In this paper, the inverse problem of identifying the unknown initial value for time fractional diffusion equation with Caputo-Hadamard derivative is considered. This problem is illposed and two regularization methods are used to solve it. Firstly, we prove that this problem is ill-posed. Secondly, the conditional stability result and the optimal error bound are given. Then, the error estimates of the Quasi-boundary regularization method and the fractional Landweber iterative regularization method under a priori and a posteriori regularization parameter selection rules are given respectively. Finally, numerical examples are given to illustrate the effectiveness of two regularization methods.
In this paper, we present a collocation method for linear Volterra–Fredholm integral equations with delay on a semi-infinite interval. The method employs orthogonal mapped Legendre basis functions together with a mapped Gauss quadrature rule adapted to the Volterra operator, leading to a stable and well-conditioned linear system. The convergence properties of both the collocation and iterated collocation solutions are investigated in the L2- and L∞-norms, and algebraic convergence rates are derived under mild regularity assumptions. Several numerical examples are presented and discussed to show the accuracy and efficiency of the methods.
We study nonlinear problems which use the generalized $\alpha(x)-$Laplacian operator under Fourier boundary conditions with $L^{1}$ data. Our research establishes both weak and entropy solutions through the framework of variable exponent Sobolev spaces. Our solution method uses monotone operator theory and appropriate approximation methods to develop a unified approach for dealing with nonlinearities that exhibit variable growth. These findings help expand knowledge about nonlinear Fourier-type problems while demonstrating how entropy formulations enable well-posedness for problems with minimal data requirements.
In this paper, we study a coefficients inversion problem of a coupled system controlled by three reaction-diffusion equations describing a simple dynamic model of a drug epidemic in an idealized community from the final measurement data. Firstly, the optimization theory is used to transform the given problem into an optimal control problem, and the existence of minimizer is established. Then the stability estimates of the Lipschitz type for the three spatially varying coefficients are proved, where the upper bounds are given by some Lebesgue norms of the final measure.
The purpose of this paper is to obtain a duality between the game put and call options assuming three component penalties – proportion of the usual option payoff, shares of the underlying asset, and a fixed amount. We examine separately the cases of finite and infinite maturities. For the perpetual options, we need to derive a polynomial-style equations for the optimal boundaries. We prove the existence and uniqueness of their solutions as well as provide a method for their deriving. This result is important in itself since the current literature in the field is based on inverting of several functions or on solving of non-linear systems which may lead to some computational difficulties and significant errors for extreme parameter values. Furthermore, the duality is established under a finite time horizon too. It is important to note that this duality does not hold under the classical assumption of a fixed penalty.
The present work investigates recurrent neural systems incorporating generalized piecewise constant delay, with particular emphasis on establishing periodic behaviors and verifying their exponential convergence on a global scale. The existence of periodic solutions is established via Mawhin’s coincidence degree in combination with sharp a priori estimates, while uniqueness and exponential attractivity are derived through a Lyapunov functional approach supported by differential inequalities adapted to the delay structure. The obtained criteria are concise, verifiable, and applicable in practice. Representative computational experiments are provided to substantiate the analytical findings.
This paper is devoted to extending two-step peer methods, for the numerical solution of ordinary differential equations, for the case where the second derivative of the solution is incorporated into the formula of the methods. The main features including consistency, zero-stability, and convergence of the proposed methods together with their order conditions and stability analysis are examined. Construction of explicit methods within the proposed class of the methods, possessing the Runge–Kutta stability property, is investigated, and examples of such methods up to order five are provided. The efficiency and accuracy of the constructed methods are validated through various numerical experiments conducted in both fixed and variable stepsize environments.
This paper introduces a uniform numerical scheme to find approximate solutions for singularly perturbed parabolic time delay reaction-diffusion problems. The scheme utilizes the Crank-Nicolson method for approximating time derivatives, combined with a novel finite difference method for spatial discretization. The stability and uniform convergence of the proposed scheme are investigated. The primary objective of this work is to demonstrate that the proposed scheme achieves a parameter-free error bound of order O(k2 + N−2). To validate the theoretical results, various numerical experiments have been conducted, showing that the proposed scheme yields superior results compared to some existing methods in the literature.
In this article, existence and uniqueness of the solution for two-dimensional non-linear singular Volterra integral equations with fractional orders in a Banach space is discussed by utilizing the concept of the measure of non-compactness and fixed-point theorem. In fact, this kind of equations is a generalization of two-dimensional Riemann-Liouville fractional non-linear integral equations. To approximate the solution of the above problem, we use modified homotopy perturbation with the help of Adomian polynomials. To validity of the derived results, we introduce an example in the field of singular non-linear integral equations. Hence, a semianalytic solution for given example is obtained ensuring satisfactory accuracy. Also, to ensure the effectiveness of the proposed method the results are compared with some other works.
The Gray-Scott (GS) model is a nonlinear reaction diffusion system widely used in applied sciences. This paper delves into the investigation of a spectral method for the GS model with homogeneous Neumann boundary conditions. The proposed numerical scheme integrates a spectral approach based on generalized Jacobi polynomials for spatial discretization with the two-step backward differentiation formula (BDF2) for temporal discretization. We prove the boundedness, generalized stability, and convergence of this new method. Extensive numerical results demonstrate the efficiency of the new proposed scheme and provide numerical validation of the theoretical analysis. The key advantages of our new approach are twofold: (i) it utilizes generalized Jacobi polynomials with indices α = β = −2, which naturally satisfy the boundary conditions and thereby not only simplify the theoretical analysis but also yield a well-conditioned discrete system that enhances computational efficiency; and (ii) the numerical errors exhibit exponential decay in space.
In this paper, we establish some new generalizations of dynamic inequalities similar to Hardy's inequality on a time scale $\mathbb{T}$, by applying Jensen's inequality, integration by parts and chain rule on time scales. In particular, when $\mathbb{T}=\mathbb{R}$, we get the classical inequalities known from the literature, while in the discrete case $\mathbb{T}=\mathbb{N}$, the obtained inequalities are essentially new. In addition, we show that our results are more accurate than some recent dynamic inequalities known from the literature. Finally, we establish the corresponding relations in quantum calculus, when $\mathbb{T}=q^{\mathbb{N}_{0}}$, $q>1$.
Let $Q$ be a positive definite $n \times n$ matrix, $n \in 2\mathbb{N}$, $n \geqslant4$. The Epstein zeta-function $\zeta(s; Q)$, defined for $\mathrm{Re}\,s > \tfrac{n}{2}$, is given by $\zeta(s; Q) = \sum_{\underline{x} \in \mathbb{Z}^n \setminus \{\underline{0}\}} (\underline{x}^T Q{\underline{x}})^{-s},$ and has a meromorphic continuation to the whole complex plane. Let $T^{{27}/{82}} \leqslant H \leqslant T^{{1}/{2}}$. In this paper, we prove a limit theorem on weak convergence for $\frac{1}{H} \mathrm{meas}\left\{t \in [T, T+H]: \zeta(\sigma + it; Q) \in A \right\},\; A \in \mathcal{B}(\mathbb{C}),$ as $T\to\infty$, where $\mathcal{B}(\mathbb{C})$ is the Borel $\sigma$-algebra on $\mathbb{C}$. The limit measure is explicitly given. The result extends a known theorem obtained for the interval $[0, T]$.
Obesity and its associated metabolic dysregulations, particularly Type 2 Diabetes Mellitus (T2DM), constitute a global health crisis. Understanding the intricate interplay of key metabolic components is crucial for effective management strategies. This study presents a novel mathematical model capturing the dynamic interactions among plasma glucose, insulin, and free fatty acids (FFAs), critically integrating the regulatory influence of Glucagon-Like Peptide-1 (GLP-1). Through qualitative analysis, we established the model’s physiological relevance and demonstrated the existence of a stable equilibrium point, confirmed by numerical simulations across various initial conditions. Sensitivity analysis revealed that FFA-related parameters (e.g., lipolysis rates and FFA-induced insulin impairment) and insulin secretion/clearance rates profoundly affect glucose homeostasis, underscoring the detrimental role of elevated FFAs in hyperglycemia. Furthermore, we applied optimal control theory, using Pontryagin’s Maximum Principle, to design GLP-1 receptor agonist intervention strategies. We evaluated two scenarios that balance the cost of intervention with the effectiveness of glucose regulation. Results show that GLP-1 agonism effectively lowers glucose and FFA levels, with greater glucose reduction achieved when control cost and glucose deviation are equally weighted. This research provides a comprehensive mathematical framework for analyzing complex glucose-insulin-FFA-GLP-1 dynamics. Our findings highlight the interconnectedness of insulin sensitivity, lipid metabolism, and incretin action in metabolic health and offer valuable insights for optimizing therapeutic interventions. The developed optimal control strategies suggest potential to improve glycemic control and to inform future clinical approaches to prevent and manage metabolic disorders.
In this study, we examine the uniqueness conditions for solutions of fractal differential equations using the Krasnoselskii-Krein uniqueness theorem. The analysis establishes sufficient criteria that guarantee the existence of unique solutions. Additionally, we employ the successive midpoint method to numerically solve chaotic systems governed by both fractal and global derivatives. To evaluate the effectiveness of the proposed approach, graphical simulations are presented for various derivative orders. These results illustrate the method’s accuracy, stability, and reliability in capturing the intricate dynamics of the considered systems.
This paper presents and analyzes robust numerical algorithms for solving inverse problems for parabolic equations, specifically focusing on the determination of an unknown time-dependent source function from an integral flux condition. The study is motivated by mathematical models based on Navier-Stokes equations, particularly those exhibiting Poiseuille-type solutions. We employ a variational approach, formulating the inverse problem as the minimization of a Tikhonov regularization cost functional. Discrete approximation schemes are rigorously derived using finite volume methods in space and both backward Euler and Crank-Nicolson schemes in time. A key contribution of this work is the strict justification of the gradient formula for the cost functional by deriving the adjoint problem directly from the fully discrete scheme, rather than discretizing the continuous adjoint problem. This methodology is extended to problems involving fractional powers of elliptic operators and two-dimensional domains. Numerical experiments are conducted to compare the efficiency of Gradient Descent and Conjugate Gradient methods. The results demonstrate that the Conjugate Gradient method significantly outperforms standard gradient descent, maintaining high accuracy and convergence rates even with the inclusion of regularization terms and complex diffusion operators.
In this article, the Newton-iteration scheme based upon iterated Galerkin operator is applied for solving non-linear Volterra Urysohn integral equations of the second kind for smooth and weakly singular kernels. A one step of improvement by iteration to the Galerkin method, named as iterated Galerkin method is a well discussed method and it gives improved convergence rates than Galerkin method. But if we iterate them one more time, then there is no guarantee that we get any improved convergence rates. The proposed Newton-iteration scheme based upon iterated Galerkin operator ensures improved convergence rates at every step of iteration. Specifically, we establish that the convergence rate in iterated Galerkin method increases by O(hr) for smooth kernel, and O(h1−α) for weakly singular kernel, in each step of reiteration, where h is the norm of the partition. Numerical examples are provided to justify the reliability and efficiency of the proposed technique.
In the literature on optimal portfolio selection problems, it is rare that closed-form solutions are found. It is even more so when liquidity risk needs to be taken into consideration. In this paper, we present a closed-form solution for the optimal weights of a portfolio that consists of a risky and riskless asset under a new key assumption that the liquidity risk is directly proportional to the wealth of the portfolio invested in the risky asset. The solution found is for the Constant Relative Risk Aversion (CRRA) utility function, after successfully solving the associated HJB (HamiltonJacobi-Bellman) equation exactly. Due to the presence of liquidity risk, the research findings reveal that the optimal weights are consistently lower than those found by [19]. Finally, the quantitative impact of the proposed solution is discussed.
The focus of this paper revolves around the initial–boundary value problem associated with a logarithmic Lamé system within a bounded domain, and incorporating a time-varying delay. We demonstrate the system’s well-posedness through the application of semigroup theory. Subsequently, we establish the existence of global solutions by employing the well-depth method. Furthermore, we establish exponential decay of solutions under adequate constraints concerning the weight of the time-varying delay and the frictional damping.
This paper is devoted to establishing novel existence criteria for weak solutions to a class of weighted quasilinear degenerate elliptic equations featuring double phase Hardy-type singular coefficients. These types of problems are rarely discussed in variable exponent Sobolev spaces in previous work. We prove the existence of at least one and at least two weak solutions via variational methods and critical point theory, under appropriate assumptions on the weight function and the nonlinearity.
The periodic zeta-function $\zeta(s; a)$, $s = \sigma + it$, $a = \{a_m \in \mathbb{C} : m \in \mathbb{N}\}$, in the half-plane $\sigma > 1$ is defined by Dirichlet series with periodic coefficients $a_m$, and has the meromorphic continuation to the whole complex plane. The function $\zeta(s; a)$ is a generalization of the Riemann zeta-function and Dirichlet $L$-functions. In the paper, using only the periodicity of the sequence $a$, we obtain that the shifts $\zeta(s + i\tau; a)$, $\tau \in \mathbb{R}$, approximate a certain class of analytic functions, defined in the strip $\{s \in \mathbb{C} : 1/2 < \sigma < 1\}$. For $T^{23/70} \leqslant H \leqslant T^{1/2}$, the set of such shifts has a positive lower density in the interval $[T, T + H]$, $T \to \infty$. The case of positive density is also discussed. For the proof, the mean square estimate in short intervals for the Hurwitz zeta-function, and probabilistic limit theorems are applied.
