Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator

This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function.     

Fundamental solution of generalized thermo-viscoelasticity using the finite element method

The present paper is aimed at an investigation of the temperature, displacement, and stress in a viscoelastic half space of Kelven–Voigt type. The formulation is applied according to three theories of generalized thermoelasticity: Lord–Shulman with one relaxation time, Green–Lindsay with two relaxation times, as well as the coupled theory. The nondimensional governing equations are solved by the finite element method. Numerical results for the temperature distribution, displacement, and thermal stress are represented graphically. Comparisons are made with the results predicted by the CD, L-S, and G-L theories in the presence and absence of the viscoelastic relaxation time.     

On the coupling of the homotopy perturbation method and Laplace transformation

In this paper, a Laplace homotopy perturbation method is employed for solving one-dimensional non-homogeneous partial differential equations with a variable coefficient. This method is a combination of the Laplace transform and the Homotopy Perturbation Method (LHPM). LHPM presents an accurate methodology to solve non-homogeneous partial differential equations with a variable coefficient. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as HPM, VIM, and ADM. The approximate solutions obtained by means of LHPM in a wide range of the problem’s domain were compared with those results obtained from the actual solutions, the Homotopy Perturbation Method (HPM) and the finite element method. The comparison shows a precise agreement between the results, and introduces this new method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.     

On Three-Dimensional Dynamic Problems of Generalized Thermoelasticity

Lord–;Shulman's system of partial differential equations of generalized thermoelasticity [1] is considered, in which the finite velocity of heat propagation is taken into account by introducing a relaxation time constant. General aspects of the theory of boundary value and initial-boundary value problems and representation of solutions by series and quadratures are considered using the method of a potential.     

State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction

This paper is concerned with the determination of the thermoelastic displacement, stress, conductive temperature, and thermodynamic temperature in an infinite isotropic elastic body with a spherical cavity. A general solution to the problem based on the two-temperature generalized thermoelasticity theory (2TT) is introduced. The theory of thermal stresses based on the heat conduction equation with Caputo’s time-fractional derivative of order α is used. Some special cases of coupled thermoelasticity and generalized thermoelasticity with one relaxation time are obtained. The general solution is provided by using Laplace’s transform and state-space techniques. It is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading (thermal shock). Some numerical analyses are carried out using Fourier’s series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically and the effects of two-temperature and fractional-order parameters are discussed.     

Forced vibration of curved beams on two-parameter elastic foundation

Forced vibration analysis of curved beams on two-parameter elastic foundation subjected to impulsive loads are investigated. The Timoshenko beam theory is adopted in the derivation of the governing equation. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method. The solutions obtained are transformed to the real space using the Durbin’s numerical inverse Laplace transform method. The static and forced vibration analysis of circular beams on elastic foundation are analyzed through various examples.     

Interaction curves for vibration and buckling of thin-walled composite box beams under axial loads and end moments

Interaction curves for vibration and buckling of thin-walled composite box beams with arbitrary lay-ups under constant axial loads and equal end moments are presented. This model is based on the classical lamination theory, and accounts for all the structural coupling coming from material anisotropy. The governing differential equations are derived from the Hamilton’s principle. The resulting coupling is referred to as triply flexural–torsional coupled vibration and buckling. A displacement-based one-dimensional finite element model with seven degrees of freedoms per node is developed to solve the problem. Numerical results are obtained for thin-walled composite box beams to investigate the effects of axial force, bending moment, fiber orientation on the buckling loads, buckling moments, natural frequencies and corresponding vibration mode shapes as well as axial-moment–frequency interaction curves.     

Numerical electroseismic modeling: A finite element approach

Electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid-saturated porous rock due to the electrokinetic phenomenon. This work presents a collection of continuous and discrete time finite element procedures for electroseismic modeling in poroelastic fluid-saturated media. The model involves the simultaneous solution of Biot’s equations of motion and Maxwell’s equations in a bounded domain, coupled via an electrokinetic coefficient, with appropriate initial conditions and employing absorbing boundary conditions at the artificial boundaries. The 3D case is formulated and analyzed in detail including results on the existence and uniqueness of the solution of the initial boundary value problem. Apriori error estimates for a continuous-time finite element procedure based on parallelepiped elements are derived, with Maxwell’s equations discretized in space using the lowest order mixed finite element spaces of Nédélec, while for Biot’s equations a nonconforming element for each component of the solid displacement vector and the vector part of the Raviart-Thomas-Nédélec of zero order for the fluid displacement vector are employed. A fully implicit discrete-time finite element method is also defined and its stability is demonstrated. The results are also extended to the case of tetrahedral elements. The 2D cases of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM-mode) and horizontally polarized shear waves coupled with the transverse electric polarization (SHTE-mode) are also formulated and the corresponding finite element spaces are defined. The 1D SHTE initial boundary value problem is also formulated and approximately solved using a discrete-time finite element procedure, which was implemented to obtain the numerical examples presented.     

The Laplace transform and polynomial Trefftz method for a class of time dependent PDEs

In this paper, we present a new method for solving 1D time dependent partial differential equations based on the Laplace transform (LT). As a result, the problem is converted into a stationary boundary value problem (BVP) which depends on the parameter of LT. The resulting BVP is solved by the polynomial Trefftz method (PTM), which can be regarded as a meshless method. In PTM, the source term is approximated by a truncated series of Chebyshev polynomials and the particular solution is obtained from a recursive procedure. Talbot’s method is employed for the numerical inversion of LT. The method is tested with the help of some numerical examples.     

Generalized thermoelastic infinite medium with spherical cavity subjected to moving heat source

This paper deals with a problem of thermoelastic interactions in an isotropic unbounded medium with spherical cavity due to the presence of moving heat sources in the context of the linear theory of generalized thermoelasticity with one relaxation time. The governing equations are expressed in the Laplace transform domain and solved in that domain. The inversion of the Laplace transform is done numerically using the Riemann-sum approximation method. The numerical estimates of the displacement, temperature, stress, and strain are obtained for a hypothetical material. The results obtained are presented graphically to show the effect of the heat source velocity and the relaxation time parameters on displacement, temperature, stress, and strain.     

A Legendre spectral element method on a large spatial domain to solve the predator–prey system modeling interacting populations

A Legendre spectral element method is developed for solving a one-dimensional predator–prey system on a large spatial domain. The predator–prey system is numerically solved where the prey population growth is described by a cubic polynomial and the predator’s functional response is Holling type I. The discretization error generated from this method is compared with the error obtained from the Legendre pseudospectral and finite element methods. The Legendre spectral element method is also presented where the predator response is Holling type II and the initial data are discontinuous.     

On a finite element method for a certain class of time-dependent problems

The finite element method is applied to solve a linear initial-boundary value problem. The basic idea is to combine this method for a disretization in space variables with the Laplace transform technique for a time variable. Formulation, existence and uniqueness of a weak solution is investigated. The convergence and the rate of convergence of the proposed approximate solution is discussed     

Modelling and simulation of acoustic pulse interaction with a fluid-filled hollow elastic sphere through numerical Laplace inversion

A detailed study is undertaken to analyze the non-steady interaction of plane progressive pressure pulses with an isotropic, homogeneous, fluid-filled and submerged spherical elastic shell of arbitrary wall thickness within the scope of linear acoustics. The formulation is based on the general three dimensional equations of linear elasticity and the wave equation for the internal and external acoustic domains. The Laplace transform with respect to the time coordinate is invoked, and the classical method of separation of variables is used to obtain the transformed solutions in the form of partial-wave expansions in terms of Legendre polynomials. The inversion of Laplace transforms have been carried out numerically using Durbin’s approach based on Fourier series expansion. Special convergence enhancement techniques are invoked to completely eradicate spurious oscillations (Gibbs’ phenomenon), and obtain uniformly convergent solutions. Detailed numerical results for the transient and vibratory responses of water-submerged steel shells of selected wall thickness parameters with various internal fluid loadings under an exponential wave excitation are presented. Many of the interesting dynamic features in the transient shell–shock interaction such as shock transparency, shell-radiated negative pressure waves, formation of triple points, and focusing of the reflected waves are examined using appropriate 2D images of the internal pressure field. Also, the temporal behavior of the specularly-reflected, the lowest symmetric S₀- and antisymmetric A₀-Lamb waves, as well as appearance of the Franz’s creeping waves are discussed through proper visualization of the external scattered field. Likelihood of cavitation is addressed and regions proned to cavitation are identified. Moreover, the effects of internal fluid impedance in addition to shell wall thickness on the dynamic stress concentrations induced within the shell are analyzed. Limiting cases are considered and fair agreements with well-known solutions are established.     

A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.     

Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating

In this work, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the theory of two-temperature generalized thermoelasticity. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating.     

Analytical layer-element solutions of Biot’s consolidation with anisotropic permeability and incompressible fluid and solid constituents

Based on the governing equations of 2D plane-strain Biot’s consolidation, the relationship between generalized displacements and stresses of a single soil layer with anisotropic permeability and incompressible fluid and solid constituents is described by an analytical layer-element, which is deduced in the Laplace–Fourier transform domain by using the eigenvalue approach. Taking the boundary conditions and the continuity of the soil layers into consideration, a global stiffness matrix is subsequently assembled and solved. As to the 3D case, the same derivation is employed after the application of a decoupling transformation. The actual solutions in the physical domain can further be acquired by inverting the Laplace–Fourier transform. Finally, numerical examples are carried out to verify the presented theory and discuss the influence of the anisotropic permeability on the consolidation behavior.     

Finite element calculations for systems with multiple Coulomb centers

The presence of multiple Coulomb centers in molecules or solids poses a challenge when solving the effective Schrödinger equation, required as a crucial ingredient in density functional or Hartree–Fock calculations. This is primarily because Kato’s cusp condition needs to be satisfied close to each nucleus and the matrix elements of the Coulomb potential at the nuclei are rather difficult to evaluate when using global basis functions. A novel method for dealing with these challenges is introduced, rewriting the wavefunction as a product of a function satisfying the nuclear cusp conditions and a smooth function, resulting in a transformed variational principle and a regularized potential. Results of three-dimensional finite element calculations based on this ansatz for the ground state of the molecule H₂⁺ in the Born–Oppenheimer approximation are presented, which were obtained using custom written Python/Fortran code.     

Three-dimensional invisible cloaks with arbitrary shapes based on partial differential equation

The method of designing electromagnetic invisible cloak is usually based on the form-invariance of Maxwell’s equations in coordinate transformation. By solving the partial differential equations (PDEs) that describe how the coordinates transform, three-dimensional (3-D) electromagnetic and acoustic invisible cloaks with arbitrary shapes can be designed provided the boundary conditions of the cloaks can be determined by the corresponding transformation. Full wave simulations based on finite element method verify the designed cloaks. The proposed method can be easily used in designing other transformation media such as matter-wave cloaks.     

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein–Uhlenbeck process associated with Bertoin’s tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Lévy processes.