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In nature, aquifers are usually composed of distinct kinds of media, i.e., heterogeneous domains rather than homogeneous domains. Groundwater level and flow changes in such domains are more complicated than those in homogeneous domains; thus, building a mathematical model for addressing groundwater flow in heterogeneous aquifers is the present research goal. In conventional research on similar topics, many one-dimensional (1D) analytical models have been presented, but it is challenging to simulate real-world scenarios. This study develops a two-dimensional (2D) analytical model for modeling groundwater flow in a conceptual sloping heterogeneous domain imposed by variable recharge. This model can consider distinct slope angles, medium heterogeneity, and any type of lateral recharge for a semi-infinite domain. The results indicate that groundwater level and flow discharge are greatly affected by the abovementioned factors. The recharge intensity significantly affects the peak of the groundwater level. For example, when the recharge rate increases by 30%, the peak water level increases by 50% as the groundwater flows from the sandy loam zone to the loam zone. The loops delineating the relationship between discharge and groundwater level for different bottom slopes cannot become close for heterogeneous aquifers. The presented 2D analytical model can simulate and better predict results of groundwater changes than previous 1D analytical models. Further, this model can simultaneously consider the effect of varying recharge over time and space on groundwater level change.

Solving boundary value problem with semi-infinite domain in a conventional way or using some of the available approximate methods poses a lot of challenges or better still seems almost impossible over the years, and some of the alternative ways for crossing this hurdle is by fixing value for infinity that is in the domain. Here in this work, we present an Integral Transform (Aboodh Transform) - Weighted Residual Based Method (AT-WRM) to address the afore-mentioned challenge. Aboodh Transform was used to transform and at the same time used to find the inverse of the given differential equations while Weighted Residual via Collocation Method was used in order to avoid fixing value for infinity as usual by introducing e-ix in trial function to decay the infinity that was part of the boundary condition. The accuracy of the presented method was authenticated by solving three different problems. The excellent results obtained from the three solved problems validate the accuracy and effectiveness of the method. [ABSTRACT FROM AUTHOR]

A semi-infinite inverse source problem in heat conduction equations is considered, where the source term is assumed to be compactly supported in the region. After introducing a suitable artificial boundary, the semi-infinite problem is transformed into a bounded one and the corresponding exact expression of the boundary condition is derived. Then we rigorously prove the uniqueness of the solution of original problem, together with the stability of the corresponding optimal control solution.

Solving boundary value problem with semi-infinite domain in a conventional way or using some of the available approximate methods poses a lot of challenges or better still seems almost impossible over the years, and some of the alternative ways for crossing this hurdle is by fixing value for infinity that is in the domain. Here in this work, we present an Integral Transform (Aboodh Transform) - Weighted Residual Based Method (AT-WRM) to address the afore-mentioned challenge. Aboodh Transform was used to transform and at the same time used to find the inverse of the given differential equations while Weighted Residual via Collocation Method was used in order to avoid fixing value for infinity as usual by introducing e-ix in trial function to decay the infinity that was part of the boundary condition. The accuracy of the presented method was authenticated by solving three different problems. The excellent results obtained from the three solved problems validate the accuracy and effectiveness of the method

This paper discusses two efficient collocation methods for solving the Hammerstein integral equations on the semi-infinite domain, where the underlying solutions decay to zero at infinity. These methods are based upon modified Legendre rational and exponential functions, and reduce the Hammerstein integral equation to a nonlinear algebraic system. The error between the approximate and exact solutions in the usual L 2 -norm is estimated. Finally, some numerical experiments are presented to examine and demonstrate the effectiveness and accuracy of the proposed methods in comparison to other approaches. [ABSTRACT FROM AUTHOR]

We consider non-Newtonian boundary-layer fluid flow, governed by a power-law Ostwald-de Waele rheology. Boundary-layer flows of non-Newtonian fluids have far-reaching applications, and are very frequently encountered in physical, as well as, engineering and industrial processes. A similarity transformation results in a BVP consisting of an ODE and some boundary conditions. Our aim is to derive highly accurate analytical relationships between the physical and mathematical parameters associated with the BVP and boundary-layer flow problem. Mathematical analyses are employed, where the results are verified at the numerical computational level, illustrating the accuracy of the derived relations. A set of "Crocco variables" is used to transform the problem, and, where appropriate, techniques are used to deal with the resulting singularities in order to establish an efficient computational setting. The resulting computational setting provides an alternative, which is different from those previously used in the literature. We employ it to carry out our numerical computations. [ABSTRACT FROM AUTHOR]

[Display omitted] • The extended Graetz problem with core-annular two immiscible liquids is considered. • Axial diffusion and semi-infinite circular ducts are considered in the formulation. • An analytical approach using integral transforms is proposed for the solution. • The solution is verified with the literature, demonstrating excellent agreement. • Obtained Nusselt numbers are presented for different parameters and axial positions. Flows involving core-annular immiscible liquids provide a wide range of useful engineering applications, such as transportation of heavy oils and cooling processes. The present article considers an analytical solution for the internal convection heat transfer in hydrodynamically developed and thermally developing laminar core-annular liquid–liquid immiscible flows in circular ducts. The axial diffusion is considered in the model and the problem is investigated for semi-infinite domain under convective boundary condition on duct walls and prescribed inlet flow temperature. The steady-state problem is solved through the generalized integral transform technique (GITT) with a single domain approach, in which, the coupled ODE system arising from the GITT is analytically solved using the matrix exponential method. Moreover, integral balance is used to compute the Nusselt number. The convergence process of the local Nusselt number is shown in different flow regions and the results are presented for multiple flow parameters. The effects of the boundary conditions, flow properties, and other parameters are discussed. An analysis of axial diffusion effects is performed, which confirms that it can interfere with the local Nusselt number, especially for lower Péclet numbers. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid method based on a Legendre wavelet spectral collocation scheme and rational approximation using Legendre rational functions is proposed. This method is utilized to solve nonlinear differential equations on semi-infinite domains. Although Legendre wavelet collocation method possesses spectral accuracy, it is not convenient to implement it on large intervals. On the other hand, Legendre rational functions are useful for approximating the solution of differential equations on large domains whilst this process is computationally cheap. The proposed hybrid method enjoys the benefits of both methods while improves their performance. The proposed method is compared with other numerical methods in the literature to demonstrate its superiority. [ABSTRACT FROM AUTHOR]

This paper introduces a straightforward traction-based approach for implementing transmitting boundaries in time-domain nonlinear seismic soil–structure interaction (SSI) analyses. The primary objective is to simplify the challenges associated with integrating appropriate boundaries into finite-element analysis software. The proposed method conceptualizes an SSI model as a free body extracted from its original domain and determines the boundary traction (i.e., stress boundary conditions in the engineering sense) at the cutting surface. This determination is based on the current velocity, free-field dynamic responses, and static equilibrium stresses. In scenarios employing the classic viscous boundary, traction is solely dependent on the free-field dynamic responses and static equilibrium stresses, facilitating predetermined calculations. Detailed implementation guidelines for Abaqus are provided to illustrate the practicality of the proposed technique. The efficacy of this technique was validated by its ability to accurately reproduce both linear and nonlinear one-dimensional shear waves in three-dimensional models. Furthermore, the nonlinear responses of the SSI models with varying soil dimensions underscored the effectiveness of the approach for domain reduction. • A simple, traction-based approach is presented for transmitting boundaries in time-domain nonlinear seismic SSI analyses. • The primary objective is to simplify the implementation of appropriate soil boundaries in finite element analysis software. • The traction can be predefined with freefield and static equilibrium responses when employing the classic viscous boundary. • The traction-based approach provides greater flexibility in selecting soil geometries and sources of free-field responses. • Implementation instructions for Abaqus, a commercial finite element analysis software, are provided as a demonstration. [ABSTRACT FROM AUTHOR]

We consider non-Newtonian boundary-layer fluid flow, governed by a power-law Ostwald-de Waele rheology. Boundary-layer flows of non-Newtonian fluids have far-reaching applications, and are very frequently encountered in physical, as well as, engineering and industrial processes. A similarity transformation results in a BVP consisting of an ODE and some boundary conditions. Our aim is to derive highly accurate analytical relationships between the physical and mathematical parameters associated with the BVP and boundary-layer flow problem. Mathematical analyses are employed, where the results are verified at the numerical computational level, illustrating the accuracy of the derived relations. A set of “Crocco variables” is used to transform the problem, and, where appropriate, techniques are used to deal with the resulting singularities in order to establish an efficient computational setting. The resulting computational setting provides an alternative, which is different from those previously used in the literature. We employ it to carry out our numerical computations.

The problem of toxic contaminants in groundwater with groundwater pollution measurement model. The advection-diffusion equation is used to describe the concerned model. The theoretical solution of the advection-diffusion equation is limited only in ideal geometries. Applications of numerical solutions are influenced in several initial and boundary conditions when dealing with complex geometries. In this research, numerical simulations for one-dimensional groundwater pollution measurement around landfills model through heterogeneous soil are focused. The forward time center space and Saulyev finite difference techniques are used to approximate the solutions. The accuracy of proposed techniques are to examine by comparing the approximated solutions with the analytical solution. The purposed technique gives good agreement approximated solution. [ABSTRACT FROM AUTHOR]

A third-order highly non-linear ODE that arises in applications of non-Newtonian boundary-layer fluid flow, governed by a power-law Ostwald–de Waele rheology, is considered. The model appears in many disciplines related to applied and engineering mathematics, in addition to engineering and industrial applications. The aim is to use a new set of variables, defined via the first and second-order derivatives of the dependent variable, to transform the problem to a bounded domain, where we study properties of solutions, discuss existence and uniqueness of solutions, and investigate some physical parameter values and limitations leading to non-existence of solutions. [ABSTRACT FROM AUTHOR]

A new collocation method, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) collocation method, is given for solving some nonlinear boundary value problems in the semi-infinite domain, such as equations of the unsteady isothermal flow of a gas, the third grade fluid, the Blasius, and the field equation determining the vortex profile. The method reduces the solution of the problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of the method, the numerical results of the present method are compared with several numerical results.

In this article, we introduce a fractional order of rational Bessel functions collocation method (FRBC) for solving the Thomas-Fermi equation. The problem is defined in the semi-infinite domain and has a singularity at $x = 0$ and its boundary condition occurs at infinity. We solve the problem on the semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration the equation is solves by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results, to show that the new method is accurate, efficient and applicable.

In this work, we introduce an analytical expression for approximating the transient melting radius during powder melting in Selective Laser Melting (SLM) assumed with a stationary laser heat source. The purpose of this work is to evaluate the suggested analytical approach in determining the melt pool geometry during laser processing, by considering heat transfer and phase change effects. This will allow for the rendering of the first findings on the way to a quasi-real time calculation of the melt pool during laser melting, which will contribute significantly to the process design and control, especially when new powders are applied. Initially, we consider the heat transfer process associated with a point heat source, releasing a continuous and constant power (in a semi-infinite powder bed. On the point of the heat source the temperature is infinite, and the material starts to melt spherically outwards, creating an interface that separates the solid from the molten material; we assume different properties between the two phases. Unlike the cases of the cartesian and cylindrical coordinates, (in a cartesian coordinate the heat source is over a plane, i.e., W/m², and in cylindrical along a line, i.e., W/m), where the melting process is proportional to the square root of time, in spherical coordinates the melting stops at a finite radius, i.e., a maximum radius, which depends only on the heat source, the conductivity of the solid and the difference between the far-field temperature and the melting temperature of the material. Here we should also point out that to achieve continuous melting in spherical coordinates the power of the source must increase with the square root of the time. The obtained analytical expression for the maximum melting radius and the approximate expression for its dependence on the time compare well with the numerical results obtained by a finite element analysis. [ABSTRACT FROM AUTHOR]

The characteristic of the mathematical solution describing linear heat diffusion from an instantaneous point source near the interface of two semi-infinite domains in perfect thermal contact is discussed. Because spherical symmetry is lacking and the solution cannot be scale-transformed into a time-independent shape, the only way to grasp the geometrical information of the solution is by ad-hoc computation. The solution which was first derived more than 65 years ago is re-arranged to permit physical interpretation. Through the term-by-term analysis, the time evolution of the solution is divided into three stages and this enables systematic understanding of its nature. [ABSTRACT FROM AUTHOR]

In this paper, a generalized Laguerre spectral Petrov–Galerkin method is proposed to solve a class of time-fractional subdiffusion equations on the semi-infinite domain. We use the generalized associated Laguerre functions of the first kind and the generalized Laguerre functions as basis functions in time and space directions separately. The respective projection error estimates can be obtained as well. We derive the projection error estimates of the solutions in time-space directions. The approximation results of the fully discrete spectral scheme are introduced in this paper. Some numerical results are presented to illustrate the efficiency of this method. [ABSTRACT FROM AUTHOR]

A novel solution method is presented which leads to an analytical model for the advective-dispersive transport in a semi-infinite domain involving a wide spectrum of boundary inputs, initial distributions, and zero-order productions. The novel solution method applies the Laplace transform in combination with the generalized integral transform technique (GITT) to obtain the generalized analytical solution. Based on this generalized analytical expression, we derive a comprehensive set of special-case solutions for some time-dependent boundary distributions and zero-order productions, described by the Dirac delta, constant, Heaviside, exponentially-decaying, or periodically sinusoidal functions as well as some position-dependent initial conditions and zero-order productions specified by the Dirac delta, constant, Heaviside, or exponentially-decaying functions. The developed solutions are tested against an analytical solution from the literature. The excellent agreement between the analytical solutions confirms that the new model can serve as an effective tool for investigating transport behaviors under different scenarios. Several examples of applications, are given to explore transport behaviors which are rarely noted in the literature. The results show that the concentration waves resulting from the periodically sinusoidal input are sensitive to dispersion coefficient. The implication of this new finding is that a tracer test with a periodic input may provide additional information when for identifying the dispersion coefficients. Moreover, the solution strategy presented in this study can be extended to derive analytical models for handling more complicated problems of solute transport in multi-dimensional media subjected to sequential decay chain reactions, for which analytical solutions are not currently available. [ABSTRACT FROM AUTHOR]

In this work, we introduce an analytical expression for approximating the transient melting radius during powder melting in Selective Laser Melting (SLM) assumed with a stationary laser heat source. The purpose of this work is to evaluate the suggested analytical approach in determining the melt pool geometry during laser processing, by considering heat transfer and phase change effects. This will allow for the rendering of the first findings on the way to a quasi-real time calculation of the melt pool during laser melting, which will contribute significantly to the process design and control, especially when new powders are applied. Initially, we consider the heat transfer process associated with a point heat source, releasing a continuous and constant power (in a semi-infinite powder bed. On the point of the heat source the temperature is infinite, and the material starts to melt spherically outwards, creating an interface that separates the solid from the molten material; we assume different properties between the two phases. Unlike the cases of the cartesian and cylindrical coordinates, (in a cartesian coordinate the heat source is over a plane, i.e., W/m2, and in cylindrical along a line, i.e., W/m), where the melting process is proportional to the square root of time, in spherical coordinates the melting stops at a finite radius, i.e., a maximum radius, which depends only on the heat source, the conductivity of the solid and the difference between the far-field temperature and the melting temperature of the material. Here we should also point out that to achieve continuous melting in spherical coordinates the power of the source must increase with the square root of the time. The obtained analytical expression for the maximum melting radius and the approximate expression for its dependence on the time compare well with the numerical results obtained by a finite element analysis.

In this paper, the nonlinear singular Thomas-Fermi differential equation on a semi-infinite domain for neutral atoms is solved by using the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind. First, this collocation method reduces the solution of this problem to the solution of a system of nonlinear algebraic equations. Second, using solve a system of nonlinear equations, the initial value for the unknown parameter L is calculated, and finally, the value of L to increase the accuracy of the initial slope is improved and the value of y'(0) = -1.588071022611375312718684509 is calculated. The comparison with some numerical solutions shows that the present solution is highly accurate. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to contribute to the construction of Green's functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green's functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green's function method, and how damping properties can be obtained. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear singular Thomas–Fermi differential equation for neutral atoms is solved using the fractional order of rational Chebyshev orthogonal functions (FRCs) of the first kind, F T n α ( t , L ) , on a semi-infinite domain, where L is an arbitrary numerical parameter. First, using the quasilinearization method, the equation be converted into a sequence of linear ordinary differential equations (LDEs), and then these LDEs are solved using the FRCs collocation method. Using 300 collocation points, we have obtained a very good approximation solution and the value of the initial slope y ′ ( 0 ) = − 1.5880710226113753127186845094239501095 , highly accurate to 37 decimal places. [ABSTRACT FROM AUTHOR]

The Dirichlet-to-Neumann finite element method (DtN FEM) has proven to be a powerful numerical approach to solve boundary-value problems formulated in exterior domains. However, its application to elastic semi-infinite domains, which frequently arise in geophysical applications, has been rather limited, mainly due to the lack of explicit closed-form expressions for the DtN map. In this paper, we present a DtN FEM procedure for boundary-value problems of elastostatics in semi-infinite domains with axisymmetry about the vertical axis. A semi-spherical artificial boundary is used to truncate the semi-infinite domain and to obtain a bounded computational domain, where a FEM scheme is employed. By using a semi-analytical procedure of solution in the unbounded residual domain lying outside the artificial boundary, the exact nonlocal boundary conditions provided by the DtN map are numerically approximated and efficiently coupled with the FEM scheme. Numerical results are provided to demonstrate the effectiveness and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

The Nye–Tinker–Barber model is a classical convection–diffusion model for nutrient uptake by plant roots in cylindrical coordinates and has one nonlinear left Robin boundary condition with Michaelis–Menten function of concentration. First the Michaelis–Menten function is fitted into a function of time by numerical concentration at root surface from difference scheme, and then the Laplace and numerical inverse Laplace transforms — Zakian inversion method are taken to obtain the approximate analytical solution. Compared with other solutions made by difference scheme, Stehfest inversion method and previous analytical methods, it is found that the analytical solution obtained by Laplace and Zakian inversion transforms has higher accuracy and computation efficiency. This analytical method can be extended to other nutrient uptake models with Michaelis–Menten function. [ABSTRACT FROM AUTHOR]

In this article, we introduce a fractional order of rational Bessel functions collocation method (FRBC) for solving the Thomas-Fermi equation. The problem is defined in the semi-infinite domain and has a singularity at x = 0 and its boundary condition occurs at infinity. We solve the problem on the semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration the equation is solves by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results, to show that the new method is accurate, efficient and applicable. [ABSTRACT FROM AUTHOR]

An analysis is carried out to discover the influence of a rotating nanofluid over a stretching surface. The two phase nanofluid model is used for this study. Two types of nanoparticles, namely copper and titanium oxide are used in our analysis with water as the base fluid. The governing system of partial differential equations along with the corresponding boundary conditions are presented and then transformed into a set of nonlinear ordinary differential equations using suitable similarity transformations. These equations are solved numerically by means of an iterative procedure called the midpoint integration scheme along with Richardson extrapolation. The results for flow and heat transfer characteristics are presented through graphs against nanoparticle volume fraction and rotation parameter for both types of nanoparticles. Quantities of physical interest such as local skin friction coefficients and local heat flux rate at the stretching surface are computed and analyzed. Numerical values for skin frictions and local heat flux rate are computed in the absence of nanoparticle volume fraction and rotation and they are found to be in very good agreement with the existing published literature. [ABSTRACT FROM AUTHOR]

An inversion framework employing a Gauss–Newton method is developed to reconstruct material profiles in heterogeneous, viscoelastic, semi-infinite domains. In particular, a full-waveform inversion approach is investigated to image the elastic and attenuating parameters of a layered media. To account for the viscoelasticity of the medium, a Generalized Maxwell Body with one spring and two Maxwell elements in parallel (GMB2) is adopted in the forward and inverse wave propagation problems. Perfectly-matched-layers were introduced as wave absorbing buffers to simulate the semi-infinite extent of the domain. Using transient wave equations endowed with the GMB2 constitutive relation and the PML, a partial-differential-equations-constrained optimization scheme was implemented that lead to classic KKT (Karush–Kuhn–Tucker) conditions including time-dependent state, adjoint, and time-invariant control problems. An optimal solution of the viscoelastic parameters was obtained using a reduced-space approach based on a line search algorithm where the search direction was computed by the Gauss–Newton method. Considerable improvements on the accuracy and convergence rate of solutions were made by the developed Gauss–Newton inversion procedure compared to previous research using the Fletcher–Reeves method. [ABSTRACT FROM PUBLISHER]

We solve by Laguerre collocation method two second order genuinely nonlinear singularly perturbed boundary value problems and a fourth order nonlinear one formulated on the half-line. First, we are concerned with solutions to a model equation for low Reynolds number flow, the so-called Cohen, Fokas, Lagerstrom model. Then we compute monotonously increasing solutions, the so-called 'bubble type solutions', to density profile equation for the description of the formation of microscopical bubbles in a nonhomogeneous fluid. We compare the results obtained in the latter case with those carried out by shooting or polynomial collocation both coupled with domain truncation. A fourth order nonlinear boundary value problem supplied with mixed boundary conditions in origin as well as at infinity is also solved. The solution and its first three derivatives are obtained and compared with their counterparts obtained by Keller's box scheme. The collocation based on Laguerre functions avoids the domain truncation, exactly impose any kind of boundary conditions at infinity and resolve with reliable accuracy the sharp interior or exterior layer of the solutions. The method is fairly easy implementable and efficient in treating various types of nonlinearities. The accuracy of the method is also tested on the classical Blasius problem (a third order one). We solve with the same method the eigenvalue problem obtained by linearization around the constant solution, which corresponds to the case of a homogeneous fluid (without bubbles), and observe that this solution is stable. [ABSTRACT FROM AUTHOR]

We investigate the solutions of a modified fractional diffusion equation which has a secondary fractional time derivative acting on a diffusion operator. We obtain analytical solutions for the modified equation in the finite and semi-infinite domains subject to absorbing boundary conditions. Most of the results have been derived by using the Laplace transform, the Fourier Cosine transform, the Mellin transform and the properties of Fox H function. We show that the semi-infinite solution can be expressed using an infinite series of Fox H functions similar to the infinite case, while the finite solution requires double infinite series including both Fox H functions and trigonometric functions instead of one infinite series. The characteristic crossover between more and less anomalous behaviour as well as the effect of absorbing boundary conditions are clearly demonstrated according to the analytical solutions. [ABSTRACT FROM AUTHOR]

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner–Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner–Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi–Davidson type methods. [ABSTRACT FROM AUTHOR]

In this paper we propose a method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. The proposed approach is based on an Unsupervised Combined Artificial Neural Networks (UCANN) method. Firstly, The trial solutions of the differential equations are written in the form of feed-forward neural networks containing adjustable parameters (the weights and biases); results are then optimized with the combined neural network. The proposed method is tested on series of Lane-Emden differential equations and the results are reported. Afterward, these results are compared with the solution of other methods demonstrating the efficiency and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we apply a DDM-FEM coupling method to reservoir simulation to evaluate the reservoir behavior over a compacting oil reservoir in half space. We use displacement discontinuity methods to account for the reservoir surroundings, and finite element methods in the fully coupled simulation of the reservoir itself. [ABSTRACT FROM AUTHOR]

• A two-dimensional generalized Oldroyd-B fluid model on a semiinfinite domain is considered. • The time direction is approximated by the second order θ scheme combined with the weighted and shifted Gruünwald difference operator. • For the case of unbounded space, the mixed Laguerre-Legendre spectral method is proposed. • The theoretical analysis and numerical implementation of the scheme are discussed in detail. • The numerical method can improve the convergence accuracy with low computational cost. In this paper, we study the numerical solution for a two-dimensional generalized Oldroyd-B fluid flowing on a semi-infinite domain. The second order θ scheme with the weighted and shifted Grünwald difference operator is derived to approximate the time derivatives with orders in (0,2). For the case of unbounded space, the Laguerre-Legendre spectral method is proposed. The fully discrete scheme is obtained and proved to be stable, convergent with accuracy O (τ 2 + N (1 − s) / 2 + M 1 − r) , where τ is the time step size, N , M are the polynomial degrees. We also implement some numerical examples to further support the theoretical analysis. [ABSTRACT FROM AUTHOR]