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A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed. [ABSTRACT FROM AUTHOR]

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin's-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul'yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach. [ABSTRACT FROM AUTHOR]

This paper is concerned with solving the problem of identifying the control vector problem for a fractional multi-order system of nonlinear ordinary differential equations (ODEs). We describe a quasilinearization approach, based on minimization of a quadratic functional, to compute the values of the unknown parameter vector. Numerical algorithm combining the method with appropriate fractional derivative approximation on graded mesh is applied to SIS and SEIR problems to illustrate the efficiency and accuracy. Tikhonov regularization is implemented to improve the convergence. Results from computations, both with noisy-free and noisy data, are provided and discussed. Simulations with real data are also performed. [ABSTRACT FROM AUTHOR]

A time stepping quasilinearization approach to the mixed (or coupled) form of one and two dimensional Richards' equations is developed. For numerical solution of the linear ordinary differential equation (ODE) for 1D case and elliptic for 2D case, obtained after this semidiscretization, a finite volume method is used for direct problems arising on each time level. Next, we propose a version of the decomposition method for the numerical solution of the inverse ODE and 2D elliptic boundary problems. Computational results for some soil types and its related parameters reported in the literature are presented. [ABSTRACT FROM AUTHOR]

In this work, we consider Cauchy-type problems for Laplace's equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green's second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations. [ABSTRACT FROM AUTHOR]

We consider an inverse problem of recovering the mortality rate in the honey bee difference equation model, that tracks a forage honeybee leaving and entering the hive each day. We concentrate our analysis to the model without pesticide contamination in the symmetric spatial environment. Thus, the mathematical problem is formulated as a symmetric inverse problem for reaction coefficient at final time constraint. We use the overspecified information to transform the inverse coefficient problem to the forward problem with non-local terms in the differential operator and the initial condition. First, we apply semidiscretization in space to the new nonsymmetric differential operator. Then, the resulting non-local nonsymmetric system of ordinary differential equations (ODEs) is discretized by three iterative numerical schemes using different time stepping. Results of numerical experiments which compare the efficiency of the numerical schemes are discussed. Results from numerical tests with synthetic and real data are presented and discussed, as well. [ABSTRACT FROM AUTHOR]

A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed. [ABSTRACT FROM AUTHOR]

The basic model motivating this work is that of contaminant transport in the Earth's subsurface, which contains layers in which analytical and semi-analytical solutions of the corresponding advection–dispersion equations could be derived. Then, using the interface relations between adjacent layers, one can streamline the study of the model to the solution to the initial boundary value problem for a coupled parabolic system on partitioned domains. For IBVPs, we set up weak formulations and prove the existence and uniqueness of solutions to appropriate Sobolev-like spaces. A priori estimates at different levels of input data smoothness were obtained. The nonnegativity preservation over time of the solution is discussed. We numerically demonstrate how to solve the reduced truncated problem instead of the original multispecies one with a large number of layers. [ABSTRACT FROM AUTHOR]

The simultaneous estimation of coefficients and the initial conditions for model fractional parabolic systems of porous media is reduced to the minimization of a least-squares cost functional. This inverse problem uses information about the pressures at a finite number of space time points. The Frechet gradient of the cost functional is derived. The application of the conjugate gradient method for numerical parameter estimation is also discussed. Computational results with noise-free and noisy data illustrate the efficiency and accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this study, a numerical solution for degenerate space–time fractional advection–dispersion equations is proposed to simulate atmospheric dispersion in vertically inhomogeneous planetary boundary layers. The fractional derivative exists in a Caputo sense. We establish the maximum principle and a priori estimates for the solutions. Then, we construct a positivity-preserving finite-difference scheme, using monotone discretization in space and L1 approximation on the non-uniform mesh for the time derivative. We use appropriate grading techniques for the time–space mesh in order to overcome the boundary degeneration and weak singularity of the solution at the initial time. The computational results are demonstrated on the Gaussian fractional model as well on the boundary layers defined by height-dependent wind flow and diffusitivity, especially for the Monin–Obukhov model. [ABSTRACT FROM AUTHOR]

We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we decouple the full inverse problem into two Dirichlet problems at each time level. On this base, we develop decomposition techniques to obtain exact formulas for the unknown boundary conditions at point measurements. A discrete version of the analytical approach is realized on time adaptive mesh for different fractional order of the equations in each of the disjoint domains. A variety of numerical examples are discussed. [ABSTRACT FROM AUTHOR]

In this paper we develop a numerical method for pricing American options under regime-switching model, whose solutions are option buyer indifference prices. The problem is formulated as a nonlinear complementarity problem. We apply interior penalty method to approximate the differential complementarity problem, which results in a system of one degenerate parabolic equation and one ordinary differential equation, weakly coupled by nonlinear exponential term. We formulate the problems in variational form in appropriate spaces and prove comparison principle. Then, as a consequence, we derive estimates about the convergence rate of the interior penalty method. We discretize the penalized problem by fully implicit finite difference scheme and prove that the early exercise constraint is strictly satisfied. We establish comparison principle, uniqueness and boundedness of the numerical solution. To solve the nonlinear system of algebraic equations we use Newton method and partial Newton method. We present numerical results that confirm the theoretical statements. To improve the computational efficiency, we unfold the two-grid idea, combining both iteration processes – Newton method and partial Newton method. We illustrate the efficiency of this approach. [ABSTRACT FROM AUTHOR]

In this work, we consider a system of weakly coupled semi-linear parabolic equations of optimal portfolio in a regime-switching model in the case of exponential utility function, suggested by A.R. Valdez and T. Vargiolu [Optimal portfolio in a regime-switching model, in Proceedings of the Ascona '11 Seminar on Stochastic Analysis, Random Fields and Applications, R.C. Dalang, M. Dozzi, F. Russo, eds., 2013, pp. 435–449]. First, we establish maximum principle for the differential problem. Then, we construct and analyse negativity preserving, flux limited finite difference schemes. Numerical experiments are discussed. [ABSTRACT FROM AUTHOR]

The focus of the present work is a one-dimensional system of weakly coupled degenerate semi-linear parabolic equations of optimal portfolio in a regime-switching with power utility function, suggested in Valdez and Vargiolu [Optimal portfolio in a regime-switching model, in Proceedings of the Ascona '11 Seminar on Stochastic Analysis, Random Fields and Applications, R.C. Dalang, M. Dozzi, and F. Russo, eds., 2013, pp. 435–449]. We extend the model, deriving a new linear parabolic system for indifference option valuation. Then, we investigate the properties of the solution of the systems. Further, we develop and analyse an adequate, flux limited finite difference discretization, which preserves the typical features of the differential problem solution. Numerical experiments are presented and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of non-linear option pricing models. The focus is on the numerical investigation of Delta equation, where the unknown solution is the first spatial derivative of the option value. We construct and analyze monotone and sign-preserving finite difference schemes for the problems. Newton’s and Picard’s iterative procedures for solving the non-linear systems of algebraic equations are proposed. On this base, in order to improve the computational efficiency, we develop fast two-grid algorithms. Numerical experiments, using also Richardson extrapolation in time, are discussed in terms of accuracy, convergence and efficiency. [ABSTRACT FROM AUTHOR]

A system of weakly coupled semilinear parabolic equations of optimal portfolio in a regime-switching model is suggested by Valdez and Vargiolu (2013). To study the effects from the typical features of the system, i.e. the boundary degeneration, exponential non-linearity and mixed derivative, we consider a representative 2D Black–Scholes semilinear equation model. We construct and analyze sign preserving, flux limited finite difference schemes for the scalar problem. Numerical experiments are discussed. [ABSTRACT FROM AUTHOR]

We present a large class of nonlinear models of European options as parabolic equations with quasi-linear diffusion and fully nonlinear hyperbolic part. The main idea of the operator splitting method (OSM) is to couple known difference schemes for nonlinear hyperbolic equations with other ones for quasi-linear parabolic equations. We use flux limiter techniques, explicit–implicit difference schemes, Richardson extrapolation, etc. Theoretical analysis for illiquid market model is given. The numerical experiments show second-order accuracy for the numerical solution (the price) and Greeks Delta and Gamma, positivity and monotonicity preserving properties of the approximations. [ABSTRACT FROM AUTHOR]

We consider a two-asset non-linear model of option pricing in an environment where the correlation is not known precisely, as it varies between two known values. First we discuss the non-negativity of the solution of the problem. Next, we construct and analyze a positivity preserving, flux-limited finite difference scheme for the corresponding boundary value problem. Numerical experiments are analyzed. [ABSTRACT FROM AUTHOR]

We construct a first order in time and second order in space, positivity preserving numerical method for a generalized Hoggard–Whalley–Wilmott, Leland’s model. We develop the hyperbolic–parabolic operator splitting method, using a kernel based algorithm for the parabolic part and van Leer flux limiter approach for the hyperbolic sub-problem. Properties of the proposed algorithms are discussed. Various numerical examples confirm the efficiency of the proposed method and verify the theoretical statements. [ABSTRACT FROM AUTHOR]

Abstract: In this work we consider Cauchy problem for the so called Gamma equation, which can be derived by transforming the fully nonlinear Black–Scholes equation for option price into a quasi-linear parabolic equation for the second derivative (Greek) of the option price V. We develop an efficient positivity preserving explicit numerical method for solving the model problem concerning different volatility terms. We prove that the obtained semi-discretization is positive and the corresponding full approximation also preserves this property, if the time step is restricted. The stability of the difference scheme in norm is shown and the existence of interface curves is investigated numerically. Results of numerical simulations are given and discussed. [Copyright &y& Elsevier]

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed. [ABSTRACT FROM AUTHOR]

Two numerical methods for a one-dimensional haptotaxis model, which exploit the use of van Leer flux limiter, are developed and analyzed. Sufficient conditions time step size and flux limiting are given for such formulation to ensure the non-negativity of the discrete solution and second-order accuracy in space. Another advantage is that we avoid solving large nonlinear systems of algebraic equations. The discrete preservation of total conservation of cell density, concentration, and logarithmic density is also verified for the numerical solution. Numerical results concerning accuracy, convergence rate, positivity, and conservation properties are presented and discussed. Similar approach could be applied efficiently in the corresponding two- and three-dimensional problems. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 [ABSTRACT FROM AUTHOR]

Abstract: In this paper, on the basis of Newton’s method, we propose a fast quasilinearization numerical scheme, coupled with Rothe’s method, for fully nonlinear parabolic equations. General conditions that provide quadratic, uniform and monotone convergence of the quasilinearization method (QLM) of solving fully nonlinear ordinary differential equations that arise on each time level, are formulated and elaborated. The convergence of QLM and its rate are examined numerically, on a simple test example with an exact solution. The first few iterations already provide extremely accurate and stable numerical results. The second goal is to consider three applications of the proposed schemes in financial mathematics. Namely, numerical results for three nonlinear problems of optimal investment are presented and discussed. The numerical experiments of the last problem are based on the data from statistic information of the Bulgarian National Bank and Bratislava Interbank. [Copyright &y& Elsevier]

Abstract: In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms. [Copyright &y& Elsevier]

In this paper we analyse three known finite difference schemes applied to the heat-diffusion equation with semilinear dynamical boundary conditions. We prove that the numerical blow-up times converge to the continuous ones. Also, the number of peaks of the solutions is studied. Numerical experiments are discussed and at the same time, certain interesting properties of the continuous solutions are predicted. [Copyright &y& Elsevier]

We consider one-dimensional systems of weakly coupled degenerate semi-linear parabolic equations for buyer's and writer's pricing of European options under regime-switching with exponential utility function. First, we prove a comparison principle and then we establish a maximum principle for the indifference pricing differential problems. The differential problems are solved numerically by fitted finite volume method with second-order of accuracy. We prove the discrete maximum principle and convergence of the numerical solutions in maximum norm. Numerical results, illustrating the theoretical statements are presented and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel exponential difference scheme for solving non-linear problems arising in variably saturated flow with discontinuous absolute permeability. First, we derive the discretization for a linear convection-diffusion-reaction problem with discontinuous coefficients. Positivity preserving property, stability and convergence of the scheme are studied. Then, the method is implemented to the transport equation and Richards' equation. Various numerical experiments on graded and uniform meshes are presented and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present numerical treatment for a compaction-driven Darcian flow viscoelastic rock magma model. This problem is a strongly coupled system of one quasi-linear parabolic equation and one integro-differential equation for the density and the porosity of the flow. The numerical discretization uses cell-centered finite difference method, combined with semi-implicit and implicit time stepping. Implicit–explicit schemes, as well as implicit–explicit iterative algorithms have been developed to solve the corresponding discrete problems. Some properties (positivity, boundness, conservation) of the numerical solutions are investigated. Convergence study of the iteration processes is also presented. The efficiency and the accuracy of the proposed methods are illustrated numerically by test examples with near-real data. [ABSTRACT FROM AUTHOR]