The increasing global demand for energy, environmental concerns, and rapid technological advancements have made future energy planning a challenging multi-criteria decision-making (MCDM) problem. The selection of appropriate energy strategies requires simultaneously considering economic, environmental, technological, and policy-related factors under uncertain and evolving conditions. Conventional MCDM approaches are often inadequate for handling complex uncertainty and temporal variations in decision information. To address these challenges, this study proposes a Temporal-Aware Complex q-Rung Orthopair Fuzzy Aczel–Alsina CRITIC-WASPAS (Cq-ROF-AA-CRITIC-WASPAS) framework for future energy planning. The proposed framework employs Cq-ROFSs to represent uncertain decision information, while the proposed Aczel–Alsina aggregation operators effectively fuse temporal decision matrices by assigning greater importance to recent evaluations. The Criteria Importance Through Intercriteria Correlation (CRITIC) method objectively determines attribute weights by considering contrast intensity and inter-criteria correlation, whereas the Weighted Aggregated Sum Product Assessment (WASPAS) method ranks the energy alternatives through a hybrid additive-multiplicative evaluation strategy. A hypothetical future energy planning case study is presented to demonstrate the applicability of the proposed mathematical model using evaluation criteria related to economic viability, environmental sustainability, technological preparedness, energy security, and policy flexibility. Furthermore, comparative and sensitivity analyses are conducted to evaluate the effectiveness and stability of the proposed framework. The obtained results demonstrate that the proposed framework provides a flexible and systematic approach for handling complex uncertainty and temporal decision information, offering reliable decision support for future energy planning and other uncertain MCDM applications.
Linear-quadratic (LQ) optimal control problems have been widely studied under symmetric information and regularity assumptions. However, in many practical systems, multiple controllers operate with different information sets, and the standard regularity condition may fail due to singular weighting matrices. This paper investigates a stochastic irregular LQ optimal control problem with dual controllers and two-layer asymmetric partial observations, in which one controller has strictly less information than the other. The main contribution is the derivation of both the solvability conditions and the explicit feedback-form solutions for this problem. To this end, we reformulate the optimization problem as a system of forward-backward stochastic differential equations (FBSDEs) that naturally captures the asymmetric information structure. By solving these FBSDEs via four interconnected Riccati equations, we obtain explicit optimal controllers for both the regular and irregular cases. In the irregular case, where the standard Riccati equation is not directly solvable, we introduce auxiliary Riccati equations and derive additional existence conditions involving singular matrix decompositions. A detailed technical comparison with existing results confirms that our solution is not a straightforward extension, as prior works either addressed irregular LQ control under full information or asymmetric information under regular conditions. This work provides a unified framework for optimal control problems where information asymmetry and singularity arise simultaneously.
This work puts forward a family of operators, which we term enriched ∇-preserving contractions, acting on a normed space carrying an arbitrary binary relation ∇. The family gathers several familiar classes under one roof: Banach contractions, order-theoretic contractions, and certain nonexpansive maps, among them averaged maps whose fixed points lie beyond the reach of ordinary contraction arguments; all arise through suitable choices of the constants and of the relation. Working in this relational framework, we show that any such operator admits a fixed point, even though its defining contractive estimate is demanded only on ∇-related pairs rather than across the whole space. A device used repeatedly is the symmetry of the norm: it forces ∇ and its symmetric closure to act compatibly with the operator, and this is what keeps the fixed-point conclusions meaningful. The fixed points themselves are located through the Krasnoselskii iteration, and a path criterion formulated in the symmetric closure of ∇ then yields their uniqueness; an almost contraction variant of the scheme is treated as well. By way of application, the theory settles the solvability, and under the stated hypotheses the uniqueness, of solutions to a Caputo fractional boundary value problem with an integral condition and to a nonlinear matrix equation. Numerical experiments over several groups of Lipschitz coefficients confirm the convergence threshold predicted by the theory, and the iteration is compared with Riemann-Hilbert analysis, Lie group integrators, physics-informed neural networks, and neural symbolic derivation tools, so that strengths and limits of the framework can be judged objectively.
The notion of a double controlled metric type space was introduced by Abdeljawad et al. as a broadening of the controlled metric type spaces of Mlaiki et al., and shortly afterwards Aydi et al. formulated the new extended b-metric spaces to widen the extended b-metric spaces of Kamran et al. Building on these developments, the present work advances the class of (θ, α)-metric spaces, a structure that simultaneously subsumes b-metric spaces, new extended b-metric spaces, controlled metric type spaces and double controlled metric type spaces. This is accomplished by attaching two control functions θ(ζ, µ, s) and α(ζ, µ, s) to the terms situated on the right-hand side of the triangle inequality in the axioms defining a (θ, α)-metric space. We first prove, through explicit propositions, that each of the aforementioned structures is recovered as a special case under an appropriate choice of θ and α, and we supply non-trivial examples confirming that the new class is strictly broader than a b-metric space. Within this setting, a collection of fixed point theorems is derived for φ-contractive and Reich–Ćirić–Rus–type operators, together with their corollaries. Finally, two applications are presented: the principal theorem is first used to guarantee that a nonlinear algebraic equation possesses a unique solution, and it is then applied to establish the existence and uniqueness of a solution to a nonlinear Caputo fractional boundary value problem, thereby strengthening the connection of the theory with differential equations.
We introduce two classes of operators on normed spaces, the enriched polynomial contractions and the almost enriched polynomial contractions. Both classes contain the enriched contractions of Berinde and Păcurar and the polynomial-type contractions of Jleli, Pacurar and Samet as special cases. In Banach spaces, we prove existence, uniqueness, and convergence theorems for the fixed points, first under continuity of the operator and then under the weaker assumption of Picard continuity, with the fixed points approximated by Krasnoselskii-type iteration. Several classical results, including the Banach contraction principle and the enriched contraction theorem, are recovered as special cases, and several worked examples are given. As the main application, we consider a nonlinear Caputo fractional boundary value problem of order 2 < N ≤ 3. Writing it as a fixed-point equation and assuming an explicit Lipschitz condition, we prove that it has a unique solution. We then examine the constructive side of this result numerically: on a manufactured problem with a known exact solution, we run the associated fractional-quadrature Krasnoselskii iteration, check that its measured geometric rate stays below the certified contraction factor, and recover the same solution with a neural fixed-point solver, a network trained to satisfy the discretized fixed-point equation of the same operator, and we relate the accuracy of both solvers to the underlying quadrature error. The numerical results are consistent with the theory and illustrate its use on a class of nonlinear fractional models of this form.
CO₂ flooding can simultaneously enhance oil recovery and enable geological carbon storage. This study develops a tNavigator-based coupled wellbore-reservoir CO₂ flooding model to examine tubing-head-to-downhole pressure conversion and early gas breakthrough through a high-permeability channel. A three-dimensional isothermal compositional model represents reservoir flow, while vertical flow performance (VFP) tables constructed using the Beggs-Brill correlation describe wellbore multiphase flow and pressure conversion. The coupled VFP relationships, well flow equations, and reservoir mass-conservation equations are solved using the adaptive implicit method (AIM) implemented in tNavigator, enabling bidirectional interactions among wellbore pressure variation, downhole boundary conditions, and reservoir dynamic parameters. A continuous high-permeability channel between injectors and producers is incorporated to characterize reservoir heterogeneity. Simulations under fixed injection rate and fixed tubing-head pressure conditions are conducted to analyze the influences of injection and production rates, porosity and streak permeability on CO₂ migration and flooding performance. The results show that injection and liquid production rates dominate the inter-well pressure difference and control gas breakthrough. Reservoir porosity primarily governs reservoir storage and pressure buffering capacity, while streak permeability determines CO₂ preferential migration velocity. Under tubing-head pressure constraints, actual injection and production performances are co-regulated by tubing-head pressure limits, wellbore pressure loss and reservoir injectivity/deliverability. The proposed model provides a numerical framework for investigating wellbore-reservoir interactions and gas-channeling risks in heterogeneous reservoirs with high-permeability channels. Quantitative validation against field measurements or controlled experimental data will be undertaken in future work.
This study presents an adaptive modified Runge-Kutta compact scheme for the numerical simulation of unsteady k − ω turbulent nanofluid flow over a heated moving surface under local thermal non-equilibrium conditions. The surface-interfacial model incorporates mixed convection, viscous dissipation, turbulence transport, and separate energy equations for the base fluid and nanoparticle phases, with the effective thermal conductivity described by Xue’s formulation. The proposed time-integration method is explicit and combined with a compact finite-difference discretization that provides fourth-order spatial accuracy. The temporal coefficients are selected to achieve second-order accuracy, and the method is further enhanced through adaptive time stepping based on local error control. Stability analysis for the scalar convection–diffusion problem and conditional convergence analysis for the corresponding system formulation are also established. Numerical comparisons show that the proposed adaptive scheme yields lower error than existing adaptive Euler- and Runge-Kutta-based schemes. The computed results further demonstrate that thermal buoyancy increases the mean velocity, whereas larger Prandtl numbers reduce the thermal boundary-layer thickness of the fluid and nanoparticle phases. In addition, stronger interphase coupling modifies the two-temperature fields in a manner consistent with local thermal nonequilibrium. A machine-learning model is also employed to predict eddy viscosity, and its reliability is confirmed through profile comparisons, contour analyses, sensitivity assessments, and Taylor diagram evaluations. Overall, the proposed framework provides an accurate and efficient computational tool for surface-associated turbulent nanofluid transport with interfacial thermal nonequilibrium.
Grid impedance has an important influence on the synchronization stability of grid-connected energy storage converters and their adaptability to weak grids. Conventional intrusive estimation methods require additional perturbations to be injected, which reduces power quality and complicates practical applications. This paper proposes a non-intrusive online grid impedance estimation method. The excitation source is the power regulation dynamics of the energy storage converter itself. The sliding-window discrete Fourier transform is used to extract the fundamental components of the point of common coupling (PCC) voltage and the converter output current from two adjacent single-cycle windows. By combining the voltage equations of the two windows, the unknown equivalent grid voltage components are eliminated, and the grid resistance and reactance can be calculated without injecting additional perturbations or using external phasor measurement equipment. Under constant-current conditions, a simulation with a sudden change in grid impedance is used to verify the effectiveness of the method. Experiments are carried out on a three-phase three-wire energy storage converter under current-step and impedance-switching conditions. The results show that the estimated grid resistance and reactance have small differences from the preset values. The estimation errors of the resistance and reactance on the α-axis are 5.78% and 5.27%, respectively, while those on the β-axis are 0.27% and 1.60%, respectively. This shows that the proposed method can achieve accurate online grid impedance estimation without interfering with the normal operation of the converter, and can improve the stability and adaptive control capability of the energy storage converter.
In this paper, we investigate a tri-trophic predator-prey system with double fear effects, where the prey is affected by fear induced by both the intermediate predator and the top predator. The top predator is assumed to feed on both the basal prey and the intermediate predator. We establish the positivity and boundedness of solutions and derive sufficient conditions for the global asymptotic stability of axial equilibria. The possible interior equilibria are characterized algebraically, and their local stability is analyzed. We further derive the Hopf bifurcation conditions and compute the first Lyapunov coefficient to determine the direction of the bifurcation and the stability of the bifurcating periodic orbit. Numerical simulations reveal several forms of complex dynamics, including a period-three orbit in the associated return dynamics, the coexistence of a stable equilibrium and a stable limit cycle, the coexistence of two stable limit cycles, and chaotic behavior. The corresponding multistability is further illustrated by basin-of-attraction diagrams. Numerical evidence for chaotic attractors in parameter regions admitting different numbers of interior equilibria is provided by positive largest Lyapunov exponents, parameter bifurcation diagrams, and bounded long-time trajectories. These results show that, within the parameter regimes examined, the interaction between the two fear mechanisms and the tri-trophic feeding structure can generate pronounced dynamical transitions, multistability, and strong sensitivity to initial population densities.
This paper compares three numerical schemes for Caputo fractional differential equations: the Shifted Chebyshev Tau Method (SCTM), the Shifted Chebyshev Collocation Method (SCCM), and the Haar Wavelet Collocation Method (HWCM). In the two Chebyshev schemes, the unknown solution is approximated by shifted Chebyshev polynomials, whereas the Haar formulation uses localized piecewise-constant basis functions and fractional integration matrices. Each method reduces the governing equation to a finite algebraic system. The original contribution is a controlled like-for-like comparison in which the three formulations use the same numbers of unknowns, common independent test grids, and the same accuracy, conditioning, sparsity, and timing diagnostics. Six benchmark problems with known exact solutions are examined using the balanced approximation sizes N = 4, 8, 16, 32. The first five examples have smooth polynomial solutions. SCTM and SCCM recover the exact profiles to the adopted working precision whenever the exact polynomial lies in the selected approximation space. The sixth example has the nonpolynomial solution u(x) = x 5/2, whose third derivative is unbounded at the left endpoint, and therefore provides a genuine low-regularity convergence test. At N = 32, the maximum errors of SCTM and SCCM are 4.39523 × 10−6 and 4.31475 × 10−6 , respectively, whereas HWCM gives 4.33493 × 10−2 . The results show that the Chebyshev methods provide the highest accuracy for the problems studied, SCCM generally achieves this accuracy with a lower assembly cost than SCTM, and HWCM produces better-conditioned and less dense systems with generally first-order-type convergence.
Nonlocal equations have emerged as a prominent research frontier in the field of nonlinear partial differential equations, while simultaneously posing pervasive challenges in theoretical modeling across disciplines including elastic vibrations and geometric analysis. To date, the academic community has developed a robust and comprehensive theoretical framework for such equations, with existing results encompassing both critical and supercritical nonlinearities. Nevertheless, when employing variational methods to investigate multiple positive solutions of nonlocal equations defined on the four-dimensional ball, the loss of embedding compactness induced by critical terms remains a core bottleneck hindering further progress. This paper explores multiple positive solutions of nonlocal equations with both critical and supercritical nonlinear terms on the four-dimensional spherical domain. To rigorously establish the existence of multiple positive solutions, we integrate the Nehari manifold framework with advanced variational techniques. We harness the Brézis–Lieb lemma to circumvent the compactness deficiency arising from critical nonlinearities, and draw upon potential function analysis to compensate for the failure of compactness conditions caused by supercritical terms, thereby rigorously proving the existence of k distinct positive solutions for the equation. This result not only generalizes some existing conclusions in the literature but also offers new insights for further research on high-dimensional nonlocal problems.
This paper develops a dynamic optimal-control framework for the coupled visitor-flow and energy-consumption processes of large-scale scenic destinations. Zone-level visitor density is modelled by nonlinear ordinary differential equations and facility loads by first-order linear equations with control inputs and occupancy disturbances, giving a single constrained plant for which we establish non-negativity, forward invariance of a compact set, and existence and uniqueness of solutions. For the associated finite-horizon problem, we prove that an optimal control exists and, applying Pontryagin's minimum principle, derive the costate equations and show that the optimal routing law is bang-bang with an explicit switching function. In contrast, the optimal energy law is saturated affine in the costate. A receding-horizon controller re-solves the problem online and, equipped with terminal ingredients, is recursively feasible and nominally asymptotically stable. An extended Kalman filter driven by an Internet of Things (IoT) sensor network supplies the state estimate, and its expected error covariance is proved uniformly bounded under Bernoulli sensor dropout. On a three-zone, nine-facility benchmark, the closed loop raises comfort compliance from 50% to 89% of slots, cuts mean waiting time by 57%, reduces total energy by 2.9% and load variance by 10%, and degrades gracefully at dropout rates up to 20%; a linear predictive variant proves competitive, so the nonlinear model's empirical advantage is not established. Robust stability under disturbance and distributed-parameter extensions remain open.
In this paper, we provide a rigorous mathematical justification for a simplified model of a piezoelectric plate stabilized around a steady state. Asymptotic analysis of a 3D piezoelectric materials model with linear feedback control laws is performed as the thickness h of the plate tends to zero. We derive a 2D piezoelectric plate model which is consistent and stable for an approximation of the 3D model, ensuring its validity for the design of thin electromechanical devices. The energy decay for the 2D and 3D systems is established. Such a dimension reduction is very important because, when the plate thickness is very small, it simplifies numerical calculations and, above all, avoids the numerical calculation problems caused by distortion between the plate dimensions. Furthermore, the study reveals that when the thickness of a piezoelectric plate is very small, we no longer need the restrictions to only eleven stabilizable types of piezoelectric materials. The core contribution of this work is the direct integration of this fully coupled dimensional reduction with control theory.
In this paper, we study the existence and uniqueness of solutions, Bielecki–Hyers–Ulam stability, and Bielecki–Hyers–Ulam–Rassias stability for non-linear fractional Volterra Fredholm Hammerstein integro-delay dynamic systems with instantaneous impulses on time scale. Such systems provide a unified framework that encompasses both continuous and discrete models, making them highly appropriate for describing complex real-world phenomena involving memory effects, hereditary properties, and sudden perturbations. Sufficient conditions are established for the existence and uniqueness of solutions to the considered systems. In particular, the Picard operator and the Banach fixed point theorem are utilized to prove the existence and uniqueness of solutions. Moreover, we analyze the qualitative behavior of solutions by proving Bielecki–Hyers–Ulam stability and Bielecki–Hyers–Ulam–Rassias stability. To obtain these stability results, Grönwall’s inequality on time scales is used as the main analytical tool. For our results, some suitable assumptions are imposed along with appropriate Lipschitz conditions on the nonlinear terms. By constructing appropriate contractive mappings in a suitably defined Bielecki-type normed space, we develop a unified and systematic framework to handle the combined effects of integral operators, fractional dynamics, delay arguments, and impulsive perturbations. Finally, an illustrative example is provided to demonstrate the effectiveness and applicability of the theoretical findings.
This study investigates the relationship between intellectual capital efficiency and the financial performance of Islamic banks in Malaysia by employing dynamic modeling and empirical panel analysis. In knowledge-driven financial systems, intellectual capital, which includes human capital, structural capital, and capital usage efficiency, has a significant impact on creating a long-term competitive edge. This study sets up a system of differential equations to anticipate how bank performance would change over time based on the parts of intellectual capital and how they are invested. Therefore, this captures both theoretical dynamics and real-world consequences. The theoretical model delineates equilibrium conditions and demonstrates the local asymptotic stability of the intellectual capital system. The study employs the Value-Added Intellectual Coefficient (VAIC) approach, examining an unbalanced panel dataset of 11 comprehensive Islamic banks in Malaysia from 2012 to 2023, resulting in 132 bank-year observations. To find out how intellectual capital affects financial success, we use fixed-effects panel regression with robust standard errors. We look at return on assets (ROA) and return on equity (ROE). The empirical findings demonstrate that the overall efficiency of intellectual capital exerts a positive and significant influence on bank profitability. When dispersed, human capital efficiency is found to be the most important factor affecting financial performance, followed by capital employed efficiency. Structural capital efficiency, on the other hand, has no statistically significant effect. The results are similar across different model setups and diagnostic tests. These results highlight the importance of knowledge resources and efficient capital deployment in enhancing the competitiveness and sustainability of Islamic banking institutions. This study contributes to the field by integrating dynamic system modeling intellectual capital.
Load Frequency Control (LFC) is a fundamental issue in modern power systems, aimed at maintaining system frequency at its nominal value (50/60 Hz) while ensuring accurate regulation of tie-line power exchanges in interconnected areas. This study develops a comprehensive state-space modeling framework for single-area, two-area, and three-area power systems to support dynamic analysis and controller design. Integral controllers and optimal control strategies based on the Riccati equation-namely, Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian (LQG)-are implemented to enhance system performance. These controllers effectively minimize frequency and voltage deviations under varying load conditions, thereby improving power quality. To achieve faster and more precise control, a Digital Deadbeat Controller (DDC) is also proposed, ensuring rapid convergence to steady-state conditions. A comparative analysis across different system configurations highlights the steady-state frequency deviations and overall performance improvements. The study also addresses persistent challenges caused by memory-dependent nonlinearities such as backlash/deadband, which induce limit cycle (LC) oscillations and degrade system stability. Backlash nonlinearity, inherent in speed governors used in LFC, plays a significant role in these oscillations. Two methods are proposed to mitigate LC behavior: signal stabilization using deterministic or random inputs. Simulation results obtained using MATLAB/Simulink demonstrate the effectiveness of these approaches. The findings confirm that both DDC and signal stabilization techniques achieve the desired performance (Delta f = 0, t -> 0), indicating robust and efficient control suitable for practical power system applications.
Based on an adaptive universe model, this perspective article presents a phenomenological framework that correlates the dark energy equation of state w with the cosmic growth index gamma via the linear relation w(a) = -1 + eta(gamma(a)-0.55). Recognizing that the coupling between dark energy and structure formation may evolve with cosmic time, the framework is extended to a redshift-dependent formulation: w(z) = -1 + eta(z)[gamma(z)-0.55] + Delta wbg(z), where eta(z) captures the structure-dependent coupling and Delta wbg(z) accounts for intrinsic background evolution. Several physically motivated parameterizations of eta(z) are proposed, including continuous forms (smooth transition and oscillatory) and a phenomenological piecewise model aligned with distinct phases of structure formation history. Built upon an interacting dark sector model that strictly conserves energy and momentum, the framework maintains the spacetime geometry of General Relativity while naturally addressing both the Hubble tension (via enhanced late-time expansion) and the S8 tension (via suppressed structure growth). A hierarchical Bayesian testing roadmap with Fisher forecasts demonstrates that upcoming surveys (DESI, Euclid, Roman) can decisively detect couplings of magnitude |eta|greater than or similar to 0.05 at high significance. The framework yields distinctive, testable predictions-including correlated w(z) and f sigma 8(z) evolution, a gravitational slip parameter eta slip = 1 that distinguishes it from modified gravity theories, and scale-dependent signatures in the nonlinear regime-providing a comprehensive path to either validate or falsify the hypothesized dark energy-structure growth connection.
This paper investigates the numerical computation of magnetic fields in a Hall thruster, a process governed by a nonlinear elliptic boundary value problem. The model rigorously accounts for the influence of ferromagnetic materials, where the relative permeability is defined as a magnetic field-dependent function exceeding unity in core regions and set to unity elsewhere. Consequently, the model equations are inherently nonlinear, and their coefficients exhibit discontinuity across material interfaces. To solve this complex system, the finite difference method is applied on a uniform staggered mesh to derive a system of nonlinear difference equations with discontinuous coefficients. An iterative algorithm featuring a nested loop structure is presented to tackle this nonlinearity: an inner loop computes the magnetic field for a fixed relative permeability, while an outer loop updates the permeability distribution based on the current field solution. A critical challenge in such simulations is the convergence difficulty under high excitation due to strong nonlinearity and magnetic saturation. To address this, we propose a robust nested iterative algorithm enhanced with Aitken extrapolation. The method is validated through numerical simulations on a miniature Hall thruster model across three distinct coil ampere-turn configurations. Results highlight a critical distinction: while the standard fixed-point iteration performs adequately under low-to-moderate excitation, it fails to converge under the high-excitation condition. In contrast, the proposed Aitken-accelerated algorithm achieves stable convergence across all test cases, successfully resolving the convergence bottleneck in high-field scenarios. This advancement provides a robust framework for the magnetic circuit design of high-power Hall thrusters.
This work presents a theoretical framework to study the non-Markovian dynamics of a two-level quantum emitter interacting with a broadband squeezed electromagnetic reservoir, and both one- and two-photon interaction processes are incorporated. Mathematical modeling uses a time-convolution less projection operator technique. This yields a time-local master equation. The coefficients of this equation are derived from integrals over the reservoir's squeezed correlation functions: and . The model is validated through rigorous numerical simulation of the resulting dynamical equations. Testing involves computing key physical observables: the transient emission spectrum and the field linear entropy . These predictions are systematically analyzed against variations in squeezing parameters , coupling strengths , and detector bandwidth . The results confirm that the model successfully captures phase-dependent decoherence, spectral modulation, and purity oscillations. Notably, two-photon processes suppress decoherence under strong squeezing. The consistency between analytical derivations and numerical outcomes validates the framework. It is established as a predictive tool for quantum optics in engineered nonclassical environments. This study directly connects engineered reservoir properties specifically its nonclassical photon statistics to observable, time-dependent quantum phenomena. The findings offer fundamental insights and a predictive tool for quantum control, sensing, and information processing in tailored electromagnetic environments.
The premise of blind image deblurring revolves around the restoration of a clear image from a blurred one without prior knowledge of the specific blur kernel employed. Within this realm, various image priors have been extensively investigated and applied to address this inherently challenging problem. Throughout the image deblurring process, ensuring the resulting image intensities remain strictly non-negative is often imperative. However, prevalent numerical methodologies utilized to solve this issue have shown instances where the outcomes are not consistently favorable, leading to undesirable negative intensities that contribute to significant areas of darkness in the restored images. This study introduces a novel model designed to tackle the blind image deblurring problem by leveraging mean curvature. The proposed model not only assures positive outcomes but also confines the upper limit of image intensity values, thereby maintaining them within a predefined range. Additionally, new numerical algorithms are introduced, which not only restore the image but also estimate the blur kernel. Comparative analyses between these proposed algorithms and existing numerical techniques have been conducted to showcase the effectiveness and feasibility of our suggested approach.
