Integro-differential Equations Research Articles

Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

<abstract><p>The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs. A rigorous convergence analysis is provided and confirmed by numerical tests. In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem.</p></abstract>

Positivity and Periodicity in Nonlinear Neutral Mixed Type Levin-Nohel Integro-Differential Equations

In this work, we give sufficient conditions for the existence of periodic and positive periodic solutions for a nonlinear neutral mixed type Levin-Nohel integro-differential equation with variable delays by using Krasnoselskii’s fixed point theorem. Also, we obtain the existence of a unique periodic solution of the posed equation by means of the contraction mapping principle. As an application, we give an example to illustrate our results. Previous results are extended and generalized.

Investigation of Integro-Differential Equations by Methods of Spectral Theory

Investigation of Integro-Differential Equations by Methods of Spectral Theory

Evolutionary dynamics of glucose-deprived cancer cells: insights from experimentally informed mathematical modelling.

Glucose is a primary energy source for cancer cells. Several lines of evidence support the idea that monocarboxylate transporters, such as MCT1, elicit metabolic reprogramming of cancer cells in glucose-poor environments, allowing them to re-use lactate, a by-product of glucose metabolism, as an alternative energy source with serious consequences for disease progression. We employ a synergistic experimental and mathematical modelling approach to explore the evolutionary processes at the root of cancer cell adaptation to glucose deprivation, with particular focus on the mechanisms underlying the increase in MCT1 expression observed in glucose-deprived aggressive cancer cells. Data from in vitro experiments on breast cancer cells are used to inform and calibrate a mathematical model that comprises a partial integro-differential equation for the dynamics of a population of cancer cells structured by the level of MCT1 expression. Analytical and numerical results of this model suggest that environment-induced changes in MCT1 expression mediated by lactate-associated signalling pathways enable a prompt adaptive response of glucose-deprived cancer cells, while fluctuations in MCT1 expression due to epigenetic changes create the substrate for environmental selection to act upon, speeding up the selective sweep underlying cancer cell adaptation to glucose deprivation, and may constitute a long-term bet-hedging mechanism.

Inverse Problem for a Hyperbolic Integro-Differential Equation with two Redefinition Conditions at the End of the Interval and Involution

Inverse Problem for a Hyperbolic Integro-Differential Equation with two Redefinition Conditions at the End of the Interval and Involution

Delay Differential Equations with Differentiable Solution Operators on Open Domains in C((−∞, 0], ℝn) and Processes for Volterra Integro-Differential Equations

Delay Differential Equations with Differentiable Solution Operators on Open Domains in C((−∞, 0], ℝn) and Processes for Volterra Integro-Differential Equations

Solvability of some Fredholm integro-differential equations with mixed diffusion in a square

Solvability of some Fredholm integro-differential equations with mixed diffusion in a square

Analysis of a class of fractional delay integro-differential equations with Riesz-Caputo derivative

Analysis of a class of fractional delay integro-differential equations with Riesz-Caputo derivative

Stability analysis of an implicit fractional integro-differential equation via integral boundary conditions

The primary objective of this research study is to analyze a boundary problem involving Caputo fractional integro-differential equations. The focus is on a differential equation with a nonlinear right-hand side composed of two terms. The stability analysis of a fractional integro-differential equation is presented using Ulam's concept. Furthermore, this research study establishes the correlation between the stated problem and the Volterra integral equation. The investigation proceeds by utilizing the renowned Banach and Krasnoselskii's fixed point theorems to explore the existence and uniqueness of solutions for the problem. Additionally, to provide tangible evidence of the abstract findings, two illustrative examples are presented.

О корректности математической модели вызванной активности первичной зрительной коры

We propose a mathematical model that formalizes the macro- and meso-level dynamics of electrical potentials in the primary visual cortex of subjects, which corresponds to the presentation of visual stimuli to them. The mathematical framework is based on a twolayer neural field model, represented by a system of integro-differential equations, where the deep layer of the neural field models electrical activity that does not depend directly on the spatial orientation of the visual stimuli, whereas the activity of the superficial layer is sensitive to spatially oriented stimuli. The experimental design of presenting a series of visual stimuli is formalised in the present study in terms of an impulse control problem for the aforementioned two-layer neural field model. We propose a special metric space for construction of a unique solution to the control problem under standard assumptions for mathematical neurobiology regarding the functions involved in the modeling equations. We formulate sufficient conditions for continuous dependence of the solutions on the impulse control.

Numerical Treatment of Volterra-Fredholm Integro- Differential Equations of Fractional Order and its Convergence Analysis

This work deals with semi-analytical and numerical methods to solve a class of fractional order Volterra-Fredholm integro-differential equations. First, a semi-analytical method is proposed using the Chebyshev and Bernstein polynomials in the Adomian decomposition method. The uniqueness of the solution and convergence of the method are proved. Further, we solve the model using a numerical scheme comparing the L1 scheme for the fractional order derivative in combination with appropriate quadrature rules for the integral parts. Numerical experiments are done by the proposed methods to show their efficiency through a few tabular data and plots. Some comparisons with the existing results show that the proposed methods are highly productive and reliable.

A tau-Gegenbauer spectral approach for systems of fractional integro-differential equations with the error analysis

<abstract><p>This research paper focused on the solution of systems of fractional integro-differential equations (FIDEs) of the Volterra type with variable coefficients. The proposed approach combined the tau method and shifted Gegenbauer polynomials in a matrix form. The investigation of the existence and uniqueness of solutions for these systems was carried out using Krasnoselskii's fixed point theorem. The equations employed Caputo-style derivative operators, and to minimize computational operations involving derivatives and multiplications, integral and product operational matrices were derived. By introducing suitable polynomial approximations and employing the tau spectral method, the original system of FIDE was transformed into an algebraic system. Solving this algebraic system provided approximate solutions to the main system. Error bounds were computed in the Gegenbauer-weighted Sobolev space. The proposed algorithm was implemented and tested on two systems of integro-fractional differential equations to demonstrate its efficiency and simplicity. By varying the parameter $ \sigma $ in the Gegenbauer polynomials, the impact of this variation on the approximate solutions can be observed. A comparison with another method utilizing the block-by-block approach was also presented.</p></abstract>

Inverse problem for non-homogeneous integro-differential equation of hyperbolic type

An inverse problem is considered, which consists in finding a solution and a one-dimensional kernel of the integral term of an inhomogeneous integro-differential equation of hyperbolic type from the conditions that make up the direct problem and some additional condition. First, the direct problem is investigated, while the kernel of the integral term is assumed to be known. By integrating over the characteristics, the given intego-differential equation is reduced to a Volterra integral equation of the second kind and is solved by the method of successive approximations. Further, using additional information about the solution of the direct problem, we obtain an integral equation with respect to the kernel of the integral k(t), of the integral term. Thus, the problem is reduced to solving a system of integral equations of the Volterra type of the second kind. The resulting system is written as an operator equation. To prove the global, unique solvability of this problem, the method of contraction mappings in the space of continuous functions with weighted norms is used. And also the theorem of conditional stability of the solution of the inverse problem is proved, while the method of estimating integrals and Gronoullo’s inequality is used.

A Generalized Hybrid Method for Handling Fractional Caputo Partial Differential Equations via Homotopy Perturbed Analysis

This article describes a novel hybrid technique known as the Sawi transform homotopy perturbation method for solving Caputo fractional partial differential equations. Combining the Sawi transform and the homotopy perturbation method, this innovative technique approximates series solutions for fractional partial differential equations. The Sawi transform is a recently developed integral transform that may successfully manage recurrence relations and integro-differential equations. Using a homotopy parameter, the homotopy perturbation method is a potent semi-analytical tool for constructing approximate solutions to nonlinear problems. The suggested method offers various advantages over existing methods, including high precision, rapid convergence, minimal computing expense, and broad applicability. The new method is used to solve the convection–reaction–diffusion problem using fractional Caputo derivatives.

Mixed sequential type pantograph fractional integro-differential equations with non-local boundary conditions

Mixed sequential type pantograph fractional integro-differential equations with non-local boundary conditions

Stability of nonlinear impulsive higher order differential – fractional integral delay equations with nonlocal initial conditions

The aim of this paper is to investigate some types of stability such as generalized Hyers-Ulam- Rassias stability(G-H-U-R-stabile) and the relation with Hyers-Ulam(H-U-stabile) stable and Hyers-Ulam-Rassias stable (H-U-R-stabile) and generalized Hyers-Ulam stable (G-H-U- stable) to obtain which one guarantee to satisfy stability of equations included a nonlinear function some of them contains a delay time of solution and the other contain a vector of different order of derivatives for the solution to n-time and vector of fractional order of integrals with different fractional orders and that was the for using a claculse of fractional calculus to satisfies the issue of this techniques. Moreover, the nonlocal initial values for the proposal equation of nonlinear impulsive higher order differential – fractional integral delay time equations which are adding more interesting for nonlinear analytic object of nonlinear higher order integro – fractional order impulsive classes, and the impulsive difference of the equation has some necessary conditions to prove the results of solution to be stable with certain type has related with other types. The necessary and sufficient conditions which assumed on this nonlinear higher order integro-differential impulsive equation have been achieved the stability with interesting certain estimates obtain through the proving technique. Also the uniqueness of solution has been studied with same conditions was presented for stability and used for that issue a contraction fixed point theorem.

Approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by Rosenblatt process via resolvent operators

In this work, we investigate the existence of a mild solution and the approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by the Rosenblatt process in Hilbert space with the Hurst parameter \(\mathsf{H} \in (1/2, 1)\). We achieve the result using the semigroup theory of bounded linear operators, Grimmer's resolvent operator theory, and stochastic analysis. Using Krasnoselskii's and Schauder's fixed point theorems, we demonstrate the existence of mild solutions and the approximate controllability of the system. Finally, an example shows the potential for significant results.

NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM FOR A SECOND ORDER IMPULSIVE SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED MAXIMA

A two-point nonlinear boundary value problem for a second order system of ordinary integro-differential equations with impulsive effects and mixed maxima is investigated. By applying some transformations is obtained a system of nonlinear functional integral equations. The existence and uniqueness of the solution of the nonperiodic two-point boundary value problem are reduced to the one valued solvability of the system of nonlinear functional integral equations in Banach space . The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional integral equations.

On the Expected Discounted Penalty Function Using Physics-Informed Neural Network

We study the expected discounted penalty at ruin under a stochastic discount rate for the compound Poisson risk model with a threshold dividend strategy. The discount rate is modeled by a Poisson process and a standard Brownian motion. By applying the differentiation method and total expectation formula, we obtain an integrodifferential equation for the expected discounted penalty function. From this integrodifferential equation, a renewal equation and an asymptotic formula satisfied by the expected discounted penalty function are derived. In order to solve the integrodifferential equation, we use a physics-informed neural network (PINN) for the first time in risk theory and obtain the numerical solutions of the expected discounted penalty function in some special cases of the penalty at ruin.

Coupled continuous time random walk with Lévy distribution jump length signifies anomalous diffusion?

The continuous time random walk (CTRW) models are highly valued for studying anomalous diffusion, a ubiquitous phenomenon in complex media whose mean squared displacement (MSD) exhibits power-law behavior. It is widely accepted that the Lévy distribution jump length of random particles with infinite second moment could be responsible for the superdiffusion phenomenon. In this paper, we study the coupled CTRW with exponentially truncated Lévy distribution jump length, deduce the corresponding integrodifferential diffusion equation, and give the propagator expression. It is interesting that we find the MSD for random particles is linearly related to time in the symmetric Lévy distribution case, and it reduces to a simple expression. Additionally, for a non-zero truncated exponent, the MSD also has a linear relationship with time, which reflects the special dependence between waiting time and jump length in this coupled CTRW model.