PDEs Of Type Research Articles

A note on a NSFD discretization of a Sobolev type PDE

Sobolev type PDEs are used to model a broad range of physical phenomena. However, their discretizations by means of finite differences have generally been implicit schemes. We demonstrate that explicit, two level discrete-time representations can exist within the context of the NSFD methodology. One consequence of our explicit construction is that the step-sizes are expressible in terms of the parameters appearing in the original PDE.

Reduced Variance Random Batch Methods for Nonlocal PDEs

Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random partitioning of the particle ensemble into smaller batches at each time step. The interaction of each particle within these batches is then evolved until the subsequent time step. This approach effectively decreases the computational cost by an order of magnitude while increasing the amount of fluctuations due to the random partitioning. In this work, we propose a variance reduction technique for RBM applied to nonlocal PDEs of Fokker-Planck type based on a control variate strategy. The core idea is to construct a surrogate model that can be computed on the full set of particles at a linear cost while maintaining enough correlations with the original particle dynamics. Examples from models of collective behavior in opinion spreading and swarming dynamics demonstrate the great potential of the present approach.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Abstract 487: ITI-1020, a novel PDE type 1 inhibitor, reduces tumor growth in triple negative breast cancer (TNBC) mouse models

Abstract Breast cancer is the most common cancer in women. Unfortunately, some patients do not benefit from most therapies and may even develop resistance to previously effective treatments leading to disease progression. Therefore, treatment resistance remains a significant challenge and new treatments are needed to overcome this problem. ITI-1020 is a potent and highly selective inhibitor of the phosphodiesterase 1 (PDE1) enzyme. Given the role of PDE1 in the regulation of essential cellular functions, including among others the transition of monocytes to macrophages and potentially the infiltration of macrophages into the tumor microenvironment (TME), we hypothesize that ITI-1020 will alter macrophage infiltration and polarization, as well as improve T cell function and recruitment into the TME, thereby promoting antitumor immunity in murine cancer models. We tested this hypothesis by studying the effect of ITI-1020 given, alone or in combination a PD-1 immune checkpoint inhibitor, on tumor growth, and TME composition of E0771 and 4T1 tumor-bearing mice, two syngeneic models of TNBC. Mice treated with ITI-1020 alone or in combination with anti-PD-1 antibody showed significant inhibition of tumor growth compared to isotype controls in both type of tumors. Flow cytometry analysis of the tumor microenvironment showed a significant shift in the polarization of macrophages toward a more inflammatory phenotype, and/or a reduction in the number of exhausted T cells. Furthermore, scRNAseq analysis of tumors from mice treated with ITI-1020 in monotherapy showed altered expression of genes involved in inflammatory processes and cell proliferation within the tumor microenvironment and in the cancer cells. These results suggest that ITI-1020 promotes antitumor immunity by a novel mechanism (PDE1 inhibition), leading to tumor growth inhibition in E0771 and 4T1 tumor models. Translating these findings to the clinic may provide a new hope for the treatment of refractory or resistant TNBC tumors. Citation Format: Maud Voisin, Aksinija A. Kogan, Stephen Goding, Gretchen L. Snyder, Robert E. Davis. ITI-1020, a novel PDE type 1 inhibitor, reduces tumor growth in triple negative breast cancer (TNBC) mouse models [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 487.

Uniformly exponential stability of semi-discrete scheme for a vibration cable with a tip mass under observer-based feedback control

In this paper, we consider uniformly exponential approximation for a vibrating cable with tip mass under a non-collocated output stabilizing feedback control. By designing an observer-based output feedback control, the closed-loop system is composed of the coupled same type of PDEs and ODEs. By order reduction method, we find a global Lyapunov functional for the closed-loop system. The closed-loop system is then semi-discretized by the finite difference method. For the discrete systems, we also construct the Lyapunov functions. The uniform exponential stability of the semi-discretized systems is then established analogously as the proof for the continuous counterpart via an indirect Lyapunov functional approach.

Canonical F-planar mappings of spaces with affine connection onto 3-symmetric spaces

We consider canonical F-planar mappings of spaces with affine connection onto 3-symmetric spaces. The main equations for the mappings have been obtained as a closed system of PDEs of Cauchy type in covariant derivatives. We have found the maximum numbers of essential parameters which the general solution of the system depends on.

Solving Some Partial Differential Equations by Using Double Laplace Transform in the Sense of Nonconformable Fractional Calculus

In this paper, we introduce the non-conformable double Laplace transform. Its properties are studied, and it is applied to solve some fractional PDEs involving the nonconformable fractional derivative. Graphical representations of the obtained solutions are shown in figures. The study shows that this transform is effective and easy to apply to create an exact solution for types of fractional PDEs.

Nonlinear Radiative Nanofluidic Hydrothermal Unsteady Bidirectional Transport with Thermal/Mass Convection Aspects

The collective effect of thermal and mass convection along with the significance of thermal radiation, heat source/sink, and magneto-nanofluid are considered. A bi-directional stretching device is used to generate the symmetry of the flowing structure. Nonlinear behavior of thermal radiation is considered here. The magnetic field is considered non-uniform and vertically upward. Significances of pedesis motion and Ludwig–Soret are also revealed in an innovative way with heat source/sink effects. The concept of symmetry is used to transmute the transport equations from PDE type to nonlinear ODE type. We solved the transformed setup numerically by adopting Keller-box method criteria with the targeted accuracy rate. Graphical interpretations are explored with code verification. It is important to conclude that friction coefficients decline for incremental values of stretching parameter (0.1≤α≤0.9), magnetic field (0.3≤M≤0.9), and unsteady parameter (0.2≤Λ≤0.9) along with the bidirectional velocity components, and the rate of heat transmission rises with temperature ratio (1.3≤Γ≤1.7) and temperature Biot number (0.3≤BiT≤0.9) amplification. Moreso, the rate of mass transfer is enhanced with growing values of pedesis motion (0.2≤Nb≤0.6), unsteady parameter and concentration Biot number (0.3≤BiC≤0.9) with opposite effect when the Ludwig–Soret parameter (0.3≤Nt≤0.6) is boosted.

Null Polygons in Conformal Gauge Theory.

We consider correlation functions of single trace operators approaching the cusps of null polygons in a double-scaling limit where so-called cusp times t_{i}^{2}=g^{2}logx_{i-1,i}^{2}logx_{i,i+1}^{2} are held fixed and the 't Hooft coupling is small. With the help of stampedes, symbols, and educated guesses, we find that any such correlator can be uniquely fixed through a set of coupled lattice PDEs of Toda type with several intriguing novel features. These results hold for most conformal gauge theories with a large number of colors, including planar N=4 SYM.

Numerical and analytical results of the 1D BBM equation and 2D coupled BBM-system by finite element method

In this paper, the numerical solutions are proposed for the 1D Benjamin–Bona–Mahony (BBM) equation and 2D coupled BBM system by using Galerkin finite element technique. In this regard, the cubic B-splines and linear triangular elements are used, respectively. In 1D space, a proposed numerical scheme is implemented to a test problem including the motion of a single solitary solution. To verify practicality and robustness of our new procedure, the error norms [Formula: see text], [Formula: see text] and three constants [Formula: see text] and [Formula: see text] are evaluated. Stability analysis of the linearized technique indicates that it is unconditionally stable. Moreover, a tsunami wave in 2D space is used to investigate the efficiency of the considered method. Also, the improved [Formula: see text]-[Formula: see text] function technique (IThChT) and the combined [Formula: see text]-[Formula: see text] function technique (ITCT) are obtained in the mentioned BBM equation. The presented methods are seen to be robust, impressive and economical to employment as compared to the existing finite difference techniques and other earlier papers for discovering the numerical solutions for numerous types of linear and nonlinear PDEs.

Barrier Option Pricing in the Sub-Mixed Fractional Brownian Motion with Jump Environment

This paper investigates the pricing formula for barrier options where the underlying asset is driven by the sub-mixed fractional Brownian motion with jump. By applying the corresponding Ito^’s formula, the B-S type PDE is derived by a self-financing strategy. Furthermore, the explicit pricing formula for barrier options is obtained through converting the PDE to the Cauchy problem. Numerical experiments are conducted to test the impact of the barrier price, the Hurst index, the jump intensity and the volatility on the value of barrier option, respectively.

Failure of Fatou type theorems for solutions to PDE of p-Laplace type in domains with flat boundaries

Let denote Euclidean n space and given k a positive integer let be a k-dimensional plane with If we first study the Martin boundary problem for solutions to the p-Laplace equation (called p-harmonic functions) in relative to We then use the results from our study to extend the work of Wolff on the failure of Fatou type theorems for p-harmonic functions in to p-harmonic functions in when Finally, we discuss generalizations of our work to solutions of p-Laplace type PDE (called -harmonic functions).

Solution of the Second Order of the Linear Hyperbolic Equation Using Cubic B-Spline Collocation Numerical Method

Wave equation is one of the second order of the linear hyperbolic equation. Telegraph equation as a special case of wave equation has interesting point to investigate in the numerical point of view. In this paper, we consider the numerical methods for one dimensional telegraph equation by using cubic B-spline collocation method. Collocation method is one method to solve the partial differential equation model problem. Cubic spline interpolation is an interpolation to a third order polynomial. This polynomial interpolate four point. B-Spline is one of spline function which related to smoothness of the partition. For every spline function with given order can be written as linear combination of those B-spline. As we known that the result of the numerical technique has difference with the exact result which we called as, so that we have an error. The numerical results are compared with the interpolating scaling function method which investigated by Lakestani and Saray in 2010. This numerical methods compared to exact solution by using RMSE (root mean square error), L2 norm error and L_∞ norm error . The error of the solution showed that with the certain function, the cubic collocation of numerical method can be used as an alternative methods to find the solution of the linear hyperbolic of the PDE. The advantages of this study, we can choose the best model of the numerical method for solving the hyperbolic type of PDE. This cubic B-spline collocation method is more efficiently if the error is relatively small and closes to zero. This accuration verified by test of example 1 and example 2 which applied to the model problem.

Observer and boundary output feedback control for coupled ODE-transport PDE

This work focuses on the observer design for a first order ODE-transport PDE coupled at the boundary points. A novel anti-collocated observer and an output feedback boundary control law are designed for an under-actuated coupled system using the backstepping method. The homo-directional hyperbolic type PDE is considered with in-domain coupling between the states. The stabilization of the coupled systems are discussed by Lyapunov theory and linear matrix inequality (LMI) approach is implemented to design the gains. The obtained results show that the observer value coincides with the actual ones and it has been demonstrated through numerical examples. The effectiveness of the output feedback controller is also illustrated.

Abundant closed-form solutions of the (3+1)-dimensional Vakhnenko-Parkes equation describing the dynamics of various solitary waves in ocean engineering

This research article is a study of the (3+1)-dimensional Vakhnenko-Parkes equation that governs the propagation of high-frequency waves in a relaxing medium. To find a variety of new analytic exact solitary wave solutions, we apply three effective schemes, such as the generalized Kudryashov method (GKM), the generalized exponential rational function method (GERFM), and the generalized Riccati equation mapping method (GREMM). Obtained solutions are crucial to understand some existing, complicated physical phenomena. Many solutions are obtained, which are expressed by the exponential, trigonometric, and hyperbolic functions, including tanh, coth, sech, csch, tan, cot, sech, csch, and their combinations. These solutions include periodic-wave solutions, kink waves, combinations of kink and multi solitons, singular solitons, singular-kink structures, and periodic solitons. The 3D and 2D plots are shown by considering the possibility of arbitrary parameters occurring in the solutions better to understand the underlying mechanisms of the stated family. The computational results of this study assure us that the proposed approaches seem useful, efficient, and reliable tools for estimating the solutions attained in the study. Moreover, the method is expected to be frequently applied to examine various types of nonlinear PDEs, which emerge in mathematical physics, ocean engineering, and science.

Linearization of nonlinear Fokker-Planck equations and applications

Let P be the space of probability measures on Rd. We associate a coupled nonlinear Fokker-Planck equation on Rd, i.e. with solution paths in P, to a linear Fokker-Planck equation for probability measures on the product space Rd×P, i.e. with solution paths in P(Rd×P). We explicitly determine the corresponding linear Kolmogorov operator L˜t using the natural tangent bundle over P with corresponding gradient operator ∇P. Then it is proved that the diffusion process generated by L˜t on Rd×P is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by L˜t in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. All this is done under merely measurability conditions on the coefficients in their measure dependence, hence in particular applies if the latter is of “Nemytskii-type”. As a consequence, we obtain the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. This realizes a programme put forward by McKean in his seminal paper of 1966 for a large class of nonlinear PDEs. As a further application we obtain a probabilistic representation of solutions to Schrödinger type PDEs on Rd×P2, through the Feynman-Kac formula for the corresponding diffusion processes.

Estimates of Mild Solutions of Navier–Stokes Equations in Weak Herz-Type Besov–Morrey Spaces

The main goal of this article is to provide estimates of mild solutions of Navier–Stokes equations with arbitrary external forces in Rn for n≥2 on proposed weak Herz-type Besov–Morrey spaces. These spaces are larger than known Besov–Morrey and Herz spaces considered in known works on Navier–Stokes equations. Morrey–Sobolev and Besov–Morrey spaces based on weak-Herz space denoted as WK˙p,qαMμs and WK˙p,qαN˙μ,rs, respectively, represent new properties and interpolations. This class of spaces and its developed properties could also be employed to study elliptic, parabolic, and conservation-law type PDEs.

An iterative algorithm for POD basis adaptation in solving parametric convection–diffusion equations

In this paper, we develop an iterative algorithm for proper orthogonal decomposition (POD) basis adaptation in solving linear parametric PDEs. Specifically, we consider the convection–diffusion equations with diffusivity as a parameter. To construct POD basis functions for the convection–diffusion equation with a small diffusivity, we need a fine-grid solver to obtain accurate solution snapshots, which leads to a large amount of computation and memory costs. Meanwhile, a coarse-grid solver is sufficient for obtaining accurate solution snapshots for the convection–diffusion equation with a large diffusivity. We aim to adapt the POD basis functions extracted from the solution snapshots of a large diffusivity for the construction of a reduced-order model at a small diffusivity without resorting to a fine-grid solver. Our POD basis adaptation method exploits the implicit dependence of solutions on diffusivity. The POD basis functions are adapted through an iterative algorithm, where the full-order model simulation at a large diffusivity and the POD-based reduced-order model simulation at a small diffusivity are implemented, alternatively. We also provide convergence analysis for our POD basis adaptation method. The algorithm and convergence analysis can be generalized to other types of linear parametric PDEs without any difficulty. Finally, we present numerical results to demonstrate the performance and accuracy of the proposed method.

Canonical almost geodesic mappings of the first type onto generalized Ricci symmetric spaces

In the paper we consider canonical almost geodesic mappings of spaces with affine connection onto m-Ricci-symmetric spaces. In particular, we studied in detail canonical almost geodesic mappings of the first type of spaces with affine connections onto 2- and 3-Ricci-symmetric spaces. In either case the main equations for the mappings have been obtained as a closed mixed system of PDEs of Cauchy type. We have found the maximum number of essential parameters which the solution of the system depends on.

Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to 10^3{times } in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).