-delay two-point BVP is extremely difficult, if not impossible. This paper provides a discrete general form of the numerical solution for the derived advance-delay system by applying a finite difference θ -method. This method is also implemented for the infinite-time horizon time-delayed optimal control problems by using a piecewise version of the θ -method. A matrix formulation and the error analysis of the suggested technique are provided. The new scheme is accurate, fast and very effective for the optimal control of linear and nonlinear time-delay systems. Various types of finite- and infinite-time horizon problems are included to demonstrate the accuracy, validity and applicability of the new technique. [ABSTRACT FROM AUTHOR]

In this paper, the Lie approximate symmetry analysis is applied to investigate new exact solutions of the singularly perturbed Boussinesq equation. The tanh-function method, is employed to solve some of the obtained reduced ordinary differential equations.We construct new analytical solutions with small parameter which is effectively obtained by the proposed method. [ABSTRACT FROM AUTHOR]

Combined operating condition usually refers to the control operating condition under which the highest and lowest water levels occur in a surge tank of hydropower station. In this paper, with the basic equations of surge analysis and nonlinear vibrational asymptotic method, analytical expressions of the worst superimposition time of surge waves in an upstream surge tank under four typical combined operating conditions (i.e. load-acceptance-then-rejection, successive load rejection, successive load acceptance and load-rejection-then-acceptance) are derived firstly. Then using these expressions, the analytical extreme water levels are determined. The analytical solutions are verified with numerical simulation results. Finally, the effect of the hydraulic resistance coefficient of surge tank on the control operating condition is investigated. The results indicate that: The analytical solutions for determining extreme water levels in surge tank under various combined operating conditions are accurate due to the good agreements between the analytical results and the numerical results. With the increase of the hydraulic resistance coefficient of surge tank, the control operating condition for the highest water level shifts from load-acceptance-then-rejection condition to successive load rejection condition, and the control operating condition for the lowest water level shifts from load-rejection-then-acceptance condition to successive load acceptance condition. [ABSTRACT FROM AUTHOR]

In this paper the celebrated second order Aw-Rascle nonhomogeneous system describing traffic flows is considered. Within the framework of the method of differential constraints, a suitable reduction procedure is developed for solving a class of Riemann problems which are of interest in traffic flows theory. In particular, for a given source term, we find the general solution of the Riemann problem in terms of shock waves, contact discontinuities and generalized rarefaction waves. The interaction between a shock wave and a generalized rarefaction wave is also studied and a related generalized Riemann problem is solved. Numerical results are in agreement with the exact analytical solution. [ABSTRACT FROM AUTHOR]

In this paper, two modified nonlinear saturation-based controllers and negative velocity feedback controllers are integrated to suppress the horizontal and vertical vibrations of a horizontally supported Jeffcott-rotor system at primary resonance excitation and the presence of 1:1 and 1:2 internal resonances. The second order approximations and the amplitude equations are obtained by applying the integral equation method to analyze the nonlinear behavior of this model. The stability of the steady-state solutions is determined based on the Floquet theory. The necessity of adding a negative velocity feedback to the main system is stated. The effects of different control parameters on the frequency-response curves and the force-response curves are investigated. Time histories of the whole system are included to show the response with and without control. It is shown that the main systems steady-state amplitude under the proposed control is reduced by about 99.8% and the negative velocity feedback can suppress the transient vibrations and prevent the main system having the large amplitude vibration. Finally, numerical simulations are provided to verify the effectiveness of the integral equation method. • Two modified saturation controllers are used for Jeffcott-rotor system. • A integral equation method is used for a four-degree-of-freedom system. • Analytical solutions based on our method agree with numerical simulations. • The saturation controller can avoid the phenomenon of double peaks. • The negative velocity feedback can suppress the transient vibrations. [ABSTRACT FROM AUTHOR]

This paper analyzes the behavior of nitrogen inside a closed cylinder with a mobile piston actuated by a nonlinear spring, a viscous damper and a control force which compensates partially the effect of the high gas pressure. Two helical heating coils are placed inside the cylinder and with their flow rates controlled by means of a linear controller of type proportional plus integral (PI) and another nonlinear control law to provide an approximately isothermal gas behavior. Based on the analysis of the mechanical and thermal subsystems and the control laws, a justification of the parameter values is presented and corroborated through analytical solutions that are obtained by approximate methods. To investigate the thermodynamic equilibrium conditions, the Soave–Redlich–Kwong and the Redlich–Kwong state equations are analyzed and compared, showing that the Soave–Redlich–Kwong equation is superior. The Melnikov method has been used to obtain sufficient conditions for chaotic behavior, which has also been investigated by means of the sensitive dependence, Lyapunov exponents and the power spectral density. The validity of the proposed model has been analyzed by using the compressibility chart for the nitrogen, and the analytical calculations have been verified through full numerical simulations. [ABSTRACT FROM AUTHOR]

With the rapid development of micro/nano-electro-mechanical systems (MEMS/NEMS), arch shaped resonators are becoming increasingly attractive for different applications. Nevertheless, the dynamics of bistable resonators is poorly understood, and the conditions for their appropriate performance are not well known. In this paper, an initially curved arch shaped MEMS resonator under combined DC and AC distributed electrostatic actuation is investigated. A reduced order model obtained from first mode Galerkin’s decomposition method is used for numerical and analytical investigations. We have used the Homotopy Analysis Method (HAM) in order to derive analytical solutions both for the amplitude and the temporal average of nonlinear vibrations. The obtained analytical expressions, validated by numerical simulations, are able to capture nonlinear behaviors including softening type vibrations and dynamic snap-through. We have used the derived analytical results in order to study the nonlinear vibrations of the bistable MEMS resonator. According to our results, in the bistability region the overall dynamic response of the system is described by means of a pair of softening type frequency responses merging together in a specific frequency band. The dynamic snap-through is then described by transitions between these two frequency responses, each of which corresponding to one of the stable configurations of the arch. This fresh insight to the problem can be used in the design and optimization of bistable resonators and determination of their sharp roll-off frequencies. A feature that can be implemented in the design of bandpass filters. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new numerical algorithm for solving linear and non linear fractional differential equations based on our newly constructed integer order and fractional order generalized hat functions operational matrices of integration. The linear and nonlinear fractional order differential equations are transformed into a system of algebraic equations by these matrices and these algebraic equations are solved through known computational methods. Further some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm. The results obtained, using the scheme presented here, are in full agreement with the analytical solutions and numerical results presented elsewhere. [Copyright &y& Elsevier]

• Use of deep neural networks to solve the time dependent Schrödinger equation to obtain good approximation to eigenenergies and eigenfunctions. • Modifications of the neural network to make it suitable for obtaining highly excited eigenfunctions, where no obvious nodal patterns exist, and quantum numbers are not well defined. • Application to typical molecular potentials. A deep neural network with a custom loss function is trained to efficiently generate ground and excited wavefunctions for different molecular potentials of interest. [Display omitted] In this paper, we design and use two Deep Learning models to generate the ground and excited wavefunctions of different Hamiltonians suitable for the study of the vibrations of molecular systems. The generated neural networks are trained with Hamiltonians that have analytical solutions and ask the network to generalize these solutions to more complex Hamiltonian functions. Since the Hamiltonians solutions used to train the neural networks are computationally cheap, the training process is fast and efficient. This approach allows to reproduce the excited vibrational wavefunctions of different molecular potentials. All methodologies used here are data-driven, therefore they do not assume any information about the underlying physical model of the system. This makes this approach versatile and can be used in the study of multiple systems in quantum chemistry. [ABSTRACT FROM AUTHOR]

• The proposed approach can quickly obtain the semi-analytical QP solution. • The ETMRM does not need multiple integration operations. • The semi-analytical solutions have high accuracy at a much lower computational cost. • The "adaptive zero-setting curve" is introduced to accelerate the convergence. • The proposed approach has a large convergence region and high computational efficiency. This paper presents a new semi-analytical approach, namely the enhanced time-domain minimum residual method to solve the semi-analytical quasi-periodic solution for nonlinear system. This approach does not require multiple numerical integration and can be applied to strongly nonlinear systems. The approach is mainly three-fold. Firstly, the semi-analytical solution of the nonlinear quasi-periodic system is expanded into a set of trigonometric series with unknown coefficients, i.e., x (t) ≈ ∑ k = 1 N [ b k cos (ω k t) + c k sin (ω k t) ]. Then, the problem of solving quasi-periodic solution can be expressed as: determining the coefficients of the trigonometric series so that the residual objective function R = M x ¨ + C x ˙ + Kx + N (x ¨ , x ˙ , x , t) − F (t) is minimum over a period, i.e., min a ∈ A ∫ 0 T R (a , t) T R (a , t) d t. Finally, the nonlinear minimum optimization problem is solved iteratively through the enhanced response sensitivity approach. Moreover, the "adaptive zero-setting curve" is introduced to accelerate the convergence. Two numerical examples, a van der Pol-Duffing system and a nonlinear energy sink system are adopted to verify the feasibility of the proposed approach. [ABSTRACT FROM AUTHOR]

• General closed-form solutions for linear constant-coefficient 2D fractional systems. • A model of fractional calculus using trivariate Mittag-Leffler kernels is constructed. • Explicit elementwise expressions are found for the 2 × 2 matrix Mittag-Leffler function. Linear systems of fractional differential equations have been studied from various points of view: applications to electric circuit theory, approximate solutions by numerical methods, and recently exact solutions by analytical methods. We discover here that, to obtain a fully closed-form solution in all cases, it is necessary to introduce a new type of Mittag-Leffler function involving triple series, and also to construct the associated fractional calculus operators, which we introduce and study in this paper. We then complete the rigorous analytical solutions for the aforesaid systems of fractional differential equations. As a consequence, comparing the solutions found here with the vector-matrix solutions known in the literature, we obtain explicit formulae for the elements of the 2 × 2 matrix Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

• European option pricing is considered under a regime switching FMLS model. • A two-step solution procedure is developed to solve the coupled FPDE system. • An exact and explicit pricing formula is presented based on Fourier cosine series. In this paper, we investigate the European option pricing problem under a regime switching FMLS (finite moment log-stable) model. This model is not only able to capture the main characteristics of asset returns, it also incorporates the effect of regime switching being consistent with market observations. However, option prices under this model are governed by a coupled FPDE (fractional partial differential equation) system, and the difficulty in seeking for analytical solution arises from the combination of the coupled system and the spatial-fractional derivative. To deal with this difficulty, we develop a two-step solution procedure; we firstly assume that the future information of the Markov chain is known, and we derive the conditional option price by analytically solving a time dependent FPDE, based on which an exact and explicit pricing formula for the unconditioned price is successfully worked out by using the Fourier cosine series expansion. It is also shown through the numerical experiments that it converges very rapidly and has potential to be applied in practice. [ABSTRACT FROM AUTHOR]

• Two efficient and accurate new Rosensweig models are proposed and compared. • One uses a phase field approach. The other one uses a modified level set approach. • They are used to investigate the ferrodroplet deformation and merging process immersed in a non-magnetic viscous medium under uniform magnetic fields. • The effect of different bond numbers and magnetic susceptibility on the ferrodroplet shape is investigated. • Furthermore, several other important aspects are illustrated, including the evolution of the flow field, comparison between the two models, the magnetic energy distribution, and the spurious flow near the interface. By using both a phase field approach and a modified level set approach, two multiphase numerical models are proposed and compared in this paper to investigate the ferrodroplet deformation and merging process in a non-magnetic viscous medium under the influence of uniform magnetic fields. The finite element method is utilized for the spatial discretization of both numerical models. The numerical results show excellent agreement with the analytical solutions in the simple axisymmetric setting. The effects of different magnetic bond numbers and magnetic susceptibility on the deformation of ferrodroplets are systematically investigated. The coalescence process, in which two small ferrodroplets merge into a single larger droplet under uniform magnetic fields, is also studied by using both the phase field approach and the modified level set approach. Moreover the attraction phenomenon between two ferrodroplets, which was previously discovered in numerical experiments, is observed in our numerical tests. By comparing with analytical solutions, our study demonstrates that the diffuse interface (phase field) approach performs better than the modified level set approach when there is large topological deformation of the ferrodroplet. Several other important aspects, including the evolution of the flow field, the magnetic energy distribution, the spurious flows near the interfaces, and conservation of mass in both approaches, are studied as well. [ABSTRACT FROM AUTHOR]

• Novel airfoil model with viscoelastic terms and random fluctuation is proposed. • Approximate analytical solutions are examined by the method of multiple scales. • Influences of the viscoelastic and noisy parameters are explored. • Bistability and stochastic jumps are observed. The purpose of this paper is to explore analytically the influences of random fluctuations on a two-degrees-of-freedom (TDOF) airfoil model with viscoelastic terms. To begin with, a convolution integral over an exponentially decaying kernel function is employed to establish a constitutive relation of the viscoelastic material. Then the corresponding TDOF airfoil model with viscoelastic terms and random excitations is introduced. Subsequently, a theoretical analysis for the proposed airfoil model is achieved through a multiple-scale method together with a perturbation technique. All of the obtained approximate analytical solutions are verified by numerical simulation results, and a good agreement is observed. Meanwhile, we also find that both high-amplitude and low-amplitude oscillations coexist within a certain range of the excitation frequency or amplitude, which is regarded as a bi-stable behavior. In addition, effects of the viscoelastic terms and the random excitations on the system responses are investigated in detail. We uncover that the viscoelastic terms have a considerable influence on the system dynamics, which can simultaneously affect the structural damping and stiffness of the airfoil system. More interestingly, stochastic jumps between high-amplitude and low-amplitude oscillations can be induced due to random fluctuations, which are further illustrated through time history and steady-state probability density function. The jumps are considered as a transition from one probable state to another or vice versa. These results indicate that the external random fluctuations have a remarkable influence on dynamics of the TDOF airfoil model with viscoelastic material property. [ABSTRACT FROM AUTHOR]

• A square discrete membrane low-amplitude dynamics may be described by acoustic vacuum equation without linear terms. • There are four nonlinear normal modes, three of which are unstable and may exchange energy with other ones. • There exists a regular regime of energy exchange between different domains of the membrane (clusters). • Analytical results are confirmed by numerical simulation data. • The possibility of usin the considered system as an effective nonlinear energy sink is supported by numerical simulations. In the presented paper we extend the results obtained earlier for unstretched string to the case of discrete square membrane. The asymptotic equations of motion are derived and their connection with the sonic vacuum problem is shown. The basic stationary solutions of asymptotic equations which are nonlinear normal modes (NNMs) are found analytically and the obtained results are confirmed by numerical integration of initial equations of motion. Within the framework of the four-particle model the non-stationary dynamics of the membrane is studied in terms of limiting phase trajectories (LPTs) and coherence domains. The analytical results are confirmed numerically by the integration of both asymptotic and initial equations of motion.The considered structure is supposed to be used as energy sink, such possibility is demonstrated by numerical simulation. [ABSTRACT FROM AUTHOR]

• For increasing order of convergence in time, we opted for anL2-1 σ formula. • For increasing order of convergence in space linear operator is implemented. • For increasing order of convergence in the fractional-order dimension, Simpson's 1/3rd rule is implemented. In this paper, we proposed a linearized second-order numerical technique for non-linear fourth-order distributed fractional sub-diffusion equation with time delay. Time fractional derivative is represented using Caputo derivative and further approximated using L 2 − 1 σ formula which gives second-order temporal convergence and compact difference operator is employed for spatial dimensions. The proposed method is unconditionally stable and convergent to the analytical solution with the order of convergence O (τ 2 + h 4 + (Δ α) 4) improvement in earlier work. Non-linear terms are linearized with the help of Taylor's series. At the last, we provided a few examples to show the efficiency of the compact difference scheme to support the theoretical results and also presented comparison with L 1-approximation of Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

• Utilizing the fractal geometry to develop instantaneous source function. • Analyzing the infinite reservoir and infinite slab reservoirs using the introduced source functions. • Developing novel models for complex real reservoirs with complex structures via different types of mathematical dimensions. This study provides basic instantaneous source functions for different source-reservoir geometries in infinite fractal reservoirs (FRs) and infinite slab FRs. The concept of fractal geometry (FG) has already been adopted to analyse and model well-reservoir systems with complex structures. All these complexities have been modelled through a partial differential equation called fractal diffusivity equation (FDE). In this paper, by use of the source/Green's function technique, we analytically solve the FDE for different boundary conditions, i.e., different source-reservoir geometries. The wellbore pressure response for horizontal wells in an infinite slab FR (synthetic example) is derived and analysed to illustrate the applications of the provided analytical source solutions and how to use these solutions. [ABSTRACT FROM AUTHOR]

• Exact fundamental resonances of the damped and forced truly nonlinear oscillator are computed. • Methodology that yields exact expression for the responses at resonance is developed. • An exact steady state response of the truly nonlinear oscillator needs a multi-harmonic excitation. • Time-periodic excitation is given as a sum of Ateb periodic functions. • For the fundamental resonance the frequency of the steady state response coincides with the applied excitation. In this paper steady states of a essentially nonlinear and damped oscillator excited with a time periodic function is investigated. The procedure for computing exact strongly nonlinear, damped resonances of a nonlinear oscillator, developed by Vakakis and Blanchard (2018), is used and extended to a more general class of strongly nonlinear oscillators. The method is applied for computing of the external excitation which produces the motion equal to that of the free strong nonlinear undamped oscillator. The obtained periodical excitation force is the sum of two Ateb periodic functions which correspond to the sum of various multi-harmonic forces. The amplitude-frequency diagrams are plotted and the resonant frequency is calculated. Motion around steady state is approximately determined by using the method of time variable amplitude and phase, An example of an excited and damped quadratic oscillator perturbed with a linear elastic force is considered. The approximate analytical solution is compared with exact numerical one. The solutions are in a good agreement. [ABSTRACT FROM AUTHOR]