Partial Integro-differential Equations Research Articles

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

In this article, we develop an efficient numerical scheme for dealing with fractional partial integro-differential equations (FPIEs) with a weakly singular kernel. The weight and shift Grünwald difference (WSGD) operator is adopted to approximate a time fractional derivative and the Sinc collocation method is applied for discretizing the spatial derivative.The exponential convergence of our proposed method is demonstrated in detail. Finally, numerical evidence is employed to verify the theoretical results and confirm the expected convergence rate.

Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs

Proposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference scheme. Also, the temporal derivative is discretized via a second-order numerical procedure. First, the spatial derivatives are approximated by a fourth-order operator to compute the second-order derivatives. This process produces a system of differential equations related to the time variable. Then, the Crank–Nicolson idea is utilized to achieve a full-discrete scheme. The kernel of the integral term is discretized by using the Lagrange polynomials to overcome its singularity. Subsequently, we prove the convergence and stability of the new difference scheme by utilizing the Rayleigh–Ritz theorem. Finally, some numerical examples in one-dimensional and two-dimensional cases are presented to verify the theoretical results.

Application of the finite element method to the multicomponent general dynamic equation of aerosols

This paper focuses on the numerical approximation of the multicomponent dynamic equation of aerosols (MCGDE), an integro-partial differential equation that describes the temporal evolution of the aerosol number distribution containing several compounds. More specifically, we compare the performance of the finite element method (FEM) and a commonly used sectional method. In addition to the standard Galerkin FEM, we introduce the Petrov–Galerkin Finite Element Method (PGFEM) to the MCGDE. These methods are compared to analytical solutions of the MCGDE and to accurate solutions given by the discrete multicomponent GDE which models the condensation process numerically at the monomer level. Results show that in cases where condensation is the dominating process, the accuracy of the FEM and PGFEM can be significantly better than the accuracy of the sectional method. They also generally achieve a given accuracy of the solution with a significantly lower number of discretization points than the sectional method. Moreover, the sparser the discretization is, the more beneficial is the use of the PGFEM, because it stabilizes the solution of condensation dominated MCGDEs. In the case of pure coagulation, all three numerical methods yield a similar accuracy.

Mild solution in the α-norm for some partial integrodifferential equations involving a nonlocal condition

Abstract In this work, we investigate the existence of mild solutions in the α \alpha -norm for a class of nonlocal integrodifferential equations. We employ the theory of resolvent operators introduced by R. Grimmer. The Leray-Schauder alternative is the principal working tool for our analysis.

A Matrix Mittag–Leffler Function and the Fractional Nonlinear Partial Integro-Differential Equation in ℝn

In this paper, we introduce the matrix Mittag–Leffler function, which is a generalization of the multivariate Mittag–Leffler function, in order to investigate the uniqueness of the solutions to a fractional nonlinear partial integro-differential equation in Rn with a boundary condition based on Banach’s contractive principle and Babenko’s approach. In addition, we present an example demonstrating applications of the key results derived using a Python code that computes the approximate value of our matrix Mittag–Leffler function.

An efficient spline technique for solving time-fractional integro-differential equations

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second order derivative approximation to obtain the numerical solution of the weakly singular kernel (SK) non-linear fractional partial integro-differential equation (FPIDE). The spatial and temporal fractional derivatives are discretized by ExCuBS and the Caputo finite difference scheme, respectively. The present study found that it is stable and convergent. The validity of the current approach is examined on a few test problems, and the obtained outcomes are compared with those that have previously been reported in the literature.

An Actuated Computational Method for Treating Parabolic Partial Delay Integro-Differential Equations Constrained by Infinite Boundary

An Actuated Computational Method for Treating Parabolic Partial Delay Integro-Differential Equations Constrained by Infinite Boundary

Generalized stimulated Raman scattering with nonlocal effects

We present a generalized quantum theory of stimulated Raman scattering, which is based on coupled Heisenberg–Langevin equations. Analytical solutions in previous models neglect the effects of the frequency dispersion, assume far-off resonance and only consider slowly varying terms in the atomic operator. In obtaining the more general theory, we have derived partial integro-differential equation containing polarization kernels that include all the temporal nonlocal effects using systematic integral method over time. Analytical solutions of the Stokes field with greater generality for arbitrary spatial-temporal laser fields have been obtained using Laplace transform in the retarded space frame with minimal approximations made. The solutions are very general and potentially useful for studying systems in which highly accurate modeling is required such as quantum memories of a wide range of Raman systems.

Adaptation of Caputo residual power series scheme in solving nonlinear time fractional Schrödinger equations

This study regards the time-fractional nonlinear Schrödinger equations (TFNLSEs). The TFNLSE is used to depict nonlocal quantum phenomena in quantum physics and investigate the quantum behaviors of either long-range relations or time-dependent processes with many scales. Basic definitions of fractional derivatives are described in the Caputo fractional derivative (CaFD) sense. We utilize the residual power series method (RPSM) to achieve the approximate solutions of the TFNLSE. The RPSM is a numeric analytic technique for solving various ordinary, partial, and fuzzy differential and integrodifferential equations of fractional order. It is a practical optimization approach, supplying solutions in the closed form of known functions. The primary benefit of this technique is the clarity in calculating the coefficients of terms of the series solution by employing only the differential operators and not the other well-known analytic approaches that require integration operators, which are challenging in the fractional case. Our main aim in this paper is to find approximate solutions using RPSM and compare these approximate solutions with exact solutions. We discuss three applications, showing the validity, precision, and efficiency of the RPSM. The outcomes are presented in graphs and tables, which display the method’s effectiveness, quality, and strength. The RPSM is entirely compatible with the complexity of this problem, and the accepted outcomes are highly favorable. The offered technique can be easily used in the spaces of higher-dimension solutions without limiting the nature of the equation or the category style. Also, this method can be implemented as a novel alternative technique to deal with many nonlinear issues occurring in quantum physics.

An adaptive finite element method for Riesz fractional partial integro-differential equations

AbstractThe Riesz fractional derivative has been employed to describe the spatial derivative in a variety of mathematical models. In this work, the accuracy of the finite element method (FEM) approximations to Riesz fractional derivative was enhanced by using adaptive refinement. This was accomplished by deducing the Riesz derivatives of the FEM bases to work on non-uniform meshes. We utilized these derivatives to recover the gradient in a space fractional partial integro-differential equation in the Riesz sense. The recovered gradient was used as an a posteriori error estimator to control the adaptive refinement scheme. The stability and the error estimate for the proposed scheme are introduced. The results of some numerical examples that we carried out illustrate the improvement in the performance of the adaptive technique.

Deep learning-based option pricing for Barndorff–Nielsen and Shephard model

This paper aims to develop a deep learning-based numerical method for option prices for the Barndorff–Nielsen and Shephard model, a representative jump-type stochastic volatility model. Using that option prices for the Barndorff–Nielsen and Shephard model satisfy a partial-integro differential equation, we will develop an effective numerical calculation method even in settings where conventional numerical methods are unavailable. In addition, we will implement some numerical experiments.

A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity

The purpose of this paper is to investigate a space-time Sinc-collocation method for solving the fourth-order nonlocal heat model arising in viscoelasticity, which is a class of partial integro-differential equations (PIDEs) whose solution typically exhibits the weak singularities at the initial time. Most of existing approximate methods for solving such kind PIDEs are unbalanced, since most work have used a low order scheme for integrating the temporal variable and a high order numerical framework for discretization of space variables. The current paper contributes to a space-time spectral-order Sinc-collocation method which is balanced in both time and space variables. In order to deal with the initial singularity of solution in the time direction, the Sinc-collocation method with single exponential (SE) transformation is used for the first time, which is an effective method to deal with the singularity of the equation. Also, we fully analyse the error bounds of the method which show the exponential convergence rate in space and time. Meanwhile, we consider several test problems for examining the suggested scheme. The experiment results highlight a good precision of the new Sinc-colloction method and the correctness of the theoretical prediction for this kind nonlocal heat problems.

Optimal Finite-Dimensional Controller of the Stochastic Differential Object’s State by Its Output. I. Incomplete Precise Measurements

The well-known problem of synthesizing the optimal on the average and on given time interval of the inertial control law for a continuous stochastic object if only a part of its state variables are accurately measured is considered. Due to the practical unrealizability of its classical infinite-dimensional Stratonovich-Mortensen solution, it is proposed to limit ourselves to optimizing the structure of a finite-dimensional dynamic controller, whose order is chosen by the user. This finiteness allows using a truncated version of the a posteriori probability density that satisfies a deterministic partial differential integrodifferential equation. Using the Krotov extension principle, sufficient optimality conditions for the structural functions of the controller and the Lagrange–Pontryagin equation for finding their extremals are obtained. It is shown that in particular cases of the absence of measurements, complete measurements and taking into account only the values of incomplete measurements, the proposed controller turns out to be static (inertialess), and the relations for its synthesis coincide with the known ones. For a dynamic controller, algorithms for finding each of its structural functions are given.

ARA-Sumudu Method for Solving Volterra Partial Integro-Differential Equations

ARA-Sumudu Method for Solving Volterra Partial Integro-Differential Equations

Combination of discrete technique on graded meshes with barycentric rational interpolation for solving a class of time-dependent partial integro-differential equations with weakly singular kernels

Combination of discrete technique on graded meshes with barycentric rational interpolation for solving a class of time-dependent partial integro-differential equations with weakly singular kernels

Primary and subharmonic resonance features of a graphene nanoplatelet strengthened beam in a thermal ambient directly attacking nonlinear integro-partial differential governing equations

The nonlinear aspects of advanced structures may be utilized for design purposes and making use of advantages and avoiding disadvantages of such structures. On the other hand, developing information from secondary resonance examinations of advanced structures is worthwhile for engineering objectives such as mass detection, energy harvesting and damage detection of systems likewise structures. Moreover, a precise solution methodology discards any possibility of qualitative errors and incorrect prejudice. Accordingly, for the first time, the nonlinear primary resonance alongside the subharmonic resonance of an advanced beam structure strengthened by means of graphene nanoplatelets in a thermal ambient is addressed in details employing the direct method of multiple scales. With the aim of defeating to the disadvantageous influences of the discretization procedure on the nonlinear dynamics of the graphene nanoplatelet strengthened beam, the nonlinear integro-partial differential equation of motion in accompanied by the in-homogeneous boundary conditions are attacked directly resorting to the method of multiple scales. It is shown that how an FGX distribution pattern of graphene nanoplatelets for a beam subjected to a subharmonic resonance excitation relative to an FGO distribution pattern effectively extends the range of excitation frequency that leads to a bistable solution. This deduction is very proficient for harvesting energy scopes. A similar finding is observed for a CC beam with respect to a CS beam. It is seen that when an FGX beam is exposed to a subharmonic resonance in a wider range of force amplitude experiences a non-trivial equilibrium state relative to an FGO beam. However, the strengthening of the beam with graphene nanoplatelets decreases the lower bound of the excitation frequency required for the activation of subharmonic resonance more significant for an FGX distribution pattern. Moreover, the primary resonance frequency response shows different trend prior to and after the situation of the external force frequency to the first natural frequency ratio equal to one for an FGX and an FGO graphene nanoplatelet strengthened beams. The presented results open a new view of the future of engineering designs for various engineering scopes such as energy harvesting, mass detection and damage detection purposes employing advanced structures.

On the Uniqueness of the Bounded Solution for the Fractional Nonlinear Partial Integro-Differential Equation with Approximations

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert the fractional nonlinear equation with variable coefficients to an implicit integral equation, which is a powerful method of studying the uniqueness of solutions. Furthermore, we construct algorithms for finding analytic and approximate solutions using Adomian’s decomposition method and recurrence relation with the order convergence analysis. Finally, an illustrative example is presented to demonstrate constructions for the numerical solution using MATHEMATICA.

Analysis of a High-Accuracy Numerical Method for Time-Fractional Integro-Differential Equations

A high-order finite difference numerical scheme based on the compact difference operator is proposed in this paper for time-fractional partial integro-differential equations with a weakly singular kernel, where the time-fractional derivative term is defined in the Riemann-Liouville sense. Here, the stability and convergence of the constructed compact finite difference scheme are proved in L∞ norm, with the accuracy order O(τ2+h4), where τ and h are temporal and spatial step sizes, respectively. The advantage of this numerical scheme is that arbitrary parameters can be applied to achieve the desired accuracy. Some numerical examples are presented to support the theoretical analysis.

A reduced-order model based on integrated radial basis functions with partition of unity method for option pricing under jump–diffusion models

The current research aims to develop a fast, stable and efficient numerical procedure for solving option pricing problems in jump–diffusion models. A backward partial integro-differential equation (PIDE) with diffusion and advection terms was investigated. Up to the best knowledge of the authors, some special numerical methods and strategies must be selected to solve advection–diffusion problems with reliable stability and accuracy. For the mentioned aims, the first- and second-order derivatives are approximated by integrated radial basis function based on partition of unity method. The IRBF-PU method is local mesh-free method that provides high order accurate result and is flexible for PDEs problems with sufficiently smooth initial conditions and also has a moderate condition number. In particular, we highlight European and American style put options, whose underlying asset follows a jump–diffusion model. For the distribution of the jumps, the Merton and Kou models are studied. Furthermore, the main model is classified in advection-x-diffusion category. As a result, we must increase the number of collocation points as well as the time steps to arrive at the final time. This procedure lengthens the execution time. To address this issue, we use the proper orthogonal decomposition (POD) method to reduce the size of the final algebraic system of equations. This numerical procedure is known as the proper orthogonal decomposition-IRBF-PU method (POD-IRBF-PU). The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the IRBF-PU and POD-IRBF-PU methods are efficient and reliable numerical methods to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering.

The relationship between controllability, optimal testing resource allocation, and incubation-latent period mismatch as revealed by COVID-19.

The severe shortfall in testing supplies during the initial COVID-19 outbreak and ensuing struggle to manage the pandemic have affirmed the critical importance of optimal supply-constrained resource allocation strategies for controlling novel disease epidemics. To address the challenge of constrained resource optimization for managing diseases with complications like pre- and asymptomatic transmission, we develop an integro partial differential equation compartmental disease model which incorporates realistic latent, incubation, and infectious period distributions along with limited testing supplies for identifying and quarantining infected individuals. Our model overcomes the limitations of typical ordinary differential equation compartmental models by decoupling symptom status from model compartments to allow a more realistic representation of symptom onset and presymptomatic transmission. To analyze the influence of these realistic features on disease controllability, we find optimal strategies for reducing total infection sizes that allocate limited testing resources between 'clinical' testing, which targets symptomatic individuals, and 'non-clinical' testing, which targets non-symptomatic individuals. We apply our model not only to the original, delta, and omicron COVID-19 variants, but also to generically parameterized disease systems with varying mismatches between latent and incubation period distributions, which permit varying degrees of presymptomatic transmission or symptom onset before infectiousness. We find that factors that decrease controllability generally call for reduced levels of non-clinical testing in optimal strategies, while the relationship between incubation-latent mismatch, controllability, and optimal strategies is complicated. In particular, though greater degrees of presymptomatic transmission reduce disease controllability, they may increase or decrease the role of non-clinical testing in optimal strategies depending on other disease factors like transmissibility and latent period length. Importantly, our model allows a spectrum of diseases to be compared within a consistent framework such that lessons learned from COVID-19 can be transferred to resource constrained scenarios in future emerging epidemics and analyzed for optimality.