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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]

• 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]

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]

• 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]

• Added missing literature references and improved literature review. • Fixed formulas going beyond the column. • Added experimental setup diagram. • Added a graph for comparing numerical results with experimental data. • Compacted part of the article with a proof of convergence, reduced the amount of information for readability. This paper presents a detailed and developed method for finding soil nonlinear thermophysical characteristics. Two-layer container complexes are constructed. The side faces of which are thermally insulated, so that one can use the one dimensional heat equation. In order not to solve the boundary value problem with a contact discontinuity and not to lose the accuracy of the solution method, a temperature sensor was placed at the junction of the two media, and a mixed boundary value problem is solved in each region of the container. In order to provide the initial data for the inverse coefficient problem, two temperature sensors are used: one sensor was placed on the open border of the container and recorded the temperature of the soil at this border, and the second sensor was placed at a short distance from the border, which recorded the air temperature. The measurements were carried out in the time interval (0 , 4 t max). At first, the initial-boundary value problem of nonlinear thermal conductivity equation with temperature dependent coefficients of thermal conductivity, heat capacity, heat transfer and material density is investigated numerically. The nonlinear initial-boundary value problem is solved by the finite difference method. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar Thomas' method, and the nonlinear difference problem is solved by the Newton's method. Assuming the constancy of all thermophysical parameters, except for the thermal conductivity of the material, on the segment (0 , t max) , the quadratic convergence of Newton's method is proved and the thermal conductivity is calculated, while the solution of the linearized difference problem was taken as the initial approximation for the Newton's method. By freezing all other thermophysical parameters, and using the measured temperatures in the segment (t max , 2 t max) , the coefficient of specific heat of the material is found. Proceeding in a similar way, using the initial data on the segment (2 t max , 3 t max) , the density of the soil is determined. Also, based on the measured initial data on the segment (3 t max , 4 t max) , the heat transfer coefficient is determined. Based on the experimentally measured data, at each time interval, the corresponding functional is minimized using the gradient descent method. The differentiation of a nonlinear difference problem with respect to the desired parameter method is used to find the gradient of the functional. This method allows you to find the functional gradient and the damping coefficient of the gradient descent in an explicit form. In this case, the compilation and solution of the conjugate problem, which is a frequently used method for solving the inverse problems, is not needed. The performed numerical calculations show that for small time intervals the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and black soil. Also the dependence of the temperature difference at the boundary between the environment and the soil for the heat transfer coefficient was shown. [ABSTRACT FROM AUTHOR]

• The proposed methods are of optimal accuracy under the L ∞ , L 2 , H 1 and H 2 norms, that is, of order k + 1 , k + 1 , k and k − 1 , respectively, where k = 2 m − 1 in W 2 m is the degree of the basis functions. • The proposed methods are still valid in W 2 2. • The proposed methods can easily deal with any kind of boundary conditions as well as interface conditions. In this paper, we present a new reproducing kernel-based collocation (RKBC) approach with optimal convergence rate for some classes of boundary value problems (BVPs). The reproducing kernel function (RKF) of the reproducing kernel space (RKS) W 2 m is a piecewise polynomial of degree 2 m − 1. We observe that the global convergence orders under L ∞ , L 2 , H 1 and H 2 -norms of our method in W 2 m (m = 2 , 3 , 4) is 2 m , 2 m , 2 m − 1 , and 2 m − 2 , which are more efficient than existing reproducing kernel methods (RKMs). Besides, our method can also be easily extended to solve interface problems without losing accuracy. Numerical experiments with detailed discussions on the figures and tables confirm the stability and convergence associated with our proposed numerical approach. [ABSTRACT FROM AUTHOR]

• This article presents a new iterative family having ninth-order convergence for computing the generalized outer inverse. • The proposed scheme requires only seven matrix multiplications at each iterations, but for the specific parameters, it uses only five matrix multi- plications. • The detailed theoretical convergence analysis is presented. • A wide range of random numerical examples is included for comparing the proposed iterations with existing methods. • It is demonstrated that the proposed methods are computationally effcient and provides an attractive alternative for computing the generalized inverse in practice. This paper presents a new iterative family for computing the generalized outer inverse with a prescribed range and null space of a given complex matrix. It is proved that the proposed methods achieve at least ninth-order of convergence. In general, the improved formulation of scheme uses only seven matrix multiplications at each iteration, but for the specific parameters, it uses only five matrix multiplications. The theoretical discussion on computational efficiency index is presented. Further, numerical results obtained are compared with existing robust methods to verify the theoretical analysis and higher computational efficiency. Several numerical examples, including rectangular, rank-deficient, large sparse, and non-singular matrices from the matrix computation tool-box (mctoolbox) are included. Further, the performance of methods is measured on randomly generated singular matrices and boundary value problem. It is demonstrated that the presented scheme gives improved results than the existing Schulz-type iterative methods for calculating the generalized inverse. [ABSTRACT FROM AUTHOR]

• To demonstrate the feasibility of the proposed system, financially relevant irregular initial data-based problems are used. • For problems with discontinuous derivative functions (Greeks in finance), the orthogonal spline collocation approach has been shown to be extremely accurate. • The Rannacher time-marching scheme is shown to be superior to the backward Euler and Crank-Nicolson schemes in terms of computational cost and accuracy. Also it is sufficient to restore the expected behavior. • An analysis of the current numerical scheme's theoretical and numerical convergence is provided. • The current research can be applied to the trading industry to reduce risk and control portfolios. In addition, the current scheme is simple to incorporate and apply to a variety of non-smooth initial data-based moving boundary value problems for future computational studies. In this paper, we extensively study the orthogonal spline collocation method known as the spline collocation at Gauss points with Rannacher's time-marching scheme for free boundary value option pricing problems. Such financial problems commonly feature non-smooth payoff functions that cause inaccuracies in approximating the solution and its derivatives. As a result, unlike the problems with the smooth initial data, the quadratic convergence is not realized by the Crank-Nicolson time-stepping scheme for these problems. Furthermore, the non-smoothness in the initial condition leads to severe degradation in the convergence rates and spurious oscillations near the discontinuity. The rationale is that classical schemes strongly rely on the smoothness of the initial data. A rigorous time-marching scheme referred to as Rannacher time-stepping scheme is introduced for the American option's price diagnosed by a linear complementarity problem to smoothen the data. Moreover, with careful analysis, second and fourth orders of convergence are established for the present scheme in temporal and spatial directions, respectively. The numerical results for three test problems are presented in tables and graphs to validate the theory. These results show that the present scheme achieves higher accuracy and sufficiently restores the expected behavior. [ABSTRACT FROM AUTHOR]

• New collocation method for solving fourth-order Boundary value problems. • Piecewise interpolation with Lidstone interpolating functions. • Lidstone first type polynomials in the interpolating function. • High accuracy solution of fourth-order BVPs both in linear and nonlinear case. • An easy and efficient method to implement. • Extension of Lidstone interpolation theory. In this paper, a new method for solving fourth-order BVPs is presented. The method is based on the first type Lidstone interpolating function. One of the greatest features of the Lidstone interpolating function is that, it can be extended to a piecewise function. Therefore, this extension led us construct piecewise Lidstone interpolating functions on the subintervals and subsequently on the collocation points to have better accuracy. We have fully discussed the method and the whole process in full details to obtain a large system of algebraic equations. The system includes numerical approximation of the problems at the collocation points. From the aspect of error estimation, we have obtained the local truncation errors of the equations and finally presented the comparisons of our approach with the other methods. [ABSTRACT FROM AUTHOR]

A direct method of identification of time dependent parameters in a linear parabolic boundary value problem with over-specified total internal energy involves the flux at the boundary, and an H 1 mixed formulation seems to be more suitable than the standard methods for such class of nonlocal problems. Therefore, this paper develops and analyses an H 1 -Galerkin mixed finite element method. Optimal error estimates in both primary and flux variables are derived in semidiscrete case. Moreover, a priori error estimate for the parameters is established. Based on linearised backward Euler method, a completely discrete scheme is proposed and optimal error analysis is derived. The results of the numerical experiments show the efficacy of the proposed method and confirm our theoretical results. [ABSTRACT FROM AUTHOR]

• We review the non-iterative method for transforming Blasius boundary value problem into initial value problem using a one-parameter scaling transformation. • We discuss in detail the iterative transformation method in solving the Falkner-Skan boundary value problem via a two parameter scaling transformation. • Since one of the two parameters has a value of unity, we show that a simple iterative procedure not requiring an extended group can be employed to solve the Falkner Skan problem, as well as any such problems for which the iterative transformation problem can be used. In this paper, a concise review is presented, of how the scaling transformation group can be employed to determine a modified boundary value problem [6,10,11], which is invariant under an extended two-parameter scaling group. The main contribution is then to show that since the non-physical scaling parameter in the extended group is unity, a simple iterative method which does not require an extended group can be deduced. [ABSTRACT FROM AUTHOR]

In this paper, we give conditions guaranteeing the existence of at least three solutions for a second-order dynamic Sturm–Liouville boundary value problem involving two parameters. In the proofs of the results, we utilize critical point theory and variational methods. In addition, an example is given in order to illustrate our results. [ABSTRACT FROM AUTHOR]