Dimensional Boundary Value Problems Research Articles

On the Solvability of a Singular Time Fractional Parabolic Equation with Non Classical Boundary Conditions

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense.

Magnetoelastic effect on semi‐infinite moving crack in an orthotropic structure influenced by anti‐plane wave

AbstractA two dimensional boundary value problem of magnetoelasticity for a semi‐infinite moving crack situated between an orthotropic half‐space and an orthotropic layer has been considered. Also, it has been considered that the crack is agitated by an anti‐plane shear wave. The contemplated problem has been reduced to a typical Wiener–Hopf equation by employing Fourier integral transform to the convenient boundary condition. The accomplished Wiener–Hopf equation has been solved only for asymptotic case to get the analytical expression of stress intensity factor (SIF) and crack tearing displacement (CTD). The dependence of several parameters including the magnetoelastic coupling factor, material constants, crack propagation velocity and crack depth from the upper surface on the SIF and CTD have been illustrated using variety of graphs.

L2 error estimates for a family of cubic finite volume methods on triangular meshes

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H1-norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L2-norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments.

A finite strain gradient theory for viscoplasticity by means of micromorphic regularization

AbstractStretching of polycarbonate films leads to the formation of shear bands in the necking zone [1]. Standard viscoplastic material models render mesh size dependent results, which requires a mathematical regularization. To this end, we present a finite strain gradient theory for a viscoplastic, isotropic material model where we extend the model presented in [2] to a micromorphic model by introducing a new micromorphic variable as an additional degree of freedom with its first gradient [3, 4]. The variable here has the meaning of a micro plastic strain, and is coupled with the macro plastic by a micro penalty term, forcing the macro‐plastic strain to be close to the micro‐plastic strain for the targeted shear band regularization effect. We have implemented the model equations as a three dimensional initial boundary value problem in an in house FE‐tool, to simulate different geometries with different thickness and to compare it the experimental tests. The analysis is performed for a uniaxial tensile geometry as well as for a biaxial tensile geometry. The numerical examples show the ability of the model to regularize the shear bands and solve the problem of localization.

The smooth extension embedding method with Chebyshev polynomials

AbstractWe propose an implementation of the smooth extension embedding method (SEEM), first described by Agress and Guidotti's study, in the setting of Chebyshev polynomials. SEEM is a hybrid fictitious domain/collocation method which solves general boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, SEEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non‐periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve elliptic and parabolic problems in complex geometries, with non‐constant coefficients, and for general boundary conditions. We consider applications to two and three dimensional boundary value problems as well as an initial boundary value problem via a genuinely space–time discretization.

Multi-domain multivariate spectral collocation method for (2+1) dimensional nonlinear partial differential equations

The novelty of this work rests upon the use of the domain partitioning technique in time variable when discretizing the domain of solution in spectral collocation algorithm. The single domain multivariate spectral collocation based methods have been proven to be effective in solving time-dependent partial differential equations (PDEs) defined over small time domains. However, there is a significant loss of accuracy as time computational domain proliferates and also when the number of grid nodes approaches a definite particular number. Therefore, the establishment of the new innovative multi-domain multivariate spectral quasilinearisation method (MDMV-SQLM) is described for the purpose of solving (2+1) dimensional nonlinear PDEs defined on large time intervals. The main output of this study is confirmation that minimizing the size of time computational domain at each subinterval assures sufficiently accurate results that are attained using minimal number of nodal points and less computational time. The solution algorithm involves partitioning the time domain into multiple non-overlapping sub-domains, simplification of the nonlinear PDEs using the quasilinearisation method and assumption of approximate solutions using triple Lagrange interpolating polynomials with Chebyshev–Gauss–Lobatto (CGL) points. The multi-domain spectral collocation procedure is executed on the linear systems of algebraic equations, where the subsequent matrix systems are solved separately in every time sub-interval with the continuity equation essentially used in obtaining initial conditions in the next subintervals. MATLAB software is used to implement the solution algorithm and numerical results are demonstrated graphically and in tabular form. To highlight the efficaciousness and accuracy of the MDMV-SQLM, error estimates, condition numbers and computational time are presented for well known (2+1) dimensional nonlinear initial-Dirichlet boundary value problems. The adoption of the domain decomposition technique is efficacious in suppressing the numerical challenges linked to large matrices and ill-conditioned nature of the resulting coefficient matrix. Also, the communicated results confirm that the numerical scheme is computationally cheap, fast and yield extremely accurate and stable results with the aid of fewer number of grid points for large time domains.

Numerical analysis of Carreau fluid flow over a vertical porous microchannel with entropy generation

This study investigates novel aspects of entropy generation in fully developed heat transport of Carreau fluid in a porous vertical microchannel. Consequences of thermal radiation and viscous heating are included in the thermal energy equation. The no-slip velocity and convective heating temperature boundary conditions are also considered. Darcy–Forchheimer model model is also considered. The system is solved using the bvp4c approach to solve a dimensional less two-point boundary value problem derived from the governing equations. The effect of effective factors on the Bejan number, entropy generation rate, temperature, and velocity are depicted in graphs. According to our analysis, increasing values of the Weissenberg number reduce entropy generation on the left and right sides of the channel, whereas the Bejan number improves on both sides of the channel and is highest in the middle. The fluid transport in a microchannel is enhanced by the pressure gradient. Because of the boundary conditions of convection heating at the microchannel walls, entropy production is at its maximum. Furthermore, the magnitude of the Bejan number is lessened due to a stronger viscous heating mechanism, whereas entropy production gets larger. The results were compared to those previously reported in the literature as a limiting case of the problem, and they were found to be in excellent agreement.

Existence of solutions for $n$-dimensional fractional order bvp with $\infty$–point boundary conditions via the concept of measure of noncompactness

In this paper, using the concept of measure of noncompactness we present some results on the existence of $n$-fixed points for a class of operators. Also as an application, we derive the sufficient conditions for the existence of solutions for $n$-dimensional fractional order functional boundary value problems.

A time–efficient variable shape parameter Kansa–radial basis function method for the solution of nonlinear boundary value problems

In this study we consider the application of a Kansa–radial basis function (RBF) collocation method for solving two– and three–dimensional nonlinear boundary value problems (BVPs) of second and fourth order. In this variable shape parameter approach, a distinct shape parameter is linked with each RBF in the approximation of the solution and the total set of unknowns in the resulting discretized nonlinear problem comprises the RBF coefficients in the approximation and the set of (distinct) shape parameters. The solution of the system of nonlinear equations is achieved using the MATLAB©optimization toolbox functions fsolve or lsqnonlin. Unlike previous applications of these routines to nonlinear BVPs, we exploit the option offered in these functions to provide the analytical expression of the Jacobian of the nonlinear systems in question and show, in several numerical applications, how this leads to spectacular savings in computational time.

Asymptotic study of unsteady mass transfer through a rigid artery with multiple irregular stenoses

The present article examines the transport of species in streaming blood through a rigid artery in the presence of multi-irregular stenosis. The carrier fluid i.e., blood is assumed to be non-Newtonian fluid (Cassons viscoplastic model is used) and the arterial wall is considered to be rigid. A robust model is developed for non-Newtonian flow and hydrodynamic dispersion with the first-order chemical reaction on the arterial boundary in multiple irregular stenosed arterial geometries. Multiple scale solutions of the non dimensional boundary value problem are presented. Asymptotic expressions are developed for velocity and shear stress. Extensive visualization of velocity, concentration, and other flow characteristics is included for various stenotic scenarios, Péclet numbers, and Damköhler numbers. Significant modification in hemodynamic characteristics is computed with viscoplasticity. Mean concentration is also dramatically modified with yield stress and Péclet and Damköhler numbers. The study is relevant to arterial disease simulation.

Geometric model of the fracture as a manifold immersed in porous media

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We develop a model to examine the flow inside fractures with complex geometries and variable thickness on a Riemannian manifold. The fracture is represented as the normal variation of a surface immersed in R3. Using operators of Laplace–Beltrami type and geometric identities, we model an equation that describes the flow in the fracture. A reduced model is obtained as a low dimensional boundary value problem. We then couple the model with the porous media. Theoretical and numerical analyses have been performed to compare the solutions between the original geometric model and the reduced model in reservoirs containing fractures with complex geometries. We prove that the two solutions are close and, therefore, the reduced model can be effectively used in large scale simulators for long and thin fractures with complicated geometry.

Singular matrices arising in the MFS from certain boundary and pseudo-boundary symmetries

We consider the application of the method of fundamental solutions (MFS) to certain two–dimensional second order boundary value problems (BVPs). We assume that the domain of the problem under consideration is a regular polygon and that the sources in the MFS are placed uniformly on a circle concentric to and surrounding the polygon. For certain uniform distributions of the collocation points we prove that, for general classes of regular polygons, the MFS discretization leads to singular coefficient matrices.

Well posedness for a singular two dimensional fractional initial boundary value problem with Bessel operator involving boundary integral conditions

<abstract> This paper studies the existence and uniqueness of solutions of a non local initial boundary value problem of a singular two dimensional nonlinear fractional order partial differential equation involving the Caputo fractional derivative by employing the functional analysis. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of obtaining the a priori bound relies on the construction of a suitable multiplicator. From the resulted a priori estimate, we can establish the solvability of the associated linear problem. Then, by applying an iterative process based on the obtained results for the associated linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the considered nonlinear problem. </abstract>

Thermal transport of bio-convection flow of micropolar nanofluid with motile microorganisms and velocity slip effects

The motivation of current research is to explore thermo-bioconvection micropolar liquid flow subject to motile microorganisms and nanomaterials. Rheological model characteristics of Maxwell viscoelasticity-based micropolar nanoliquid are considered for analysis. Slip mechanisms and stratification phenomenon are accounted. Heat and concentration diffusions are characterized by exploiting the Cattaneo-Christov fluxes for heat and mass. Aspects of thermophoresis, thermal radiation and Brownian motion are also accounted. The dimensional non-linear boundary value problems are rendered into the dimensionless ODE’s by utilizing admissible transformations and then tackled numerically by utilizing bvp4c technique via computational commercial software MATLAB. Significance of sundry variables against velocity, temperature, concentration of nanoparticles and microorganism’s concentration are described through graphs and numeric data.

The method of single expression (MSE) as a prospective modeling tool for boundary value problems: an extension from nano-optics to quantum mechanics

Mathematical description of the wave phenomena in nano-optics and quantum mechanics is similar and requires wavelength-scale analysis of wave interaction with nano-layers in optics and micro-particle interaction with potential barriers or wells in quantum mechanics. Traditionally, when dealing with boundary problems in nano-optics and quantum mechanics, the same fundamental approach of counter-propagating waves is often being used, when general solutions of the wave equations are presented as a sum of counter-propagating waves. This type of solution presentation relies on the superposition principle restricting correct description of strong intensity-dependent nonlinear wave-matter interaction. The non-traditional method of single expression (MSE) does not exploit the superposition principle, but rather uses resulting field representation and backward-propagation algorithm allowing to obtain correct steady-state solutions of boundary value problems without approximations and at any value of wave intensity by taking into account correctly intensity-dependent nonlinearity, loss or gain in a medium. In the present work a detailed description of the MSE approach extended for one dimensional quantum mechanical boundary value problems is presented. Results of numerical simulations by the MSE of electron tunneling through rectangular single and double potential barriers are presented and discussed.

Positive solutions of a nonlinear multiple point for one dimensional p-Laplacian boundary value problems on time scales

Positive solutions of a nonlinear multiple point for one dimensional p-Laplacian boundary value problems on time scales

Geometrically nonlinear response of a fractional-order nonlocal model of elasticity

This study presents the analytical and finite element formulation of a geometrically nonlinear and fractional-order nonlocal model of an Euler–Bernoulli beam. The finite nonlocal strains in the Euler–Bernoulli beam are obtained from a frame-invariant and dimensionally consistent fractional-order (nonlocal) continuum formulation. The finite fractional strain theory provides a positive definite formulation that results in a mathematically well-posed formulation which is consistent across loading and boundary conditions. The governing equations and the corresponding boundary conditions of the geometrically nonlinear and nonlocal Euler–Bernoulli beam are obtained using variational principles. Further, a nonlinear finite element model for the fractional-order system is developed in order to achieve the numerical solution of the integro-differential nonlinear governing equations. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the geometrically nonlinear response of a nonlocal beam subject to various loading and boundary conditions. Although presented in the context of a 1D beam, this nonlinear f-FEM formulation can be extended to higher dimensional fractional-order boundary value problems.

Numerical simulation of an anisotropic heat transfer in magnetized neutron stars with 3D basic operators method

We have solved numerically a three dimensional boundary-value problem for a heat transfer equation in a magnetized neutron star crust with an updated tensorial heat conductivity coefficient. The temperature distribution in a neutron star crust in presence of a magnetic field was simulated. To calculate the surface temperature distribution, we have constructed a local one-dimensional plane-parallel model of a magnetized outer envelope of the neutron star and used it as an outer boundary condition for 3D problem to find a self-consistent solution.This problem was solved with our extension of a basic (support) operators numerical method on a tetrahedral mesh. The idea of operator approach consists of inclusion the boundary conditions into difference form of the solving problem and its formulation as one operator equation. The finite difference operators are constructed in the way to fulfill corresponding relations between continuous operators (for example, div(curl)=0, div is conjugated to -grad so div(grad) is self-conjugated etc.). Such approach allows to obtain completely conservative implicit finite difference schemes. Efficient iterative methods can be used to find the solution because constructed matrixes have good properties, such as symmetry and positive definiteness.

Lie symmetries, reduction and exact solutions of the (1+2)-dimensional nonlinear problem modeling the solid tumour growth

The well known nonlinear model for describing the solid tumour growth (Byrne et al. 2003) is under study using an approach based on Lie symmetries. It is shown that the model in the two-dimensional (in space) approximation forms a (1+2)-dimensional boundary value problem, which admits a highly nontrivial Lie symmetry. The special case involving the power-law nonlinearities is examined in details. The symmetries derived are applied for the reduction of the nonlinear boundary value problem in question to problems of lower dimensionality. Finally, the reduced problems with correctly-specified coefficients were exactly solved and the exact solutions derived were analysed, in particular, some plots were build in order to understand the time-space behaviour of these solutions and to discuss their biological interpretation.

Fracture model reduction and optimization for Forchheimer flows in reservoirs

In this study, we analyze the flow filtration process of slightly compressible fluids in fractured porous media. We model the coupled fractured porous media system, where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. Flow in the fracture is modeled as a reduced low dimensional boundary value problem which is coupled with an equation for flow in the reservoir. We prove that the solution of the reduced model can serve very accurately to approximate the solution of the actual high-dimensional flow in reservoir fracture system because the thickness of the fracture is small. In the analysis, we consider two types of Forchheimer flows in the fracture: isotropic and anisotropic, which are different in their nature. Using the method of reduction, we develop a formulation for an optimal design of the fracture, which maximizes the capacity of the fracture in the reservoir with fixed geometry. Our method, which is based on a set point control algorithm, explores the coupled impact of the fracture geometry and β Forchheimer coefficient.