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This paper presents the extension of Improved Residual Power Series Method (IRPSM) towards the system of ordinary differential equations (ODEs). The present system of ODEs is based on the thin film flow over an inclined planar surface. The proposed system is highly nonlinear. Additionally, some embedded factors are taken into the flow analysis in order to investigate the impacts of these parameters on the flow profiles. The method is coded in MATHEMATICA 12.0 software. The results of the present analysis show that the IRPSM has a fast convergence. The impacts of embedded parameters have been successfully investigated and have comparable effects on the flow profiles. Comparison of the present results with the results present in the literature has confirmed the validity of IRPSM. The IRPSM is also applicable to the systems of both linear and highly nonlinear ordinary and partial differential equations. [ABSTRACT FROM AUTHOR]

A discrete numerical model of a pressuremeter test was developed in this paper. The model is constituted of spherical particles with a contact law involving an inter-particle rolling resistance in addition to the classical Coulomb friction model. The model is calibrated from a pressuremeter test performed in the Hostun sand. An adaptive size of discretization of the soil domain is considered to limit the computational cost. Different parameter sets are defined to identify the initial density of the numerical granular assembly. Then, each parameter set leads to a different calibration of the mechanical parameters of the model in order to reproduce the pressuremeter test results. Simulations show that experimental data from a pressuremeter test alone are not enough to fully calibrate the discrete model. Other laboratory test results need to be considered in the calibration methodology. Eventually, the prediction capability of the model is assessed by comparing the simulated constitutive responses to those measured experimentally on different triaxial loading paths. [ABSTRACT FROM AUTHOR]

In this paper, a high-order numerical scheme based on quartic B-spline functions is proposed to solve derivative-dependent nonlinear singular boundary value problems. Convergence of the method is analysed. Five test problems are considered to illustrate the accuracy and efficiency of the method. The results obtained by present method are compared with those obtained by the uniform mesh cubic B-spline collocation (UCS) method [P. Roul and V.M.K. Prasad Goura, B-spline collocation methods and their convergence for a class of nonlinear derivative-dependent singular boundary value problems, Appl. Math. Comput. 341 (2019), pp. 428–450] and non-uniform mesh cubic B-spline collocation (NCS) method [P. Roul and V.M.K. Prasad Goura, B-spline collocation methods and their convergence for a class of nonlinear derivative-dependent singular boundary value problems, Appl. Math. Comput. 341 (2019), pp. 428–450]. The CPU time of proposed method is compared with that of the NCS method. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonlocal boundary value problems for the Laplace operator in a ball, which are a multidimensional generalisation of the Samarskii–Ionkin problem. The well-posedness of the problems are investigated, and Fredholm property of the problems are studied. Moreover, we obtain integral representations of their solutions in explicit forms. [ABSTRACT FROM AUTHOR]

This paper is devoted to elliptic fractional boundary value problems of Sturm–Liouville type on an interval and on a general star graph. We first give some existence and uniqueness results on an open bounded real interval, prove the existence of solutions to a quadratic boundary optimal control problem and provide a characterization via optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem on a star graph with mixed Dirichlet and Neumann boundary controls. [ABSTRACT FROM AUTHOR]

In this paper, we study a boundary value problem of fractional convection–diffusion equation with general two-sided fractional derivative. The well-posedness of the variation formulation is investigated under some properly assumptions. A Petrov-Galerkin method is developed, which employs the Jacobi poly-fractonomials for the trial and test space. The new approach allows the derivation of optimal error estimates in properly weighted Sobolev space and the matrix of the leading term is diagonal. We show that even for smooth data, we only obtain algebraic convergence due to the regularity of the solution. Some numerical examples are presented to demonstrate the validity of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, by employing the invariant sets of descending flow in combination with variational methods, we establish some new criteria to ensure that a class of second-order nonlinear difference equations containing both advance and retardation with different boundary conditions admit at least four distinct solutions which are composed of sign-changing solutions, positive solutions, negative solutions and trivial solution. Furthermore, as an application of our theoretical results, two examples are presented. [ABSTRACT FROM AUTHOR]

We study the stationary problem of the drift–diffusion model with a mixed boundary condition. For this problem, the existence of solutions was established in general settings, while the uniqueness was investigated only in some special cases which do not entirely cover situations that semiconductor devices are used in integrated circuits. In this paper, we prove the uniqueness in a physically relevant situation. The key to the proof is to derive two-sided uniform estimates for the densities of the electron and hole. We establish a new technique to show the lower bound. This together with the Moser iteration method leads to the upper bound. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the forward and inverse problems for a fractional boundary value problem with Dirichlet boundary conditions. The existence and uniqueness of solutions for the forward problem is first proved. Then an inverse source problem is considered. [ABSTRACT FROM AUTHOR]

In this paper, we study a second-order p-Laplacian dynamic equation on a periodic time scale subject to certain boundary value conditions. The variational structure of the above-mentioned boundary value problem is presented. Some new results on the existence of at least one or three distinct periodic solutions are established by means of the saddle point theorem, the least action principle as well as the three critical point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we will consider a wide class of singularly perturbed problems described by the differential equation of fractional multi-order with small parameter multiplying the highest derivative and the appropriate boundary conditions. We construct the linear B-spline operational matrix of fractional derivative in the Caputo sense and introduce a new operational method to solve the mentioned problems. The main characteristic behind this method is that it converts such problems to a system of algebraic equations and overcomes the difficulty and computational complexity induced by the problem. Some illustrative examples are included to demonstrate the validity and applicability of the method. [ABSTRACT FROM AUTHOR]

In the present paper, the positivity of the second order of approximation of the difference operator approximates the second-order differential operator with nonlocal conditions is established. The equivalence of norms of these fractional spaces generated by this operator and Hölder spaces is proved. In applications, the stability of difference schemes for the approximate solution of the boundary value problem for elliptic differential equations is provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions for systems of nonlinear p-Laplacian fractional differential equations whose nonlinearity contains the first-order derivative explicityin Banach spaceE, where𝜃is the zero element ofE,φpis the p-Laplacian operator, i.e.,withp>1, its inverse function is denoted byφqwith, andp,qsatisfy,is standard Caputo derivative andf:I×E×E→Eis continuous. Our main tool is the Sadovskii fixed point theorem. [ABSTRACT FROM AUTHOR]

The present paper deals with a nonlinear boundary value problem derived from a model introduced by Rotenberg describing the growth of a cell population. Each cell of this population is distinguished by its degree of maturityand its maturation velocity. The biological boundary atandare fixed and tightly coupled through the mitosis. We present existence results for the problem in the case daughter cells and mother cells are related by a general reproduction rule and the maturation velocity is allowed to be infinite, i.e.. Our results are based on the classical topological fixed point theorems and use a compactness result established by below. [ABSTRACT FROM AUTHOR]

We consider the problem of interpolating a multivariable function defined on a bounded domain using its traces on parametric hypersurfaces. Our approach is based on the theory of operator polynomial interpolation. We construct the corresponding operator interpolation polynomial for a given function and analyse in detail particular two- and three-dimensional cases. Numerical examples presented in the paper show the flexibility of the proposed approach. [ABSTRACT FROM PUBLISHER]

In this paper, we use discrete cubic spline based on central differences to obtain approximate solution of a second-order boundary value problem. It is shown that the method is of order 4 if a parameter takes a specific value, else it is of order 2. Two numerical examples are included to illustrate our method as well as to compare the performance with other numerical methods proposed in the literature. [ABSTRACT FROM AUTHOR]

We provide criteria for the existence and uniqueness of solutions to a class of discrete fractional boundary value problems of order. An example illustrating our results is presented at the end of the paper. [ABSTRACT FROM PUBLISHER]

In this paper, we study the existence of multiple solutions of boundary value problems for second-order discrete equations [image omitted] , [image omitted] , [image omitted] , [image omitted] bounded on [image omitted] . The proofs are based on the upper and lower solutions, degree theory and sequential argument. [ABSTRACT FROM AUTHOR]

In this paper, we study a double barrier option when the underlying asset price follows a regime-switching exponential mean-reverting process. Our method is a combination of analysis of a deterministic boundary value problem with a probabilistic approach. In this setting, the double barrier option prices satisfy a system of m linear second-order differential equations with variable coefficients and with Dirichlet boundary conditions, where m is the number of regimes considered for the economy. We prove the existence of a smooth solution of the boundary value system by the method of upper and lower solutions; we proceed to construct monotonic sequences of upper and lower solutions that converge to true solutions, respectively. The uniqueness of the solution is established by applying Dynkin's formula. This proof by construction also provides a numerical procedure to compute approximate option values. An important feature of the proposed numerical method is that the true option values are bracketed by the upper and the lower solutions. Examples are provided to illustrate the method. [ABSTRACT FROM AUTHOR]

This paper deals with discrete second order Sturm-Liouville problems where the parameter that is part of the Sturm-Liouville difference equation appears nonlinearly in the boundary conditions. The operator is extended to a self-adjoint operator and an expansion theorem is proved. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple positive solutions of boundary value problems for second-order discrete equations ? 2 x ( n - 1) - p ? x ( n - 1) - qx ( n - 1)+ f ( n , x ( n )) = 0, n ? {1,2,...}, ax (0) - ß? x (0) = 0, x (8) = 0. The proofs are based on the fixed point theorem in Fréchet space (see Agarwal and O'Regan, 2001, Cone compression and expansion and fixed point theorems in Fréchet spaces with application, Journal of Differential Equations , 171, 412–42). [ABSTRACT FROM AUTHOR]

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved. [ABSTRACT FROM AUTHOR]

In this paper we present the theoretical and numerical procedure for solving a Boundary problem of class M . Of particular interest are the works of [ 6 , 9 ] . The method of solution which includes the modality or the procedure for the construction of the difference scheme used in solving such problems is also presented. [ABSTRACT FROM AUTHOR]