Integrable Partial Differential Equations Research Articles

Mechanisms of resistance to CAR-T cell immunotherapy: Insights from a mathematical model

Chimeric Antigen Receptor (CAR)-T cell therapy long-term follow-up studies revealed non-durable remissions in a significant number of patients. Some of the mechanisms underlying these relapses include poor CAR T cell cytotoxicity or persistence, as well as antigen loss or lineage switching in tumor cells. In order to investigate how antigen-mediated resistance mechanisms affect therapy outcomes, we develop a mathematical model based on a set of integral-partial differential equations. Using a continuous variable to describe the level of antigen expression of tumor cells, we recapitulated important cellular mechanisms across patients with different therapeutic responses. Fitted with clinical data, the model successfully captured the dynamics of tumor and CAR-T cells for several hematological cancers. Furthermore, the role played by these mechanisms are explored with regard to different biological scenarios, such as pre-existing or aquired mutations, providing a deeper understanding of key factors underlying resistance to CAR-T cell immunotherapy.

Vortex-Induced Nonlinear Bending Vibrations of Suspension Bridges with Static Wind Loads

A low stiffness makes long-span suspension bridges sensitive to loads, and this sensitivity is particularly significant for wind-induced nonlinear vibrations. In the present paper, nonlinear vibrations of suspension bridges under the combined effects of static and vortex-induced loads are explored using the nonlinear partial differential–integral equation that models the plane bending motion of suspension bridges. First, we discretized the differential–integral equation through the Galerkin method to obtain the nonlinear ordinary differential equation that describes the vortex-induced vibrations of the bridges at the first-order symmetric bending mode. Then, the approximate analytical solution of the ordinary differential equation was obtained using the multiple scales method. Finally, the analytical solution was applied to reveal the relationships between the vibration amplitude and other parameters, such as the static wind load, the frequency of dynamic load, structural stiffness, and damping. The results show that the static wind load slightly impacts the bridge’s vibrations if its influence on the natural frequency of bridges is ignored. However, the bridge’s vibrations are sensitive to the load frequency, structural stiffness, and damping. The vibration amplitude, as a result, may dramatically increase if the three parameters decrease.

Multi-harmonic resonance of pipes conveying fluid with pulsating flow

The parametric excitation analysis of pipes conveying pulsating fluid usually focuses on the principal parametric resonance. In this paper, the principal parametric resonance and the combination resonance under parametric excitation are compared for the first time. A dynamic model of a fluid-conveying pipe with elastic torsional constraints at both ends is established. The nonlinear partial differential-integral equation is derived for governing the transverse vibration. The natural frequencies and modes of pipe are obtained. Moreover, the analytical solutions of stable region and nonlinear amplitude-frequency responses for three kinds of parametric resonances are derived. Then, the analytical results are verified numerically. The results show that there is no differential-type combination parametric resonance for pipes supported at both ends. Increasing material damping or torsional stiffness and decreasing mean velocity can increase the stable region of the pipe. However, it is interesting to note that when combination parametric resonance occurs between symmetric and antisymmetric modes, the effect of mean velocity on stability is reduced. In addition, more than one form of resonance occurs in some regions. The sum-type combination resonance is easier to be excited than the principal parametric resonance. Among them, the combination of first-fourth mode is the most sensitive to the torsional boundary, and its response is the largest. In summary, this paper demonstrates that the combination parametric resonance, which has not been researched deeply yet, can be more destructive than the principal parametric resonance.

Exact Solutions and Conservation Laws of A (2+1)-dimensional Combined Potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili Equation

This article investigates a sixth order integrable nonlinear partial differential equation model that fulfills the Hirota N-soliton. Space and time-dependent shift, rotation and space-dependent shift, time and space translations, and time and space dilations Lie point symmetries are presented methodically. Under a specific point symmetries, the Lie point symmetries lead to group invariant solutions. The significance of conservation laws of the underlying equation are shown. The results are quite accurate in recreating complex waves and the dynamics of their interactions.

Non-homogeneous impulsive time fractional heat conduction equation

This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.

Higher-dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of an integrable evolution equation in one spatial dimension. Do there exist integrable analogs of the modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.

Interpolation synthesis of thecontrollers for a multi-motorelectric drive system containingan elastically linked element

Partial differential equations, integral, differential, or other equations describe multi-motor automatic electric drive systems containing elastic conveyor belts. Because of the elastic and distributive nature of the system parameters, the transfer function describing them is often a complex expression, containing not only the arguments as a linear system but also the inertial and transcendental components. This makes the precise control of tension and speed synchronously much more complicated than the centralized parameter system. A promising numerical solution based on the real interpolation method will simplify the procedure for synthesizing control loops while preserving the characteristic properties of objects with distributed parameters. The objective of the study is to propose a feasible solution for synthesizing the regulators based on the real interpolation method; it allows direct operation with the original transfer function containing the inertial and transcendental components. In this paper, we proposed an approach to synthesize the control system for objects with distributed parameters using the real interpolation method to reduce computational capacity and synthesis error while preserving the properties of this object class. Building an experimental model of the two-motor electric drive system containing an elastic conveyor to verify the effectiveness of the proposed algorithm. The simulation and experimental results indicate that the control system with the received regulators operating stably and meets the required quality criteria. It proves the efficiency of the synthesis algorithm based on the real interpolation method.

An optimal System of Lie Subalgebras and Group-Invariant Solutions with Conserved Currents of a (3+1)-D Fifth-Order Nonlinear Model I with Applications in Electrical Electronics, Chemical Engineering and Pharmacy

AbstractHigher-dimensional nonlinear integrable partial differential equations are significant as they often describe diverse phenomena in nonlinear systems like laser radiations in a plasma, optical pulses in the glass fibres, fluid mechanics, radio waves in the ion sphere, condensed matter and electromagnetics. This article shows an analytical investigation of a (3+1)-dimensional fifth-order nonlinear model with KdV forming its main part. Lie group analysis of the model is performed through which its infinitesimal generators are obtained. These generators are engaged in the construction of an optimal system of Lie subalgebra in one dimension. Moreover, members of the system secured are utilized in reducing the underlying model to ordinary differential equations (ODEs) for possible exact solutions. In consequence, we achieve various functions, ranging from trigonometric, logarithmic, rational, to hyperbolic. In addition, the results found constitute diverse solitonic solutions such as complex, topological kink and anti-kink, trigonometric and bright. We utilize the power series technique to obtain a series solution of the most complicated ordinary differential equation with forty-four terms. In addition, we reveal the dynamics of these solutions via graphical depictions. In the end, we constructed conserved currents of the underlying equation through the use of the multiplier technique. Further, we utilize the optimal system of the underlying model to derive more conserved vectors using Ibragimov’s theorem for conservation laws.

On the Solutions of the Fractional-Order Sawada–Kotera–Ito Equation and Modeling Nonlinear Structures in Fluid Mediums

This study investigates the wave solutions of the time-fractional Sawada–Kotera–Ito equation (SKIE) that arise in shallow water and many other fluid mediums by utilizing some of the most flexible and high-precision methods. The SKIE is a nonlinear integrable partial differential equation (PDE) with significant applications in shallow water dynamics and fluid mechanics. However, the traditional numerical methods used for analyzing this equation are often plagued by difficulties in handling the fractional derivatives (FDs), which lead to finding other techniques to overcome these difficulties. To address this challenge, the Adomian decomposition (AD) transform method (ADTM) and homotopy perturbation transform method (HPTM) are employed to obtain exact and numerical solutions for the time-fractional SKIE. The ADTM involves decomposing the fractional equation into a series of polynomials and solving each component iteratively. The HPTM is a modified perturbation method that uses a continuous deformation of a known solution to the desired solution. The results show that both methods can produce accurate and stable solutions for the time-fractional SKIE. In addition, we compare the numerical solutions obtained from both methods and demonstrate the superiority of the HPTM in terms of efficiency and accuracy. The study provides valuable insights into the wave solutions of shallow water dynamics and nonlinear waves in plasma, and has important implications for the study of fractional partial differential equations (FPDEs). In conclusion, the method offers effective and efficient solutions for the time-fractional SKIE and demonstrates their usefulness in solving nonlinear integrable PDEs.

Numerical solution for a class of evolution differential equations with [formula omitted]-Laplacian and memory

In this paper, we study a partial integral differential equation with p-Laplacian using a mixed finite element method. Two stable and convergent fixed point schemes are proposed to solve the nonlinear algebraic system. Using the implementation of the method in Matlab environment, we numerically analyse the convergence with an example. Some other examples, which illustrate several asymptotic behaviours and some localization effects of the solutions, are presented.

Position Measurement Based Slave Torque Feedback Control for Teleoperation Systems With Time-Varying Communication Delays

Bilateral teleoperation system is referred to as a promising technology to extend human actions and intelligence to manipulating objects remotely. For the tracking control of teleoperation systems, velocity measurements are necessary to provide feedback information. However, due to hardware technology and cost constraints, the velocity measurements are not always available. In addition, the time-varying communication delay makes it challenging to achieve tracking task. This paper provides a solution to the issue of real-time tracking for teleoperation systems, subjected to unavailable velocity signals and time-varying communication delays. In order to estimate the velocity information, immersion and invariance (I&I) technique is employed to develop an exponential stability velocity observer. For the proposed velocity observer, a linear relationship between position and observation state is constructed, through which the need of solving partial differential and certain integral equations can be avoided. Meanwhile, the mean value theorem is exploited to separate the observation error terms, and hence, all functions in our observer can be analytically expressed. With the estimated velocity information, a slave-torque feedback control law is presented. A novel Lyapunov-Krasovskii functional is constructed to establish asymptotic tracking conditions. In particular, the relationship between the controller design parameters and the allowable maximum delay values is provided. Finally, simulation and experimental results reveal that the proposed velocity observer and controller can guarantee that the observation errors and tracking error converge to zero.

Study on the symmetries and conserved quantities of flexible mechanical multibody dynamics

<abstract> <p>In this paper, in order to provides a powerful new tool for quantitative and qualitative analysis of dynamics properties in flexible mechanical multibody systems, the symmetry theory and numerical algorithms for preserving structure in modern analytical mechanics is introduced into flexible multibody dynamics. First, taking the hub-beam systems as an example, the original nonlinear partial differential-integral equations of the system dynamics model are discretized into the finite-dimensional Lagrange equations by using the assumed modal method. Second, the group analysis theory is introduced and the criterion equations and the corresponding conserved quantities of Noether symmetries are given according to the invariance principle, which provide an effective way for analytic integral theory of dynamic equations. Finally, a conserved quantity-preserving numerical algorithm is constructed by coordinates incremental discrete gradient, which makes full use of the invariance of conserved quantity to eliminate the error consumption for a long time. The simulation results show that the deeper mechanical laws and motion characteristics of flexible mechanical multibody systems dynamics can be obtained with the help of symmetries and conserved quantities, which can provide reference for more precise dynamic optimization design and advanced control of systems.</p> </abstract>

Reflected generalized discontinuous BSDEs with rcll barrier and an obstacle problem of IPDE with nonlinear Neumann boundary conditions

Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.

Spectral Parameter as a Group Parameter

A large class of integrable non-linear partial differential equations is characterized by the existence of the associated linear problem (in the case of two independent variables, known as a Lax pair) containing the so-called spectral parameter. In this paper, we present and discuss the conjecture that the spectral parameter can be interpreted as the parameter of some one-parameter groups of transformation, provided that it cannot be removed by any gauge transformation. If a non-parametric linear problem for a non-linear system is known (e.g., the Gauss–Weingarten equations as a linear problem for the Gauss–Codazzi equations in the geometry of submanifolds), then, by comparing both symmetry groups, we can find or indicate the integrable cases. We consider both conventional Lie point symmetries and the so-called extended Lie point symmetries, which are necessary in some cases. This paper is intended to be a review, but some novel results are presented as well.

Integrable partial differential equations and Lie–Rinehart algebras

We develop the method for constructing Lax representations of pdes via the twisted extensions of their algebras of contact symmetries by generalizing the construction to the Lie–Rinehart algebras. We present examples of application of the proposed technique.

A Hybrid SIE-PDE Formulation Without Boundary Condition Requirement for Transverse Magnetic Electromagnetic Analysis

A hybrid surface integral equation partial differential equation (SIE-PDE) formulation without the boundary condition requirement is proposed to solve the transverse magnetic (TM) electromagnetic problems. In the proposed formulation, the computational domain is decomposed into two overlapping domains: the SIE and PDE domains. In the SIE domain, complex structures with piecewise homogeneous media, e.g., highly conductive media, are included. An equivalent model for those structures is constructed by replacing them with the background medium and introducing a surface equivalent electric current density on an enclosed boundary to represent their electromagnetic effects. The remaining computational domain and homogeneous background medium replaced domain consist of the PDE domain, in which inhomogeneous or non-isotropic media are included. Through combining the surface equivalent electric current density and the inhomogeneous Helmholtz equation, a hybrid SIE-PDE formulation is derived. It requires no boundary conditions, and is mathematically equivalent to the original physical model. Through careful construction of basis functions to expand electric fields and the equivalent current density, the discretized formulation is made compatible with the SIE and PDE domain interface. The accuracy and efficiency are validated through two numerical examples. Results show that the proposed SIE-PDE formulation can obtain accurate results, and significant performance improvements in terms of CPU time and memory consumption compared with the FEM are achieved.

On the Double ARA-Sumudu Transform and Its Applications

The main purpose of this work is to present a new double transform called the double ARA-Sumudu transform (DARA-ST). The application of the new double transform to some basic functions and the master properties are introduced. The convolution and existence theorems are also presented and proved. These new results are implemented to obtain the solution of partial differential equations (PDEs), integral equations (IEs) and functional equations. We obtain new formulas for solving families of PDEs. The latter ones are used to obtain exact solutions of some familiar PDEs such as the telegraph equation, the advection–diffusion equation, the Klein–Gordon equation and others. Moreover, a simple formula for solving a special kind of integral equations is presented and implemented in some applications. The outcomes show that DARA-ST is useful and efficient in handling such kinds of equations.

Extensions of Gronwall-Bellman type integral inequalities with two independent variables

Abstract In this paper, we establish several kinds of integral inequalities in two independent variables, which improve well-known versions of Gronwall-Bellman inequalities and extend them to fractional integral form. By using these inequalities, we can provide explicit bounds on unknown functions. The integral inequalities play an important role in the qualitative theory of differential and integral equations and partial differential equations.

Analysis of Applied Mathematics

Mathematics applied to applications involves using mathematics for issues that arise in various fields, e.g., science, engineering, engineering, or other areas, and developing new or better techniques to address the demands of the unique challenges. We consider it applied math to apply maths to problems in the real world with the double purpose of describing observed phenomena and forecasting new yet unknown phenomena. Thus, the focus is on math, e.g., creating new techniques to tackle the issues of the unique challenges and the actual world. The issues arise from a variety of applications, including biological and physical sciences as well as engineering and social sciences. They require knowledge of different branches of mathematics including the analysis of differential equations and stochastics. They are based on mathematical and numerical techniques. Most of our faculty and students work directly with the experimentalists to watch their research findings come to life. This research team investigates mathematical issues arising out of geophysical, chemical, physical, and biophysical sciences. The majority of these problems are explained by time-dependent partial integral or ordinary differential equations. They are also accompanied by complex boundary conditions, interface conditions, and external forces. Nonlinear dynamical systems provide an underlying geometrical and topological model for understanding, identifying, and quantifying the complex phenomena in these equations. The theory of partial differential equations lets us correctly formulate well-posed problems and study the behavior of solutions, which sets the stage for effective numerical simulations. Nonlocal equations result from the macroscopically modeling stochastic dynamical systems characterized by Levy noise and the modeling of long-range interactions. They also provide a better understanding of anomalous diffusions.

Detailed balance for particle models of reversible reactions in bounded domains.

In particle-based stochastic reaction-diffusion models, reaction rates and placement kernels are used to decide the probability per time a reaction can occur between reactant particles and to decide where product particles should be placed. When choosing kernels to use in reversible reactions, a key constraint is to ensure that detailed balance of spatial reaction fluxes holds at all points at equilibrium. In this work, we formulate a general partial-integral differential equation model that encompasses several of the commonly used contact reactivity (e.g., Smoluchowski-Collins-Kimball) and volume reactivity (e.g., Doi) particle models. From these equations, we derive a detailed balance condition for the reversible A + B ⇆ C reaction. In bounded domains with no-flux boundary conditions, when choosing unbinding kernels consistent with several commonly used binding kernels, we show that preserving detailed balance of spatial reaction fluxes at all points requires spatially varying unbinding rate functions near the domain boundary. Brownian dynamics simulation algorithms can realize such varying rates through ignoring domain boundaries during unbinding and rejecting unbinding events that result in product particles being placed outside the domain.