Space-dependent Source Research Articles

Coupling RBF-based meshless method and Landweber iteration algorithm for approximating a space-dependent source term in a time fractional diffusion equation

The problem of determining a space-dependent source term in a time fractional diffusion equation is considered from the measured data at the final time. To find an approximation of the source term, a methodology involving minimization of the cost functional is applied. Also, in order to construct the Landweber iteration algorithm, an explicit formula for the gradient of the cost functional J is given via the solution of an adjoint problem. The resulting adjoint problem is treated by a radial basis function method for spatial dimension and a finite difference scheme for the time fractional derivative followed by an iterative domain decomposition method to achieve a desired accuracy. In addition, Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of Landweber iteration algorithm are proved. At the end, a numerical example is given to show the validation of the iterative algorithm.

A fast iterative method for identifying the radiogenic source for the helium production-diffusion equation

ABSTRACT In this paper, we consider an inverse source problem for the helium production-diffusion equation. That is to determine a space-dependent source term in the helium production-diffusion equation from a noisy final data. Since the problem is ill-posed, we propose an effective iterative scheme to deal with the problem. More specifically, we use an acceleration technique to speed up a simple iterative scheme. We also give two kinds of convergence rates by using an a priori and a posteriori regularization parameter choice rule, respectively. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

Determination of the Space-Dependent Source Term in a Fourth-Order Parabolic Problem

Determination of the Space-Dependent Source Term in a Fourth-Order Parabolic Problem

Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

This article is devoted to identifying a space-dependent source term in linear parabolic equations. Such a problem is ill posed, i.e., a small perturbation in the input data may cause a dramatically large error in the solution (if it exists). The conditional stability of the solution is analyzed. Based on a convoluting equation method, we can deal with the problem under the a priori parameter choice rule. Meanwhile, a modified version of Morozov’s discrepancy principle is provided to decide on an a posteriori regularization parameter choice strategy and a log-type error estimate is obtained. Two numerical results show that our proposed method works well.

Characteristics of variable plate conditions on the time-dependent flow of polar fluid over an expanding/contracting surface

The present scenario intended for the flow phenomena of micropolar fluid past an expanding/contracting surface is carried out to reveal the impact of a drag coefficient. Free convection due to the inclusion of the buoyant forces along with the radiative heat energy and non-uniform heat source/sink encourages the flow properties. The novelty of the present investigation is the use of variable plate conditions that affect the flow properties greatly. The transformation of the governing flow phenomena is obtained with the use of suitable similarity transformation and numerical treatment based upon Runge-Kutta fourth-order followed by shooting is imposed to get the solution of this transformed nonlinear system. Further, the simulation of the characterizing parameters is obtained and presented via graphs and tables. The major findings are; the enhancement in the axial velocity is characterized by the non-Newtonian behavior of the fluid and both the space and temperature dependent heat source favors for the augmentation in the fluid temperature.

Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems

Abstract We study the uniqueness of some inverse source problems arising in thermoelastic models of type-III. We suppose that the source terms can be decomposed as a product of a time dependent and a space dependent function, i.e. g ⁢ ( t ) ⁢ 𝐟 ⁢ ( 𝐱 ) {g(t)\mathbf{f}(\mathbf{x})} for the load source and g ⁢ ( t ) ⁢ f ⁢ ( 𝐱 ) {g(t)f(\mathbf{x})} for the heat source. In the first inverse source problem, the source 𝐟 ⁢ ( 𝐱 ) {\mathbf{f}(\mathbf{x})} has to be determined from the final in time measurement of the displacement 𝐮 ⁢ ( 𝐱 , T ) {\mathbf{u}(\mathbf{x},T)} , or from the time-average measurement ∫ 0 T 𝐮 ⁢ ( 𝐱 , t ) ⁢ d t {\int_{0}^{T}\mathbf{u}(\mathbf{x},t)\,\mathrm{d}t} . In the second inverse source problem, the source f ⁢ ( 𝐱 ) {f(\mathbf{x})} has to be determined from the time-average measurement of the temperature ∫ 0 T θ ⁢ ( 𝐱 , t ) ⁢ d t {\int_{0}^{T}\theta(\mathbf{x},t)\,\mathrm{d}t} . We show the uniqueness of a solution to these problems under suitable assumptions on the function g ⁢ ( t ) {g(t)} . Moreover, we provide some examples showing the necessity of these assumptions. Finally, we conclude the article by studying two combined problems of determining both sources.

A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equation

This paper is devoted to identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. By the series expression of the solution of the direct problem, the inverse source problem can be formulated by a first kind of Fredholm integral equation. The existence and uniqueness, ill-posedness and a conditional stability in Hilbert scale for the considered inverse problem are provided. We propose a generalized quasi-boundary value regularization method to solve the inverse source problem and also prove that the regularized problem is well-posed. Further, two kinds of convergence rates in Hilbert scale for the regularized solution can be obtained by using an a priori and an a posteriori regularization parameter choice rule, respectively. The numerical examples in one-dimensional case and two-dimensional case are given to confirm our theoretical results for the constant coefficients problem. We also propose a finite difference method based on a variant of L1 scheme to solve the regularized problem for the variable coefficients problem and give its convergence rate. One finite difference method based on a convolution quadrature is provided to solve the regularized problem for comparison. The numerical results for three examples by two algorithms are provided to show the effectiveness and stability of the proposed algorithms.

Recovering a source term in the higher-order pseudo-parabolic equation via cubic spline functions

In this paper, we considered an inverse problem of recovering the space-dependent source coefficient in the third-order pseudo-parabolic equation from final over-determination condition. This inverse problem appears extensively in the modelling of various phenomena in physics such as the motion of non-Newtonian fluids, thermodynamic processes, filtration in a porous medium, etc. The unique solvability theorem for this inverse problem is supplied. However, since the governing equation is yet ill-posed (very slight errors in the final input may cause relatively significant errors in the output source term), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown force term. The third-order pseudo-parabolic problem is discretized using the Cubic B-spline (CB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. Both perturbed data and analytical solutions are inverted. Numerical outcomes are reported and discussed. The computational efficiency of the method is investigated by small values of CPU time. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.

Identification of time-varying source term in time-fractional diffusion equations

This paper is concerned with the inverse problem of determining the time and space dependent source term of diffusion equations with constant-order time-fractional derivative in $(0,2)$. We examine two different cases. In the first one, the source is the product of two spatial and temporal terms, and we prove that both of them can be retrieved by knowledge of one arbitrary internal measurement of the solution for all times. In the second case, we assume that the first term of the product varies with one fixed space variable, while the second one is a function of all the remaining space variables and the time variable, and we show that both terms are uniquely determined by two arbitrary lateral measurements of the solution over the entire time span. These two source identification results boil down to a weak unique continuation principle in the first case and a unique continuation principle for Cauchy data in the second one, that are preliminarily established. Finally, numerical reconstruction of spatial term of source terms in the form of the product of two spatial and temporal terms, is carried out through an iterative algorithm based on the Tikhonov regularization method.

Inverse Problems for Diffusion Equation with Fractional Dzherbashian-Nersesian Operator

Abstract Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives named after Riemann-Liouville, Caputo and Hilfer are shown to be special cases of the earlier one. The expression for Laplace transform of fractional Dzherbashian-Nersesian operator is constructed. Inverse problems of recovering space dependent and time dependent source terms of a time fractional diffusion equation with involution and involving fractional Dzherbashian-Nersesian operator are considered. The results on existence and uniqueness for the solutions of inverse problems are established. The results obtained here generalize several known results.

Inverse problems for a multi-term time fractional evolution equation with an involution

This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The fractional derivatives are defined in Caputo's sense. The ISPs are proved to be ill-posed in the sense of Hadamard. Recovering a space dependent source term from over-specified data given at some time constitute the first ISP, while in the second ISP determination of a time dependent component of the source term is considered when over-specified condition of integral type is given. The solution of ISPs are constructed by using Fourier's method. The time-dependent components of the solutions are presented in terms of the multinomial Mittag-Leffler function. Under certain conditions, the solutions of ISPs for the multi-term time-fractional evolution equation are shown to be classical solutions. In addition, some particular examples are formulated to illustrate the obtained results for the ISPs.

Uniqueness for Inverse Source Problems of Determining a Space-Dependent Source in Time-Fractional Equations with Non-Smooth Solutions

In this contribution, we investigate an inverse source problem for a fractional diffusion and wave equation with the Caputo fractional derivative of the space-dependent variable order. More specifically, we discuss the uniqueness of a solution when reconstructing a space-dependent source from a time-averaged measurement, or a final in time measurement. Weakly singular solutions are included in the class of admissible solutions. The obtained results are also valid if the order of the fractional derivative is constant.

A Non-iterative Inverse Method for the Identification of Time and Space-Dependent Heat Sources of Multi-dimensional GMs by Boundary Temperature Data

Background: Detection of heat sources is frequently encountered in many fields of sci-ence and engineering and plays a significant role in monitoring and control of many engineering thermal systems. Objective: The objective of this paper is to estimate the space and time-dependent heat sources in multi-dimensional functionally graded materials using boundary temperature data. Methods: First, the dimensionless temperature at the measurement points on the boundary is ob-tained by a direct process. Then, the objective function is obtained by a series of matrix operations, and the relationship between the temperature at the measurement points and the unknown parame-ters is established. After that, the strength of the heat sources can be inversed directly by the least-square error method, then the location and number of the heat sources can be obtained directly from the strength distribution of heat sources. The coefficient expansion method and truncated singular value decomposition method are applied to reduce the ill-posed degree of the inverse process. Results: Through the analysis of typical 2-dimensional and 3-dimensional examples, the influence of various factors such as the type of basis functions, the circular supported radius and the meas-urement noise on the inversion results is discussed, which shows that this method can identify the strength, location and number of heat sources of FGMs well. Conclusion: A non-iterative inverse method based on precise integration finite element method and least-square error method is established to estimate the strength, location and number of the un-known heat sources of functionally graded materials using boundary temperature data.

Activation energy effect on the magnetized-nanofluid flow in a rotating system considering the exponential heat source

The variation of properties with temperature have many practical applications in the area of chemical engineering and metallurgy, in the process of extrusion, the materials moving between a feed roll and on carrier belt. Here our intention is to explore the consequences of temperature dependent viscosity and variable thermal conductivity on flow of nanoliquid in a rotating system. Fluid between parallel plates is taken. Energy expression incorporates the exponential heat source. This analysis also explores the attributes of modified Arrhenius energy function with chemical reaction. Boundary layer problems have been computed via NDSolve approach. The obtained solutions are sketched for various estimations of embedded variables of interest. Results pointed out that both axial and transverse velocities show decaying behavior for higher rotation parameter. Thermal field boosts up with an enhancement of thermophoretic and space dependent heat source variables. Moreover the rate of mass transport decays through dimensionless activation energy.

On the simultaneous reconstruction of boundary Robin coefficient and internal source in a slow diffusion system

Consider an inverse problem of simultaneous reconstruction of boundary impedance coefficient and space-dependent source term from the final measurement data for a slow diffusion system, which is governed by a diffusion equation with time-fractional order derivative and Robin boundary condition. We firstly prove the uniqueness of this inverse problem by the maximum principle for the slow diffusion system. Then a regularizing scheme combining the mollification method and the Tikhonov regularization is proposed to recover the two unknowns, with a rigorous analysis on the choice strategies for the regularizing parameters and the error estimates on the regularizing solutions, revealing the error propagation effects due to recovering the boundary Robin coefficient firstly. Numerical examples are presented to illustrate the validity of the proposed method and support our theoretical analysis.

An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

On the convergence rate of an improved quasi-reversibility method for an inverse source problem of a nonlinear parabolic equation with nonlocal diffusion coefficient

This paper investigates the identification of the space-dependent source term in a nonlinear parabolic equation with nonlocal diffusion coefficient, which means that the diffusion process modelled by the above nonlinear parabolic equation depends on the population itself in biology and ecology fields. The nonlinear expansion of the solution to direct problem is established firstly, in terms of which an improved quasi-reversibility method for the inverse problem is proposed. Then to overcome the difficulty of nonlinear and nonlocal of the problem, the reasonable convergence rate of the regularizing solution under the bounded conditions is derived rigorously, with the choice strategy for the regularizing parameter.

Finite-time blow-up in a higher-dimensional quasilinear parabolic-elliptic chemotaxis system with space dependent logistic source

This paper deals with a quasilinear parabolic-elliptic chemotaxis system with space dependent logistic source{ut=∇⋅(φ(u)∇u)−χ∇⋅(ψ(u)∇v)+κ(|x|)u−μ(|x|)uθ,0=Δv−m(t)|Ω|+u,m(t):=∫Ωu(⋅,t) under homogeneous Neumann boundary conditions in a ball Ω=BR(0)⊂Rn, n≥2, where R>0 for θ≥1, χ>0, sufficiently smooth functions κ,μ:[0,R]→[0,∞) and the nonlinear diffusivity φ(u) and chemosensitivity ψ(u) are supposed to extend the prototypesφ(u)=(u+1)−p,ψ(u)=uq,p≥0,q∈R. We proved that whenever μ′,−κ′>0 and μ(s)≤μ1sα for all s∈[0,R] and some μ1>0, α≥n(θ−1), as well as 0≤p<1,1≤q<2−p, then for all ∫Ωu0>4(n2−n)n1−p−qa(n)qχ, there exist nonnegative radially symmetric initial data such that the corresponding solutions blow up in finite time, where a(n) is the volume of the unit ball in Rn.

Unsteady mixed convection flow of magneto-Williamson nanofluid due to stretched cylinder with significant non-uniform heat source/sink features

This analysis reports an unsteady and incompressible flow of Williamson nanoliquid in presence of variable thermal characteristics are persuaded by a permeable stretching cylinder. The flow field investigation is established with the effect of mixed convection and non-uniform heat source/sink on flow and heat transfer. On the cylinder surface, the analysis is inspected with utilization of zero mass flux constraints. By using the appropriate similarity variables, the framed equations for the energy, momentum and mass is converted into non-linear ODEs. The numerical communication of the boundary value problem is successfully implemented using a computer algorithm programmed into the fifth Runge-Kutta scheme. Additionally, the wall shear factor and rate of heat transfer are calculated in two different cases namely, with curvature and without curvature. In addition, the results obtained are confirmed by making comparisons with previously published articles and we found an excellent match that guarantees the indemnity of current communication. A comprehensive change in velocity, temperature and concentration is examined for involved parameters like local Weissenberg number, space dependent heat source constant, magnetic number, curvature constant, thermophoretic parameter, buoyancy parameter, Brownian motion parameter, Prandtl number, Schmidt number, unsteadiness parameter, reaction rate parameter, activation energy parameter and temperature difference parameter. A reduction in velocity is observed for unsteady parameter and buoyancy constant. An enhanced nanofluid temperature is noted for space dependent heat source parameter, time dependent heat source parameter and unsteady parameter. Moreover, the nanofluid concentration is increases for temperature difference parameter while reverse observations are noticed for chemical reaction rate.

Second Law Analysis of MHD Micropolar Fluid Flow through a Porous Microchannel with Multiple Slip and Convective Boundary Conditions

The impact of space dependent heat source in the transport of micropolar fluid in the existence of magnetic dipole, Joule heating, viscous heating, thermal radiation, hydrodynamic slips and convective condition effects has been numerically investigated. The dimensioned governing equations are non-dimensionlzed by using dimensionless variables then non-dimensional forms of the corresponding equations are than tackled by the versatile Finite Element Method (FEM). The effects of pertinent physical parameters characterize the flow phenomena are presented through graphs and discussed. It is found that, the impact of thermal based heat source advances the heat transfer characteristics significantly than exponential to space dependent. The thermal performance can be improved through the effects of magnetic dipole, viscous heating, Joule heating and convective condition. Further, the present numerical results are compared with previously published results in the literature as a limiting case of the considered problem and found to be in good agreement with the existing results.