Space-dependent heat source determination problem with nonlocal periodic boundary conditions

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.

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Determination of a spacewise dependent heat source

The nonlocal boundary conditions (NLBCs) for high-order finite-difference parabolic equations (PEs) are obtained by Z transformation of the discrete PE in a homogeneous medium. The considered NLBCs include the free-space radiation condition, possibly with a density jump at the NLBC interface, the NLBC at an arbitrary impedance interface, and the NLBCs for sources and the starting field beyond the NLBC interface. The derivation is presented for the multiterm Padé PE model OWWE (one-way wave equation), but the developed technique is applicable to a broad class of finite-difference PEs. The obtained NLBCs are exact for the given finite-difference scheme. They are not limited by the order of Padé approximation or by the PE steps in range and depth. The NLBC convolution coefficients are calculated by the numerical inverse-Z transformation. The accuracy and performance of the algorithm are analyzed for several benchmark problems. The solution is robust for the range steps over 25 wavelengths and/or the approximations up to the tenth order. The NLBCs are faster and more accurate for large steps because fewer previous range steps contribute to the convolution. Precomputation of the NLBC coefficients may be required in time-demanding applications. The results are compared with earlier proposed NLBCs for high-order PEs.

Transportation planning and path planning are hot research topics which can be evidenced by their frequent and repeated presence in the benchmark test in International Planning Competition. However, most of the benchmark test examples only consider the static planning problems, whose state transitions are only caused by the actions of the planners. In this paper, we will discuss the dynamic version and introduce the formalization of the Period/Dynamic planning problem. Then, we will prove that the computational complexity is EXPSPACE-Complete.

Numerical solution of an inverse source problem for a time-fractional PDE via direct meshless local Petrov–Galerkin method

In this paper, a numerical method is proposed to solve the Fredholm-Volterra fractional integro-differential equation with nonlocal boundary conditions by using Haar wavelets. A collocation based Galerkin's method is applied by using Haar wavelets as basis functions over the interval [0, 1). It converts the Fredholm-Volterra fractional integro-differential equation into a system of m linear equations. On incorporating q nonlocal boundary conditions, it leads to further q equations. All together it will give a system of (m + q) linear equations in (m + q) variables which can be solved. A variety of test examples are considered to illustrate the proposed method. The actual error is also measured with respect to a norm and the results are validated through error bounds.

ABSTRACTIn this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.

Laplacian decomposition and the boundary element method for solving Stokes problems

Advanced Dynamics has developed an Integrated Variable-Fidelity Tool Set – ASTE-P for Modeling and Simulation of Aero-Servo-Thermo-Elasticity and Propulsion (ASTE-P) of Aerospace Vehicles Ranging from Subsonic to Hypersonic Flights. The ASTE-P software tool set is developed in the state-of-the-art and commercial standard and enables accurate integration and tight/loose coupling of the fluid, structural and control field simulation with variable fidelity options available. The ASTE-P software tool can be applicable to modeling and simulation of aerodynamics, structural dynamics, flight control and propulsion dynamics as well as more important interactions of these dynamics. All flight regimes from subsonic to hypersonic are covered. The interface of structural/control surface motion and vibration modes with fluid flows is modeled using either unified particle-based methods (MPM/PPM) or FVM/FEM based tight/loose coupled fluid/structure solving algorithms. The Euler and RANS based solvers enable the accurate prediction of nonlinear coupled fluidstructure problems in aeroelasticity and the embedded fluid and structural dynamics solvers make the software self-contained and not require the integration of two separate third-party fluid and structure solvers for aeroelastic modeling and simulation. Three levels of simulation environments are included in ASTE-P tool set: (1) the bottom level of high-fidelity and full-order simulation environment, (2) middle level of fast analysis and evaluation environment which is based upon reduced order models (ROM) and provides fast turn-around time, and (3) top level of rapid design and optimization environment. This paper provides the comprehensive validation and verification (V&V) of ASTE-P toolset recently done at Advanced Dynamics which resulted in over 600 pages report for over 40 benchmark test examples that were selected from industry and academic research and design . The validation cases presented in this paper indicate that the capability of ASTE-P is compatible or even more powerful than the similar software that is currently available.

Application of meshfree methods for solving the inverse one-dimensional Stefan problem

A diffuse interface–lattice Boltzmann model for conjugate heat transfer with imperfect interface

This paper is devoted to a Bitsadze-Samarskii type overdetermined multipoint nonlocal boundary value problem (NBVP). This inverse problem is reduced to an auxiliary multipoint NBVP. Stability estimates for the solution of the auxiliary NBVP are established. The finite difference method is applied to get the first and second order of accuracy of approximate solutions of the abstract overdetermined problem. Stability estimates for the solution of difference problems are proved. Then the established abstract results are applied to get stability estimates for the solution of the Bitsadze-Samarskii type overdetermined elliptic multidimensional differential and difference problems with multipoint nonlocal boundary conditions (NBVC). Finally, numerical results with explanation on the realization for two dimensional and three dimensional elliptic overdetermined multipoint NBVPs in test examples are presented.

Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions

Partial differential equations are central to describing many physical phenomena. In many applications these phenomena are observed through a sensor network, with the aim of inferring their underlying properties. Leveraging from certain results in sampling and approximation theory, we present a new framework for solving a class of inverse source problems for physical fields governed by linear partial differential equations. Specifically, we demonstrate that the unknown field sources can be recovered from a sequence of, so called, generalised measurements by using multidimensional frequency estimation techniques. Next we show that---for physics-driven fields---this sequence of generalised measurements can be estimated by computing a linear weighted-sum of the sensor measurements; whereby the exact weights (of the sums) correspond to those that reproduce multidimensional exponentials, when used to linearly combine translates of a particular prototype function related to the Green's function of our underlying field. Explicit formulae are then derived for the sequence of weights, that map sensor samples to the exact sequence of generalised measurements when the Green's function satisfies the generalised Strang-Fix condition. Otherwise, the same mapping yields a close approximation of the generalised measurements. Based on this new framework we develop practical, noise robust, sensor network strategies for solving the inverse source problem, and then present numerical simulation results to verify their performance.