Space-dependent Source Research Articles

Simultaneous determination of the space-dependent source and initial value for a two-dimensional heat conduction equation

Simultaneous determination of the space-dependent source and initial value for a two-dimensional heat conduction equation

Filter Regularization Method for Inverse Source Problem of the Rayleigh–Stokes Equation

In this paper, we consider a problem of recovering a space-dependent source term for the Rayleigh–Stokes equation, where the additional data is the observation at a final moment $t = T$, which is ill-posed in the sense of Hadamard. Firstly, the uniqueness, ill-posedness and the conditional stability of inverse source problem is given. Next, we develop a filter regularization method to overcome the ill-posedness of the problem. Under reasonable a priori bound assumption about the source function, a Hölder-type error estimate of the regularized solution is proved for a priori regularization parameter choice rule. Furthermore, a logarithmic-type error estimate between the exact solution and the regularized solution is established based on a posteriori regularization parameter choice rule.

Operational calculus for Hilfer-Prabhakar operator Applications to inverse problems

Mikusiński's operational calculus has proven to be a powerful tool for tackling various fractional order differential equations. Notably, it has been extended to encompass the Hilfer-Prabhakar fractional order operator. By employing this operational calculus, we have been able to recover and connect previously derived results pertaining to the Hilfer, Riemann-Liouville, and Caputo operators. Furthermore, we address two inverse problems that involve determining space and time dependent source terms and diffusion concentration. Using our operational calculus results, we obtain series solutions for these inverse problems. It is essential to note that the inverse problems belong to the class of ill-posed problems according to Hadamard's definition. To establish the validity and uniqueness of the solutions to both inverse problems, we rigorously prove the existence and uniqueness results. Additionally, we provide specific examples of inverse problems, thereby illustrating the practical applications and versatility of the operational calculus approach.

Heat transfer analysis of magnetized Cu-Ag-H2O hybrid nanofluid radiative flow over a spinning disk when the exponential heat source and Hall current are substantial: Optimization and sensitivity analysis

The main motive of the instigated mathematical model is to observe the impact of Hall current on the hybrid nanofluid flow over a disk that is rotating. The copper and silver metal nanoparticles have been considered with volume fraction φ1=φ2=0.01(0.01)0.04 and are suspended in water to form the hybrid nanofluid. Diverse characteristics like magnetic field, thermal radiation, and (ESHS) exponential space dependent heat source are incorporated to investigate the nature of the flow. The present mathematical model is initiated with partial derivative equations (PDEs) which are redrafted as ordinary derivative equations (ODEs) with appropriate transformations of similarity. The results are attained through a blend of the Runge-Kutta method, shooting procedure, and the influences of parameters on the flow of nanofluid and hybrid nanofluid are compared and illustrated both as tables and graphs. The present numerical research is unique because by employing a complete quadratic CCD framework using the RSM strategy, the sensitivity and optimization analysis of the heat transmission improvement for the volume fraction, ESHS, and thermal radiation parameters have been performed. The R-squared and adjusted R-Squared are obtained as 100%. The residual graphs and contour diagrams of the same are also shown. The current study establishes that the Hall parameter increases the radial velocity, but it also controls the energy and cross-radial velocity. The rate of heat transmission is increased by thermal radiation even at low levels of ESHS. The rate of heat transmission is more sensitive (0.024670) to the volume fraction of the hybrid nanofluid when ESHS is at an intermediate level. The lowest sensitivity (-1.269967) value towards ESHS is observed For thermal radiation and ESHS parameter values, the heat transmission rate of the mono nanofluid is not as great as that of hybrid nanofluid. The current study finds applications in the generation of hydroelectric power, air cleansing and rotating equipment, healthcare devices, and many other industries.

The Landweber Iterative Regularization Method for Identifying the Unknown Source of Caputo-Fabrizio Time Fractional Diffusion Equation on Spherically Symmetric Domain

In this article, the inverse problem for identifying the space-dependent source of time fractional diffusion equation on a spherically symmetric domain with Caputo–Fabrizio fractional derivative is discussed. This problem is a typical ill-posed problem and the Landweber iterative regularization method is used to obtain the approximation solution. The convergent error estimates under a priori regularization parameter choice rule and a posteriori regularization parameter choice rule are given, respectively. The numerical examples are given to show the effectiveness of the Landweber iterative regularization method.

Landweber Iterative Method for an Inverse Source Problem of Time-Space Fractional Diffusion-Wave Equation

Abstract In this paper, we apply a Landweber iterative regularization method to determine a space-dependent source for a time-space fractional diffusion-wave equation from the final measurement. In general, this problem is ill-posed, and a Landweber iterative regularization method is used to obtain the regularization solution. Under the a priori parameter choice rule and the a posteriori parameter choice rule, we give the error estimates between the regularization solution and the exact solution, respectively. Some numerical results in the one-dimensional and two-dimensional cases show the utility of the used regularization method.

Identifying a Space-Dependent Source Term and the Initial Value in a Time Fractional Diffusion-Wave Equation

This paper is focused on the inverse problem of identifying the space-dependent source function and initial value of the time fractional nonhomogeneous diffusion-wave equation from noisy final time measured data in a multi-dimensional case. A mollification regularization method based on a bilateral exponential kernel is presented to solve the ill-posedness of the problem for the first time. Error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical experiments of interest show that our proposed method is effective and robust with respect to the perturbation noise in the data.

An iterative method based on Nesterov acceleration for identifying space-dependent source term in a time-fractional diffusion-wave equation

In this paper, we investigate an inverse problem of identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. Since the problem is ill-posed, we propose an iterative method based on the Nesterov acceleration strategy to deal with it. Two kinds of convergence rates for the regularized solution are given under both the a priori and a posteriori regularization parameter choice rules. Compared with the classical regularization methods, our method is easy to implement with a fewer number of iterations and can always yield the order optimal error estimates provided the iteration parameter is chosen large enough. Some numerical examples including one-dimensional and two-dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.

EMHD flow of second-grade fluid through a heated permeable disk with space dependent heat source

In this article, von Kárman MHD viscoelastic fluid flow with second-grade model through a heated permeable disk on electric field is examined thoroughly. The effects of viscous dissipation and elasticity on fluid motion around the disk, temperature and space-dependent heat source/sink are also examined. Governing partial differential equations are transmitted into ordinary differential equations with similarity functions. Later, nonlinear system of ordinary differential equations was solved with the help of Galerkin finite element method. Accuracy and validity of the method were noticed by comparing our results with the previous literature. Simulations are performed in order to capture the dynamics of the physical situations against the variation of the pertinent parameters. Nature of heat flux and stresses of the physical parameters are discussed. Physical influence of heat flux, heat transfer is exhibited graphically at the surface of disk and increases when the values of Prandtl number are increased, whereas there is a decreasing trend in the rate of heat transfer when the value of Eckert number is increased.

Simultaneous Inversion of the Space-Dependent Source Term and the Initial Value in a Time-Fractional Diffusion Equation

Abstract The inverse problem for simultaneously identifying the space-dependent source term and the initial value in a time-fractional diffusion equation is studied in this paper. The simultaneous inversion is formulated into a system of two operator equations based on the Fourier method to the time-fractional diffusion equation. Under some suitable assumptions, the conditional stability of simultaneous inversion solutions is established, and the exponential Tikhonov regularization method is proposed to obtain the good approximations of simultaneous inversion solutions. Then the convergence estimations of inversion solutions are presented for a priori and a posteriori selections of regularization parameters. Finally, numerical experiments are conducted to illustrate effectiveness of the proposed method.

FRACTIONAL TIKHONOV REGULARIZATION METHOD FOR SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL DATA IN A TIME-FRACTIONAL DIFFUSION EQUATION

This paper is concerned with the problem of identifying the space-dependent source term and initial value simultaneously for a time-fractional diffusion equation. The inverse problem is ill-posed, and the idea of decoupling it into two operator equations is applied. In order to solve this inverse problem, a fractional Tikhonov regularization method is proposed. Furthermore, the corresponding convergence estimates are presented by using the a priori and a posteriori parameter choice rules. Several numerical examples compared with the classical Tikhonov regularization are constructed for verifying the accuracy and efficiency of the proposed method.

Existence and regularity in inverse source problem for fractional reaction-subdiffusion equation perturbed by locally Lipschitz sources

<p style='text-indent:20px;'>In this paper, we consider an inverse problem of determining a space-dependent source in the time fractional reaction-subdiffusion equation involving locally Lipschitz perturbations, where the additional measurements take place at the terminal time which are allowed to be nonlinearly dependent on the state. By providing regularity estimates on both time and space of resolvent operator and using local estimates on Hilbert scales, we establish some results on the existence and uniqueness of solutions and the Lipschitz type stability of solution map of the problem under consideration. In addition, when the input data take more regular values, we obtain results on regularity in time of solution for both the direct linear problem and the inverse problem above.</p>

Identifying a space-dependent source term in distributed order time-fractional diffusion equations

<p style='text-indent:20px;'>The aim of this paper is to investigate an inverse problem of recovering a space-dependent source term governed by distributed order time-fractional diffusion equations in Hilbert scales. Such a problem is ill-posed and has important practical applications. For this problem, we propose a general regularization method based on the idea of the filter method. With a suitable source condition, we prove that the method is of optimal order under various choices of regularization parameter. One is based on the a priori regularization parameter choice rule and another one is the discrepancy principle. Finally, the capabilities of our method are illustrated by both the Tikhonov and the Landweber method.</p>

Fast matrix exponential-based quasi-boundary value methods for inverse space-dependent source problems

<abstract><p>In this paper, we study the well-established quasi-boundary value methods for regularizing inverse state-dependent source problems, where the convergence analysis of three typical cases is presented in the framework of filtering regularization method under suitable source conditions. Interestingly, the quasi-boundary value methods can be interpreted as certain Lavrentiev-type regularization, which was not known in literature. As another major contribution, efficient numerical implementation based on matrix exponential in time is developed, which shows much improved computational efficiency than MATLAB's backslash solver based on the all-at-once space-time discretization scheme. Numerical examples are reported to illustrate the promising computational performance of our proposed algorithms based on matrix exponential techniques.</p></abstract>

Determination of a space-dependent source in the thermal-wave model of bio-heat transfer

Determination of a space-dependent source in the thermal-wave model of bio-heat transfer

A direct parallel-in-time quasi-boundary value method for inverse space-dependent source problems

Inverse source problems arise often in real-world applications, such as localizing unknown groundwater contaminant sources. Being different from Tikhonov regularization, the quasi-boundary value method has been studied as an effective way for regularizing such inverse source problems, which was shown to achieve an optimal order convergence under suitable assumptions. However, fast direct or iterative solvers for the resulting large-scale all-at-once linear systems have been rarely studied in the literature. In this work, we propose and analyze a modified quasi-boundary value method that leads to a fast diagonalization-based parallel-in-time (PinT) direct solver, which can achieve a dramatic speedup in CPU times when compared with MATLAB’s sparse direct solver. In particular, the time-discretization matrix B is shown to be diagonalizable, and the condition number of its eigenvector matrix V is proven to exhibit only quadratic growth, which guarantees that the roundoff errors due to diagonalization is well-conditioned. Several 1D and 2D examples are presented to demonstrate the very promising computational efficiency of our proposed method, where the CPU times in 2D cases can be speeded up by three orders of magnitude.

Stability of the time-dependent identification problem for delay hyperbolic equations

Time-dependent and space-dependent source identification problems for partial differential and difference equations take an important place in applied sciences and engineering, and have been studied by several authors. Moreover, the delay appears in complicated systems with logical and computing devices, where certain time for information processing is needed. In the present paper, the time-dependent identification problem for delay hyperbolic equation is investigated. The theorems on the stability estimates for the solution of the time-dependent identification problem for the one dimensional delay hyperbolic differential equation are established. The proofs of these theorems are based on the Dalambert’s formula for the hyperbolic differential equation and integral inequality.

Experience of non-uniform heat source/sink on the flow of micropolar nanofluid fluid with heterogeneous and homogenous chemical reaction

Due to the varied applications in industries, chemical reaction has a significant role in the perfect shape and size of the product that is dependent upon the proper cooling of the system. Therefore, the existing analysis reveals the impression of non-uniform heat source/sink on the micropolar nanofluid through a nonlinear vertical stretching surface for the interaction of heterogeneous and homogeneous chemical reactions. Oxide nanoparticle embedding with the traditional water is considered by introducing operational properties such as thermal conductivity and viscosity. The free convective water-based nanofluid past through a permeable medium interacts with the transverse magnetic field and the thermal buoyancy enriches the present discussion. Further, the convective heat transfer condition is useful to enhance the thermal properties as well. Imposing suitable transformations the present problem turned into ordinary and solved by employing the bvp5c built-in MATLAB code. Further, the important results are; inclusion of particle concentration favors for the enhancement in the shear stress as well as the heat transfer rate in conjunction with the resistive forces such as magnetic and porosity but the impact is reversed in case of couple stress coefficient however, retardation in the fluid temperature occurs for the increasing space and temperature dependent heat source.

Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative

This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source problem can be given. Further, a fractional Tikhonov regularization method was applied to regularize the inverse source problem. In the theoretical results, we propose a priori and a posteriori regularization parameter choice rules and obtain the convergence estimates.

Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model

Abstract This paper is devoted to recovering simultaneously the fractional order and the space-dependent source term from partial Cauchy’s boundary data in a multidimensional time-fractional diffusion equation. The uniqueness of the inverse problem is obtained by employing analytic continuation and the Laplace transform. Then a modified non-stationary iterative Tikhonov regularization method with a regularization parameter chosen by a sigmoid-type function is used to find a stable approximate solution for the source term and the fractional order. Numerical examples in one-dimensional and two-dimensional cases are provided to illustrate the efficiency of the proposed algorithm.