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This paper is concerned with higher odd-order boundary value problems. We first prove that the operators associated with the problems are symmetric and the corresponding eigenvalues are real, and we give the resolvent operators related to the differential operators. Then we obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem. Moreover, the differential expressions of the eigenvalues as regards these parameters are given. In particular, we give the Frechet derivatives of the eigenvalues for the leading coefficient function q 0 and coefficient functions q 1 , ⋯ , q n. • This paper is concerned with dependence of eigenvalues of higher odd-order boundary value problems. • We give a brief overview of the studies related to the problems, and show that the eigenvalues of the problem depend not only continuously but also smoothly on all parameters of the problem. • By some derivation methods and techniques, we give the differential expressions of the eigenvalues with respect to the coefficient functions q 0 , q 1 , ⋯ , q n , which are some interesting and concise results. And these results have not been done by the predecessors. [ABSTRACT FROM AUTHOR]

• The technique bypasses technically challenging derivation of stochastic representative solutions underlying Monte Carlo solution of initial boundary value problems. • In contrast to MC solutions, the technique estimates non-problem-specific impulse-response functions, i.e., Green's functions, which can be used to solve IBVP's subject to an infinite array of boundary and internal forcing conditions. • A random walker swarm is launched from a chosen response (or impulse) point, well-developed RWboundary interaction models can be implemented, and a simple estimator is used to estimate the transition density - and corresponding Green's function - between the response/impulse point and the impulse/response point. • Deleterious depletion of RW's at absorbing Dirichlet boundaries is solved using a RW respawning algorithm, while spatial variance in estimated GF's, produced by our naĩve transition density estimator, is minimized using a simple area-averaging technique. • The proposed technique is tested against three problems having known Green's functions. • The paper is presented in recipe fashion, and is aimed at non-specialists. A stochastic method is described for estimating Green's functions (GF's), appropriate to linear advection-diffusion-reaction transport problems, evolving in arbitrary geometries. By allowing straightforward construction of approximate, though high-accuracy GF's, within any geometry, the technique solves the central challenge in obtaining Green's function solutions. In contrast to Monte Carlo solutions of individual transport problems, subject to specific sets of conditions and forcing, the proposed technique produces approximate GF's that can be used: a) to obtain (infinite) sets of solutions, subject to any combination of (random and deterministic) boundary, initial, and internal forcing, b) as high fidelity direct models in inverse problems, and c) as high quality process models in thermal and mass transport design, optimization, and process control problems. The technique exploits an equivalence between the adjoint problem governing the transport problem Green's function, G (x , t | x ′ , t ′) , and the backward Kolmogorov problem governing the transition density, p (x , t | x ′ , t ′) , of the stochastic process used in Green's function construction. We address nonspecialists and report four contributions. First, a recipe is outlined for diagnosing when stochastic Green's function estimation can be used, and for subsequently estimating the transition density and associated Green's function. Second, a naive estimator for the transition density is proposed and tested. Third, Green's function estimation error produced by random walker absorption at Dirichlet boundaries is suppressed using a simple random walker splitting technique. Last, spatial discontinuity in estimated GF's, produced by the naive estimator, is suppressed using a simple area averaging method. The paper provides guidance on choosing key numerical parameters, and the technique is tested against two simple unsteady, linear heat conduction problems, and an unsteady groundwater dispersion problem, each having known, exact GF's. [ABSTRACT FROM AUTHOR]

Highlights • An approximate analytical solution to the one-dimensional Bratu boundary value problem is introduced in this paper. • The solution is based on some perturbation methods where artificial parameter is embedded to expand a dependent variable. • Modifications are introduced to implement some techniques to solve a boundary value problem with different nonlinearities. • The current solution gives good convergence compared with the exact lower branch solution at any value of Bratu constant. • Its recursive nature allows for more iteration with more correction terms added to the final solution, i.e. more accuracy. Abstract An approximate analytical solution to the one-dimensional Bratu boundary value problem is introduced in this paper. The solution is based on some perturbation expansion methods. The first step is taken from the linearized perturbation technique which was developed to solve initial value problem with a nonlinear term and no small parameter. An artificial small parameter is embedded and the dependent variable can then be expanded in terms of this parameter. However, necessary modifications are introduced to implement some techniques to solve a boundary value problem in order to allow for different nonlinearities. The current solution showed good convergence at any value of the Bratu constant, when compared with the exact lower branch of the solution to the one-dimensional Bratu problem. Also its recursive nature allows for more iterations and adding more correction terms to the final approximate solution which increases the accuracy. [ABSTRACT FROM AUTHOR]

In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical simulation of the flow field in a turbine cascade, which corresponds to the tip section of a last low-pressure steam turbine rotor. Considered cascade consists of very thin profiles with high stagger angle. The resulting flow field is complex with interactions of strong shock waves, shear layers and shock reflections. The paper proposes a proper numerical approximation of boundary conditions suitable for cases with supersonic inlet and outlet flow velocities and compares the flow field for two cascade configurations: the first one corresponding to real experiment (cascade with finite number of blades located in the wind tunnel test section) and the second one corresponding to annular cascade. The experimental configuration includes the complicated geometry of wind tunnel. The annular configuration leads to blade to blade periodicity, which is not guaranteed for the experimental configuration. Numerical simulations are based on the Favre-averaged Navier–Stokes equations with SST k – ω turbulence model and the in-house implicit finite volume solver with AUSM-type discretization. This method considers structured multi-block grid. Results are compared with experimental data. [ABSTRACT FROM AUTHOR]

The backward substitution method is a newly developed meshless method that has been used for the simulation of many problems in science and engineering with high accuracy and efficiency. In this paper, we explore the feasibility of employing the backward substitution method for simulating two-dimensional incompressible flows. Two non-increment pressure correction projection methods are considered to decompose the original velocity and pressure coupling system into two boundary value problems of intermediate velocity and pressure. Then, two boundary value problems are solved by the backward substitution method in each time iteration step. Five numerical examples are provided to demonstrate the accuracy, computational efficiency and convergence of the method. Comparisons with some existing meshless methods verify the method's advantages and potential applications to engineering. • The backward substitution method is first raised for the simulation of incompressible flows. • Investigated effects of the time step and the number of collocation points on the accuracy. • Verification benchmark comprising several fluid flows problems. • Two projection methods are considered to verify the algorithm. • The algorithm is easily extended to three-dimensional incompressible flows. [ABSTRACT FROM AUTHOR]

Combining the simplified interpolating moving least-squares (IMLS) method with the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method is presented for the numerical study of the regularized long wave (RLW) equation. The trial function in the IMLS method has one less unknown coefficient than in the traditional MLS approximation, and the shape functions satisfy the interpolating property at nodes. Then the IEFG method can apply the essential boundary conditions naturally and directly, and has high computational accuracy. And then this paper studies the error estimates of the IEFG method for RLW equation. The error estimate of the semi-discrete IEFG method for the RLW equation on the influence domain radius is first studied, and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied. The theoretical results show that the IEFG method for RLW equation has good convergence rate. For the purpose of demonstration, three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study one class of generalized convolution-type singular integral equations in class {0}. Such equations are turned into complete singular integral equations with nodal points and further turned into boundary value problems for analytic function with discontinuous coefficients by Fourier transforms. For such equations, we will propose one method different from classical one and obtain the general solutions and their conditions of solvability in class {0}. Thus, this paper generalizes the theory of classical equations of convolution type. [ABSTRACT FROM AUTHOR]

• In this paper, the weak Galerkin finite element method is studied for a singularly perturbed problem with two parameters. On a piecewise uniform Shishkin mesh, we prove uniform convergence of the weak Galerkin finite element method with piecewise higher order discontinuous polynomials. • One important feature of the WG-FEM is that it is possible to eliminate the element basis functions on each local interval by static condensation with the help of the weak derivatives and using the basis functions in the element and element boundary separately. Then, the resulting global coupled linear system will only involve the degrees of freedom on the skeleton of the mesh. Therefore, the degrees of freedom in the WG-FEM is comparable to the standard finite element method and the WG-FEM has much less unknown than the discontinuous Galerkin methods. • Using a special interpolation operator and its good features on anisotropic mesh, we established a robust uniform optimal convergence in the corresponding energy norm. Moreover, we presented error analysis in a stronger balanced norm with a locally defined L2-projection for the first time. • Extensive numerical experiments were carried out to support the theoretical findings in the paper. A weak Galerkin finite element method is proposed for solving singularly perturbed problems with two parameters. A robust uniform optimal convergence has been proved in the corresponding energy and a stronger balanced norms using piecewise higher order discontinuous polynomials on a piecewise uniform Shishkin mesh. Finally, we give some numerical experiments to support theoretical results. [ABSTRACT FROM AUTHOR]

Numerical methods for solving boundary value problems, despite their popularity, are also affected by defects. The most troublesome are: the necessity of discretization, the calculation of singular integrals and large computational complexity. The aim of the paper is to develop a general strategy for approximation of the derivatives of solutions. The versatility of the approach is ensured by performing approximation in a normalized domain. In order to obtain efficient and accurate tool for solving the variety of boundary problems, we should combine the mentioned strategy with the method developed by the authors, called parametric integral equations system (PIES). The effectiveness of the proposed approach lies in: lack of the discretization of both a boundary and a domain, less computational complexity, possibility of determining the derivatives of solutions in a continuous manner at any point of the boundary and the domain without the necessity of calculating singular integrals. The paper presents the different variants of the proposed strategy. Tests were performed on the examples of elastic bodies with various shapes and boundary conditions. The reliability of the proposed approach is shown in comparison to existing analytical solutions. The normalized approximation strategy for derivatives in combination with the PIES method gives accurate results, which are obtained in elementary and efficient way. [ABSTRACT FROM AUTHOR]

• What Is New of This Paper In this paper, sixth-order QCD schemes are proposed to solve the boundary value problems of multi-dimensional Helmholtz equations with variable wavenumber. To the best of our knowledge, there were no any explicit sixth-order compact difference schemes without containing the derivatives of given functions whose global convergence order can be theoretically guaranteed in the meantime. • Significance of This Paper In this paper, sixth-order QCD schemes are constructed to solve the boundary value problems of two-dimensional and three-dimensional Helmholtz equations with variable wavenumber. Our strategy is to discretize the equation by the fourth-order compact scheme at the improper interior grid points that adjoin the boundary, while the sixth-order scheme, where it is compact only for the unknowns, is exploited to the proper interior grid points that are without adjoining the boundary. Different from the classic sixth-order schemes in the literature, this treatment makes the proposed schemes only require the values of source term and wavenumber on the grid points instead of the derivatives of given functions. Theoretically, we prove the QCD schemes can achieve global sixth-order accuracy for non-positive constant wavenumber. Finally, numerical experiments are carried out to manifest the accuracy and convergence order of proposed numerical methods for general wavenumber. • Highlights Sixth-order QCD schemes are constructed to solve multi-dimensional Helmholtz equations with variable wavenumber. The proposed schemes do not contain the derivatives of given functions and without involving the additional calculation of the second boundary layer for unknowns. Theoretically, we prove the QCD schemes can achieve the global sixth-order accuracy. Sixth-order quasi-compact difference (QCD) schemes are proposed for the two-dimensional (2D) and the three-dimensional (3D) Helmholtz equations with the variable parameter. Our approach provides the compact mesh stencil for the unknowns, while the noncompact mesh stencil is employed for the source term and the parameter function without involving their derivatives. For the proper interior grid points that are without adjoining the boundary, the sixth-order truncated errors are obtained by the QCD method. Yet the compact scheme is utilized for both of the source term and the parameter function on the improper interior grids that neighbor the boundary, which only reaches the fourth-order local truncated errors. Theoretically, the sixth-order accuracy of the global error by the proposed QCD method is strictly proved for the non-positive constant parameter. Numerical examples are given to demonstrate that the QCD method achieves the global sixth-order convergence for general variable parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study four classes of discrete type equations with harmonic singular operator and convolution. Such equations are turned into boundary value problems for analytic function with discontinuous coefficients by discrete Fourier transform. The general solutions and the conditions of solvability are obtained in class h by our method. Thus, this paper generalizes the theory of classical equations of convolution type. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Caputo type sequential fractional differential equations supplemented with nonlocal Riemann–Liouville integral boundary conditions. The existence of solutions is derived by applying Leray–Schauder’s alternative, while the uniqueness of solution is established via Banach’s contraction principle. An illustrative example is also included. The paper concludes with some interesting observations. [ABSTRACT FROM AUTHOR]

In the present paper we apply a sinc-Gaussian technique to compute approximate values of the eigenvalues of Dirac systems and Dirac systems with eigenvalue parameter in one or two boundary conditions. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc-method. Numerical worked examples with tables and illustrative figures are given at the end of the paper showing that this method gives us better results in comparison with the classical sinc-method in Annaby and Tharwat (2007, 2012) [5,6]. [ABSTRACT FROM AUTHOR]

This paper deals with a concept and description of a RC network as an electro-analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions. It is based on well-known analogy between heat and electrical conduction. In the paper are compared analytical solution together with numerical solution and experimentally measured data. For the first time a fractional order model of diffusion process and its modeling via lumped RC network has been used. Simple examples of simulations, measurements and their comparison are shown. [ABSTRACT FROM AUTHOR]

In this paper, the initial-boundary-value problems for linear and non-linear multi-term fractional diffusion equations with the Riemann–Liouville time-fractional derivatives are considered. To guarantee the uniqueness of solutions, we employ the weak and the strong maximum principles for the equations under consideration that are formulated and proved in this paper for the first time. An essential element of our proof of the maximum principles is an estimation for the value of the Riemann–Liouville fractional derivative of a function at its maximum point that is established in this paper for a wider space of functions compared to those used in our previous publications. In the linear case, the solutions to the problems under consideration are constructed in form of the Fourier series with respect to the eigenfunctions of the corresponding eigenvalue problems. [ABSTRACT FROM AUTHOR]

In a recent paper [1], we presented two B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems. The first method is based on uniform mesh and the second method is based on non-uniform mesh. Convergent results for both methods were established. It is shown that the first method is of second order accuracy, while the second method is of fourth order accuracy. It is worth pointing out that there were some errors in the proof of Theorem 3 of [1], where the convergence proof of the first method is provided. In this erratum, we correct the mistakes occurred in the proof of Theorem 3. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a variable-territory predator–prey model with Dirichlet boundary condition. We establish a necessary and sufficient condition for the existence of positive solutions to steady state. Furthermore, we also prove that the local bifurcation positive solutions are unconditional stable. By regular perturbation theorem, we investigate the convergence and stability of positive solutions when handling time m is large enough. At the end of this paper, there are some numerical simulations and biological significance to check and complement our theoretical analysis results. [Copyright &y& Elsevier]

Abstract: The paper deals with the analysis of the mathematical model of the two dimensional stationary viscous incompressible flow through a rotating radial blade machine. The flow is described and studied in the rotating frame. The paper provides the classical and weak formulation of the corresponding boundary value problem. The boundary condition on the outflow is the so called “natural” boundary condition, with the nonlinear term proposed by Bruneau and Fabrie (1996) [1], and also modified by a term arising from the rotation of the machine. The existence of a weak solution is proved. [Copyright &y& Elsevier]

Abstract: In this paper we develop the eigenvalue theory for general time scale symplectic systems, in which the dependence on the spectral parameter λ is allowed to be nonlinear. At the same time we do not impose any controllability or strict normality assumptions. We prove the oscillation theorems for eigenvalue problems with Dirichlet, separated, and jointly varying endpoints, including the periodic boundary conditions. We also allow the boundary conditions depending on the spectral parameter. Our new theory generalizes and unifies recently published results on continuous-time linear Hamiltonian systems and discrete symplectic systems with nonlinear dependence on λ and on time scale symplectic systems with linear dependence on λ. The results of this paper are also new in the special case of linear Hamiltonian systems with variable endpoints, as well as for Sturm–Liouville dynamic equations. [Copyright &y& Elsevier]

Abstract: In this paper we propose a new discontinuous Galerkin finite element (DG) method to solve Troesch’s problem, which is highly sensitive for large values of the parameter. This two-point boundary value problem has been heavily studied since 1960, however, only a few papers have provided a reliable solution for high sensitivity. Therefore, we developed the DG method which has proved its efficiency for many decades to be a new numerical solver. We demonstrate through computational results compared with those computed by other methods, that the discontinuous Galerkin method provides a quite efficient, accurate and reliable solution. Thus, the DG method is an attractive and competitive alternative to other numerical and semi-analytical techniques to solve highly sensitive nonlinear problems. [Copyright &y& Elsevier]

Abstract: Initial boundary value problems for the heat conduction equation with the constant and piecewise constant coefficients are considered in the paper. The Dirac delta function appears in the initial data. The considered problems are mathematical models of the temperature distribution arising from a pulse point heat source in homogeneous or two composite materials. The Fourier series expansion method is applied for solving these problems. The formal solutions are constructed in the form of the infinite series which are not classical functions. We show that these formal solutions are generalized functions (distributions). We prove that the partial sum of the constructed infinite series is a classical function which can be considered as a regularization of the generalized solution. The method of the computation of this regularized solution is suggested in the paper. The computational examples confirm the robustness of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the initial-boundary value problem of the advection–diffusion equation with rapidly oscillating coefficients in both time and space domain. A multiscale asymptotic expansion of the solution for this kind of problem is presented. A full discrete finite element method for computing above mentioned problem is introduced. The proposed method not only can greatly save computer memory and CPU time, but also has high precision for approximating the solution. The numerical results show that the method presented in this paper is effective and reliable. [Copyright &y& Elsevier]

Abstract We consider the initial boundary value problems for a homogeneous Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain Ω. The memory kernel of the equation is made up in a complicated way from the (distinct) moduli of stress relaxation for compression and shear, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will study the second order continuous time Galerkin approximation for its space discretization and propose a fully discrete scheme employing the Crank–Nicolson method for the time discretization. Then we derive the uniform long time error estimates in the norm L t 1 (0 , ∞ ; L x 2) for the finite element solutions. Some numerical results are presented to illustrate our theoretical error bounds. [ABSTRACT FROM AUTHOR]

Abstract We study the three-point boundary value problems for the fractional differential coupled systems with both left and right fractional derivatives in the paper. Firstly, the upper-lower solutions method for the systems is presented and some existence results to the systems solutions are also obtained. Secondly, we also give out an iterative approach for finding approximate solutions of the systems. Finally, we consider a concrete example where we obtain the existence result of the solution and also give out its numerical solution and the graphical simulation. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we develop and analyze a high order compact finite difference method (CFDM) for solving a general class of two-point nonlinear singular boundary value problems with Neumann and Robin boundary conditions arising in various physical models. Convergence result of this method is established through matrix analysis approach. To illustrate the applicability and accuracy of the method, we consider nine numerical examples, including heat conduction in the human head, equilibrium of isothermal gas sphere, oxygen-diffusion in a spherical cell and reaction–diffusion process in a spherical permeable catalyst. It is shown that the computational order of convergence of the proposed CFDM is four. The obtained results are compared with those obtained by other existing numerical methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose Fourier spectral method to solve space fractional Klein–Gordon–Schrödinger equations with periodic boundary condition. First, the semi-discrete scheme is given by using Fourier spectral method in spatial direction, and conservativeness and convergence of the semi-discrete scheme are discussed. Second, the fully discrete scheme is obtained based on Crank–Nicolson/leap-frog methods in time direction. It is shown that the scheme can be decoupled, and preserves mass and energy conservation laws. It is proven that the scheme is of the accuracy O (τ 2 + N − r). Last, based on the numerical experiments, the correctness of theoretical results is verified, and the effects of the fractional orders α , β on the solitary solution behaviors are investigated. In particular, some interesting phenomena including the quantum subdiffusion are observed, and complex dynamical behaviors are shown clearly by many intuitionistic images. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a new numerical approach for a quad-curl model problem which arises in the inverse electromagnetic scattering problems and magnetohydrodynamics (MHD). We first split the quad-curl problem with homogeneous boundary conditions into a system of second order equations, and then apply a mixed finite element method to solve the resulting system. The perturbed mixed finite element method is constructed by using edge elements. The well posedness of the numerical scheme is derived. The optimal error estimates in H (curl) and L 2 norms for the primal and auxiliary variables are obtained, respectively. The theoretical results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion. [ABSTRACT FROM AUTHOR]

Abstract The theory of nonlinear difference equations and discrete boundary value problems has been widely used to study discrete models in many fields such as computer science, economics, mechanical engineering, control systems, artificial or biological neural networks, ecology, cybernetics, and so on. Fourth-order difference equations derived from various discrete elastic beam problems. In this paper, we seek further study of the multiplicity results for discrete fourth-order boundary value problems with four parameters. In fact, using a consequence of the local minimum theorem due Bonanno we look for the existence one solution under algebraic conditions on the nonlinear term and two solutions for the problem under algebraic conditions with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear term. Furthermore, by employing two critical point theorems, one due Averna and Bonanno, and another one due Bonanno we guarantee the existence of two and three solutions for our problem in a special case. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider some boundary value problems generated by third-order formally symmetric (self-adjoint) regular differential expression and separated, real-coupled and complex-coupled boundary conditions. It is shown that these problems generate self-adjoint operators. Moreover, the dependence of eigenvalues of these problems on the data are studied and some derivatives of the eigenvalues with respect to some elements of data are introduced. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the construction and convergence analysis of two B-spline collocation methods for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The first method is based on uniform mesh, while the second method is based on non-uniform mesh. For the second method, we use a grading function to construct the non-uniform grid. We prove that the method based on uniform mesh is of second-order accuracy and the method based on non-uniform mesh is of fourth-order accuracy. Three nonlinear examples with derivative dependent source functions are considered to verify the performance and theoretical rate of convergence of present methods. Moreover, we consider some special cases of the problem under consideration in order to compare our methods with other existing methods. It is shown that our second method based on cubic B-spline basis functions has the same order of convergence as quartic B-spline collocation method [1]. Moreover, our methods yield more accurate results and are computationally attractive than the methods developed in [1–8]. The proposed methods are applied on three real-life problems, the first problem describes the distribution of radial stress on a rotationally shallow membrane cap, the second problem arises in the study of thermal explosion in cylindrical vessel and the third problem arises in astronomy. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce an effective finite element scheme for the Poisson problem with inhomogeneous Dirichlet boundary condition on a non-convex polygon. Due to the corner singularities, the solution is composed of a singular part not belonging to the Sobolev space H 2 and a smoother regular part. We first provide a generalized extraction formula for the coefficient of singular part, which is depending on the inhomogeneous boundary condition and the regular part. We then propose a stable finite element method for the regular part by the use of the derived formula. We show the H 1 and L 2 error estimates of the finite element solution for the regular part and the absolute error estimate of the approximation for the coefficient of singular part. Finally, we give some numerical examples to confirm the efficiency and reliability of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the blow-up of positive solutions for a nonlinear reaction–diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied. [Copyright &y& Elsevier]

Abstract: In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper. [Copyright &y& Elsevier]

Abstract: An improved meshless local Petrov–Galerkin method (MLPG) for stress analysis of two-dimensional solids is presented in this paper. The MLPG method based on the moving least-squares approximation is one of the recent meshless approaches. However, accurate imposition of essential boundary conditions in the MLPG method often presents difficulties because the MLPG shape functions does not possess the Kronecker delta property. In order to eliminate this shortcoming, this approach uses the moving Kriging interpolation instead of the traditional moving least-square approximation to construct the MLPG shape functions, and then, the Heaviside step function is used as the test function over a local sub-domain. In this method, the essential boundary conditions can be enforced as the FEM, no domain integration is needed and only regular boundary integration is involved. In addition, the sensitivity of several important parameters of the present method is mainly studied and discussed. Comparing with the original meshless local Petrov–Galerkin method, the present method has simpler numerical procedures and lower computation cost. The effectiveness of the present method for two-dimensional solids problem is investigated by numerical examples in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the initial-boundary value problem of parabolic type equation with rapidly oscillating coefficients in both time and space. A multiscale asymptotic expansion of solution for this kind of problem is presented. The full discrete finite element method for computing above problem is introduced. This method can apply to heat conduction analysis of composite materials. The main advantages of this method are that it can greatly save computer memory and CPU time, and it has good precision at the same time. Finally numerical results show that the method presented in this paper is effective and reliable. [Copyright &y& Elsevier]

Abstract: In this paper, we shall present a new second-order and two-scale approximate solution of boundary value problems for the linear elasticity systems with quasi-periodic structures by multi-scale analysis. The computation of strain and stress is presented here. This idea can apply to the damage computation of composite materials. Because the computation of the local stress by this method having more precision, it can provide some inspiration for construct optimal design. Finally numerical results show that the method presented in this paper is effective and reliable. [Copyright &y& Elsevier]

Abstract This paper deals with the study of the onset of convection in a thin magnetic nanofluid layer which is heated from below under the influence of an applied magnetic field and a variable gravity field. The model used in this study includes the effect of Brownian diffusion, thermophoresis, and magnetophoresis. We applied the Chebyshev pseudospectral method to determine the numerical solutions and discussed the results for Rigid–Rigid, Rigid–Free and Free–Free boundary conditions for water and ester based magnetic nanofluids. The effect of significant parameters affecting the instability of the system has been investigated at the onset of convection. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the adaptive fashion of the Ciarlet–Raviart mixed method for biharmonic equation/eigenvalue problem with simply supported boundary condition in R d. We propose an a posteriori error indicator of the Ciarlet–Raviart approximate solution for the biharmonic equation and an a posteriori error indicator of the Ciarlet–Raviart approximate eigenfuction, and prove the reliability and efficiency of the indicators. We also give an a posteriori error indicator for the approximate eigenvalue and prove its reliability. We design an adaptive Ciarlet–Raviart mixed method with piecewise polynomials of degree less than or equal to m , and numerical experiments show that numerical eigenvalues obtained by the method can achieve the optimal convergence order O (d o f − 2 m d ) (d = 2 , m = 2 , 3 ; d = 3 , m = 3). [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with normal solutions of a two-point boundary-value problem of second order which arises in the steady free convection boundary-layer flow over a vertical permeable flat plate being embedded in a saturated porous medium with both prescribed heat flux and suction rate of the plate. We use a very simple argument to prove that there exists a λ min ∈ (1.3782407, 1.4166499) such that this problem has no normal solution for all λ < λ min , a unique normal solution for λ = λ min , and exactly two normal solutions θ 1 (η) and θ 2 (η) with θ 1 (η) < θ 2 (η) on [ 0 , + ∞) for each fixed λ > λ min , which is applied to the reduced boundary-layer system. [ABSTRACT FROM AUTHOR]

This paper is devoted to establish the existence and non-existence of the traveling waves for the nonlocal Holling–Tanner predator–prey model. By applying the Schauder’s fixed point theorem, we can obtain the existence of the traveling waves. Moreover, in order to prove the limit behavior of the traveling waves at infinity, we construct a sequence that converges to the coexistence state. For the proof of the nonexistence of the traveling waves, we use the property of the two-sided Laplace transform. Finally, we give the effect of the nonlocal diffusion term for the traveling waves. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation. Due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational effort. In the present paper, we describe an alternative approach, based on the reduction of the original problem to an integro-differential equation. Though the problem is posed on the half-line, we just need to approximate the solution on a finite interval. By analyzing the behavior of the numerical approximation on this interval, we identify the solution that satisfies the prescribed boundary condition. Although the numerical algorithm is much simpler than the ones proposed before, it provides accurate approximations. We illustrate the proposed methods with some numerical examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to the application of the method of lines to solve one-dimensional diffusion equation where the classical (integer) second derivative is replaced by a fractional derivative of the Caputo type of order α less than 2 as the space derivative. A system of initial value problems approximates the solution of the fractional diffusion equation with spline approximation of the Caputo derivative. The result is a numerical approach of order O ( Δ x 2 + Δ t m ) , where Δx and Δt denote spatial and temporal step-sizes, and 1 ≤  m  ≤ 5 is an integer which is set by an ODE integrator that we used. The convergence and numerical stability of the method are considered, and numerical tests to investigate the efficiency and feasibility of the scheme are provided. [ABSTRACT FROM AUTHOR]

Motivated by a specific application in seismic reflection, the goal of this paper is to present a modified version of the Castillo–Grone mimetic gradient operators that allows for a high-order accurate solution of the Eikonal equation with Soner boundary conditions. The modified gradient operators utilize a non-staggered grid. In dimensions other than 1D, the modified gradient operators are expressed as Kronecker products of their corresponding 1D versions and some identity matrices. It is shown, that these modified 1D gradient operators are as accurate as the original gradient operators in terms of approximating first-order partial derivatives. It turns out, that in 1D one requires to solve two linear systems for finding a numerical solution of the Eikonal equation. Some examples show that the solution obtained by utilizing the modified operators increases its accuracy when incrementing the order of their approximation, something that does not occur when using the original operators. An iterative scheme is presented for the nonlinear 2D case. The method is of a quasi-Newton-like nature. At each iteration a linear system is built, with progressively higher-order stencils. The solution of the Fast Marching method is the initial guess. Numerical evidence indicates that high-order accurate solutions can be achieved. [ABSTRACT FROM AUTHOR]

We develop a new semi-analytical method for solving multilayer diffusion problems with time-varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers. The convergence rate of the semi-analytical method, relative to the number of eigenvalues, is investigated and the effect of varying the interface conditions on the solution behaviour is explored. Numerical experiments demonstrate that solutions can be computed using the new semi-analytical method that are more accurate and more efficient than the unified transform method of Sheils [ Appl. Math. Model. , 46:450–464, 2017]. Furthermore, unlike classical analytical solutions and the unified transform method, only the new semi-analytical method is able to correctly treat problems with both time-varying external boundary conditions and a large number of layers. The paper is concluded by replicating solutions to several important industrial, environmental and biological applications previously reported in the literature, demonstrating the wide applicability of the work. [ABSTRACT FROM AUTHOR]

Strang splitting is a well established tool for the numerical integration of evolution equations. It allows the application of tailored integrators for different parts of the vector field. However, it is also prone to order reduction in the case of non-trivial boundary conditions. This order reduction can be remedied by correcting the boundary values of the intermediate splitting step. In this paper, three different approaches for constructing such a correction in the case of inhomogeneous Dirichlet, Neumann, and mixed boundary conditions are presented. Numerical examples that illustrate the effectiveness and benefits of these corrections are included. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the multiplicity of solutions for second-order impulsive differential equation with a parameter on the half-line. By using a variational method and a three critical points theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. Also an example is given in this paper to illustrate the main results. [ABSTRACT FROM AUTHOR]

Abstract: The problem of the construction of an object functioning in the regime of optimum performance at the design stage is reduced to the solution of the problem of multicriterion optimization, where the quality criteria are chosen to be its most essential characteristics (parameters). At the same time in all methods of multicriterion optimization the vector quality criterion is considered basically in the linear Euclidean space. Actually, in most cases, the criterion space is non-Euclidean – it is curved. Therefore, such setting cannot give results adequately reflecting the processes running in real systems. In order for the design system to really satisfy the optimality requirements the authors of the given paper offer an absolutely new approach to the solution of the problems of multicriterion optimization based on the definition of the quality criteria space and on finding an invariant corresponding to the distance between any two points of that space. The idea of the study of the metric properties of the quality criteria space and their use in solving problems of multicriterion optimization was offered in the work . But that idea, due to its complexity, has not been completely realized until now. When solving such problems the quality criteria space was automatically identified with the Euclidean space with corresponding metrics. In the general case this could not give results adequately reflecting the processes occurring in real systems. In the present paper metric properties of space criteria are studied for the first time, using as the main instrument the mathematical apparatus of tensor analysis, Riemannian geometry, differential equations in partial derivatives etc. Boundary problems relative to the components of the metric tensor of the n-dimensional space of the phenomenon states enabling to determine its metric properties are posed. The knowledge of the metric tensor furthers the objective appraisal of the phenomenon state and the definition of the optimal state. [Copyright &y& Elsevier]

Abstract: This paper introduces the notion of mixed leadership in nonzero-sum differential games, where there is no fixed hierarchy in decision making with respect to the players. Whether a particular player is leader or follower depends on the instrument variable s/he is controlling, and it is possible for a player to be both leader and follower, depending on the control variable. The paper studies two-player open-loop differential games in this framework, and obtains a complete set of equations (differential and algebraic) which yield the controls in the mixed-leadership Stackelberg solution. The underlying differential equations are coupled and have mixed-boundary conditions. The paper also discusses the special case of linear-quadratic differential games, in which case solutions to the coupled differential equations can be expressed in terms of solutions to coupled Riccati differential equations which are independent of the state trajectory. [ABSTRACT FROM AUTHOR]