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Journal articles on the topic 'Asymptotic analysis of solutions of PDEs'

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Author: Grafiati

Published: 10 March 2023

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1

Rabier, Patrick J. "Fredholm operators, semigroups and the asymptotic and boundary behavior of solutions of PDEs." Journal of Differential Equations 193, no. 2 (September 2003): 460–80. http://dx.doi.org/10.1016/s0022-0396(03)00094-9.

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2

Costin, O., and S. Tanveer. "Nonlinear evolution PDEs inR+×Cd: existence and uniqueness of solutions, asymptotic and Borel summability properties." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 24, no. 5 (September 2007): 795–823. http://dx.doi.org/10.1016/j.anihpc.2006.07.002.

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3

Malek, Stephane. "On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities." Abstract and Applied Analysis 2020 (January 10, 2020): 1–32. http://dx.doi.org/10.1155/2020/7985298.

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A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.

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4

Papageorgiou, Demetrios T., and Saleh Tanveer. "Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2252 (August 2021): 20210307. http://dx.doi.org/10.1098/rspa.2021.0307.

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This article studies a coupled system of model multi-dimensional partial differential equations (PDEs) that arise in the nonlinear dynamics of two-fluid Couette flow when insoluble surfactants are present on the interface. The equations have been derived previously, but a rigorous study of local and global existence of their solutions, or indeed solutions of analogous systems, has not been considered previously. The evolution PDEs are two-dimensional in space and contain novel pseudo-differential terms that emerge from asymptotic analysis and matching in the multi-scale problem at hand. The one-dimensional surfactant-free case was studied previously, where travelling wave solutions were constructed numerically and their stability investigated; in addition, the travelling wave solutions were justified mathematically. The present study is concerned with some rigorous results of the multi-dimensional surfactant system, including local well posedness and smoothing results when there is full coupling between surfactant dynamics and interfacial motion, and global existence results when such coupling is absent. As far as we know such results are new for non-local thin film equations in either one or two dimensions.

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5

Fiza, Mehreen, Hakeem Ullah, Saeed Islam, Qayum Shah, Farkhanda Inayat Chohan, and Mustafa Bin Mamat. "Modifications of the Multistep Optimal Homotopy Asymptotic Method to Some Nonlinear KdV-Equations." European Journal of Pure and Applied Mathematics 11, no. 2 (April 27, 2018): 537–52. http://dx.doi.org/10.29020/nybg.ejpam.v11i2.3194.

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In this article we have introduced the mathematical theory of multistep optimal homotopy asymptotic method (MOHAM). The proposed method is implemented to different models having system of partial differential equations (PDEs). The results obtained by proposed method are compared with Homotopy Analysis Method (HAM) and closed form solutions. The comparisons of these results show that MOHAM is simpler in applicability, effective, explicit, control the convergence through optimal constants, involve less computational work. The MOHAM is independent of the assumption of initial conditions and small parameters like Homotopy Perturbation Method (HPM), HAM, Variational Iteration Method (VIM), Adomian Decomposition Method (ADM) and Perturbation Method (PM).

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6

BECHOUCHE, PHILIPPE, NORBERT J. MAUSER, and SIGMUND SELBERG. "ON THE ASYMPTOTIC ANALYSIS OF THE DIRAC–MAXWELL SYSTEM IN THE NONRELATIVISTIC LIMIT." Journal of Hyperbolic Differential Equations 02, no. 01 (March 2005): 129–82. http://dx.doi.org/10.1142/s0219891605000415.

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We study the behavior of solutions of the Dirac–Maxwell system (DM) in the nonrelativistic limit c → ∞, where c is the speed of light. DM is a nonlinear system of PDEs obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the moving charge of the spinor. The limit c → ∞, sometimes also called post-Newtonian, yields a Schrödinger–Poisson system, where the spin and magnetic field no longer appear. We prove that DM is locally well-posed for H1 data (for fixed c), and that as c → ∞ the existence time grows at least as fast as log(c), provided the data are uniformly bounded in H1. Moreover, if the datum for the Dirac spinor converges in H1, then the solution of DM converges, modulo a phase correction, in C([0,T];H1) to a solution of a Schrödinger–Poisson system. Our results also apply to a mixed state formulation of DM, and give also a convergence result for the Pauli equation as the "semi-nonrelativistic" limit. The proof relies on modifications of the bilinear null form estimates of Klainerman and Machedon, and extends our previous work on the nonrelativistic limit of the Klein–Gordon–Maxwell system.

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7

Rabier, Patrick J. "Corrigendum to “Fredholm operators, semigroups and the asymptotic and boundary behavior of solutions of PDEs” [J. Differential Equations 193 (2003) 460–480]." Journal of Differential Equations 237, no. 1 (June 2007): 257. http://dx.doi.org/10.1016/j.jde.2007.03.010.

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8

Pravica, David W., Njinasoa Randriampiry, and Michael J. Spurr. "Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations." Axioms 9, no. 3 (July 21, 2020): 83. http://dx.doi.org/10.3390/axioms9030083.

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A family of Schwartz functions W ( t ) are interpreted as eigensolutions of MADEs in the sense that W ( δ ) ( t ) = E W ( q γ t ) where the eigenvalue E ∈ R is independent of the advancing parameter q > 1 . The parameters δ , γ ∈ N are characteristics of the MADE. Some issues, which are related to corresponding q-advanced PDEs, are also explored. In the limit that q → 1 + we show convergence of MADE eigenfunctions to solutions of ODEs, which involve only simple exponentials and trigonometric functions. The limit eigenfunctions ( q = 1 + ) are not Schwartz, thus convergence is only uniform in t ∈ R on compact sets. An asymptotic analysis is provided for MADEs which indicates how to extend solutions in a neighborhood of the origin t = 0 . Finally, an expanded table of Fourier transforms is provided that includes Schwartz solutions to MADEs.

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9

Kong, De-Xing, and Tong Yang. "Asymptotic Behavior of Global Classical Solutions of Quasilinear Hyperbolic Systems." Communications in Partial Differential Equations 28, no. 5-6 (January 7, 2003): 1203–20. http://dx.doi.org/10.1081/pde-120021192.

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10

Perelman, Galina. "Asymptotic Stability of Multi-soliton Solutions for Nonlinear Schrödinger Equations." Communications in Partial Differential Equations 29, no. 7-8 (January 11, 2004): 1051–95. http://dx.doi.org/10.1081/pde-200033754.

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11

Herrmann, L., and C. Schwab. "Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 5 (August 6, 2019): 1507–52. http://dx.doi.org/10.1051/m2an/2019016.

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We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ ℝd. The multilevel algorithm QL* which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411–449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63–102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ ℝd, d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.

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12

Fowler, Paul J., Xiang Du, and Robin P. Fletcher. "Coupled equations for reverse time migration in transversely isotropic media." GEOPHYSICS 75, no. 1 (January 2010): S11—S22. http://dx.doi.org/10.1190/1.3294572.

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Reverse time migration (RTM) images reflectors by using time-extrapolation modeling codes to synthesize source and receiver wavefields in the subsurface. Asymptotic analysis of wave propagation in transversely isotropic (TI) media yields a dispersion relation describing coupled P- and SV-wave modes. This dispersion relation can be converted into a fourth-order scalar partial differential equation (PDE). Increased computational efficiency can be achieved using equivalent coupled second-order PDEs. Analysis of the corresponding dispersion relations as matrix eigenvalue systems allows one to characterize all possible coupled linear second-order systems equivalent to a given linear fourth-order PDE and to determine which ones yield optimally efficient finite-difference implementations. Setting the shear velocity along the axis of symmetry to zero yields a simpler approximate TI wave equation that is more efficient to implement. This simpler approximation, however, can become unstable for some plausible combinations of anisotropic parameters. The same eigensystem analysis can be applied using finite vertical shear velocity to obtain solutions that avoid these instability problems.

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13

Biage, M., and J. C. C. Campos. "THE ASSIMPTOTICAL SOLUTION FOR THE STUDY OF THE TRANSITION OF A GAS-LIQUID FLOW FROM COUNTER-CURRENT TO CO-CURRENT." Revista de Engenharia Térmica 6, no. 1 (June 30, 2007): 89. http://dx.doi.org/10.5380/reterm.v6i1.61822.

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The transition of a flow of a liquid film from counter-current to co-current to a gas flow is known as the flooding phenomenon. In this paper is shown a quite criterions mathematical formulation in the form to identify with success the flooding point. One applies thus the conservation equations for a bi-dimensional and isothermical flow. Using the theorem of the PI of Vashy-Buckingham one makes a dimensional analysis in order to obtain parameters that make it possible to establish an asymptotic analysis for the a-dimensional equations and reduce a set of PDEs to a unique PDE for the thickness of the liquid film. This PDE will be decomposed in an equation for the permanent problem and another referring to the transient. It will be developed only the non–linear permanent equation applying the spectral method of collocation of Chebychev for its discretization. One intends in this paper to stress the physical interpretation of the equations for the thickness of the liquid film obtained through the asymptotic expansion and describe the characteristics of the used numeric method (spectral technique). The results are compared to others found in the literature proving that the spectral method of collocation is a very powerful technique for the solving of this kind of problem.

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14

Barnafi, Nicolás A., Luis Miguel De Oliveira Vilaca, Michel C. Milinkovitch, and Ricardo Ruiz-Baier. "Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning." Mathematics 10, no. 21 (November 3, 2022): 4096. http://dx.doi.org/10.3390/math10214096.

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In this paper we propose a new mathematical model for describing the complex interplay between skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media describing feather primordia patterning. Tissue growth, in turn, modifies the transport of morphogens (described by reaction-diffusion equations) through diverse mechanisms such as advection from the solid velocity generated by mechanical stress, and mass supply. By performing an asymptotic linear stability analysis on the coupled poromechanical-chemotaxis system (assuming rheological properties of the skin cell aggregates that reside in the regime of infinitesimal strains and where the porous structure is fully saturated with interstitial fluid and encoding the coupling mechanisms through active stress) we obtain the conditions on the parameters—especially those encoding coupling mechanisms—under which the system will give rise to spatially heterogeneous solutions. We also extend the mechanical model to the case of incompressible poro-hyperelasticity and include the mechanisms of anisotropic solid growth and feedback by means of standard Lee decompositions of the tensor gradient of deformation. Because the model in question involves the coupling of several nonlinear PDEs, we cannot straightforwardly obtain closed-form solutions. We therefore design a suitable numerical method that employs backward Euler time discretisation, linearisation of the semidiscrete problem through Newton–Raphson’s method, a seven-field finite element formulation for the spatial discretisation, and we also advocate the construction and efficient implementation of tailored robust solvers. We present a few illustrative computational examples in 2D and 3D, briefly discussing different spatio-temporal patterns of growth factors as well as the associated solid response scenario depending on the specific poromechanical regime. Our findings confirm the theoretically predicted behaviour of spatio-temporal patterns, and the produced results reveal a qualitative agreement with respect to the expected experimental behaviour. We stress that the present study provides insight on several biomechanical properties of primordia patterning.

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15

Bilal, Hazrat, Hakeem Ullah, Mehreen Fiza, Saeed Islam, Muhammad Asif Zahoor Raja, Muhammad Shoaib, and Ilyas Khan. "A Levenberg-Marquardt backpropagation method for unsteady squeezing flow of heat and mass transfer behaviour between parallel plates." Advances in Mechanical Engineering 13, no. 10 (October 2021): 168781402110408. http://dx.doi.org/10.1177/16878140211040897.

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In this study, a new computing model by developing the strength of feed-forward neural networks with Levenberg-Marquardt Method (NN-BLMM) based backpropagation is used to find the solution of nonlinear system obtained from the governing equations of unsteady squeezing flow of Heat and Mass transfer behaviour between parallel plates. The governing partial differential equations (PDEs) for unsteady squeezing flow of Heat and Mass transfer of viscous fluid are converting into ordinary differential equations (ODEs) with the help of a similarity transformation. A dataset for the proposed NN-BLMM is generated for different scenarios of the proposed model by variation of various embedding parameters squeeze Sq, Prandtl number Pr, Eckert number Ec, Schmidt number Sc and chemical-reaction-parameter [Formula: see text]. Physical interpretation to various embedding parameters is assigned through graphs for squeeze Sq, Prandtl Pr, Eckert Ec, Schmidt Sc and chemical-reaction-parameter [Formula: see text]. The processing of NN-BLMM training (T.R), Testing (T.S) and validation (V.L) is employed for various scenarios to compare the solutions with the reference results. For the fluidic system convergence analysis based on mean square error (MSE), error histogram (E.H) and regression (R.G) plots is considered for the proposed computing infrastructures performance in term of NN-BLMM. The results based on proposed and reference results match in term of convergence up to 10-02 to 10-08 proves the validity of NN-BLMS. The Optimal Homotopy Asymptotic Method (OHAM) is also used for comparison and to validate the results of NN-BLMM.

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16

Bonorino, Leonardo, and Jaime Ripoll. "On the nature of isolated asymptotic singularities of solutions of a family of quasi-linear elliptic PDE's on a Cartan–Hadamard manifold." Communications in Analysis and Geometry 27, no. 4 (2019): 791–807. http://dx.doi.org/10.4310/cag.2019.v27.n4.a2.

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17

Colbrook, Matthew J. "Computing Spectral Measures and Spectral Types." Communications in Mathematical Physics 384, no. 1 (April 11, 2021): 433–501. http://dx.doi.org/10.1007/s00220-021-04072-4.

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AbstractSpectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous and singular continuous parts) often characterise relevant physical properties such as the long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts have focused on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as the linear Schrödinger equation on $$L^2({\mathbb {R}}^d)$$ L 2 ( R d ) . Computational spectral problems in infinite dimensions have led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. We classify the computation of measures, measure decompositions, types of spectra, functional calculus, and Radon–Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from orthogonal polynomials on the real line and the unit circle (giving, for example, computational realisations of Favard’s theorem and Verblunsky’s theorem, respectively), and are applied to evolution equations on a two-dimensional quasicrystal.

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18

GAETA, GIUSEPPE, and ROSARIA MANCINELLI. "ASYMPTOTIC SCALING SYMMETRIES FOR NONLINEAR PDES." International Journal of Geometric Methods in Modern Physics 02, no. 06 (December 2005): 1081–114. http://dx.doi.org/10.1142/s0219887805000983.

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In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large x and/or t) invariant under a group G which is not a symmetry of the equation. After recalling the geometrical meaning of symmetries of differential equations — and solution-preserving maps — we provide a precise definition of asymptotic symmetries of PDEs; we deal in particular, for ease of discussion and physical relevance, with scaling and translation symmetries of scalar equations. We apply the general discussion to a class of "Richardson-like" anomalous diffusion and reaction-diffusion equations, whose solution are known by numerical experiments to be asymptotically scale invariant; we obtain an analytical explanation of the numerically observed asymptotic scaling properties. We also apply our method to a different class of anomalous diffusion equations, relevant in optical lattices. The methods developed here can be applied to more general equations, as shown by their geometrical construction.

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19

Benhamou, Mabrouk. "Applied Quantum Field Theory to General Diffusion-Reaction Phenomena." Conference Papers in Mathematics 2013 (July 17, 2013): 1–8. http://dx.doi.org/10.1155/2013/684643.

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Diffusion-reaction phenomena are generally described by parabolic differential equations (PDEs), and I am interested in those possessing solutions that fail at large time. A sophisticated method to study the large-time behavior is the Renormalization Group usually encountered in Particles-Physics and Critical Phenomena. In this paper, I review the application of such an approach. In particular, attention is paid to Quantum Field Theory techniques used for the extraction of the asymptotic solutions to PDEs. Finally, I extend discussion to the fractional-time PDEs and with noise.

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20

BABIN, A., and A. FIGOTIN. "LINEAR SUPERPOSITION IN NONLINEAR WAVE DYNAMICS." Reviews in Mathematical Physics 18, no. 09 (October 2006): 971–1053. http://dx.doi.org/10.1142/s0129055x06002851.

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We study nonlinear dispersive wave systems described by hyperbolic PDE's in ℝd and difference equations on the lattice ℤd. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi–Pasta–Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully-developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.

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21

Čiegis, Raimondas, and Remigijus Čiegis. "Numerical Stability Analysis of Solutions of PDEs." Computational Methods in Applied Mathematics 4, no. 1 (2004): 23–33. http://dx.doi.org/10.2478/cmam-2004-0002.

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Abstract In this paper we investigate two iterative methods for solving one problem of nonlinear optics. The main goal is not only to find a stationary solution but also to investigate its stability. It is shown that both methods have very different stability properties and the less stable algorithm is close to the approximation of the physically important non-stationary problem. We also propose a new iterative algorithm for solving a more complicated problem which describes the optical conjugation in stimulated Brillouin backscattering with pump depletion. This algorithm is based on a symmetrical splitting scheme and the nonlinear interaction is approximated by using the special mass conservation property of the discrete problem. Thus, we obtain a conservative iterative algorithm. The results of the numerical experiments are presented and they confirm our theoretical conclusions.

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22

Shen, Luhang, Daolun Li, Wenshu Zha, Li Zhang, and Jieqing Tan. "Physical Asymptotic-Solution nets: Physics-driven neural networks solve seepage equations as traditional numerical solution behaves." Physics of Fluids 35, no. 2 (February 2023): 023603. http://dx.doi.org/10.1063/5.0135716.

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Deep learning for solving partial differential equations (PDEs) has been a major research hotspot. Various neural network frameworks have been proposed to solve nonlinear PDEs. However, most deep learning-based methods need labeled data, while traditional numerical solutions do not need any labeled data. Aiming at deep learning-based methods behaving as traditional numerical solutions do, this paper proposed an approximation-correction model to solve unsteady compressible seepage equations with sinks without using any labeled data. The model contains two neural networks, one for approximating the asymptotic solution, which is mathematically correct when time tends to 0 and infinity, and the other for correcting the error of the approximation, where the final solution is physically correct by constructing the loss function based on the boundary conditions, PDE, and mass conservation. Numerical experiments show that the proposed method can solve seepage equations with high accuracy without using any labeled data, as conventional numerical solutions do. This is a significant breakthrough for deep learning-based methods to solve PDE.

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23

MADUREIRA, ALEXANDRE L. "HIERARCHICAL MODELING BASED ON MIXED PRINCIPLES: ASYMPTOTIC ERROR ESTIMATES." Mathematical Models and Methods in Applied Sciences 15, no. 07 (July 2005): 985–1008. http://dx.doi.org/10.1142/s0218202505000662.

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We analyze approximation properties of dimension reduction models that are based on mixed principles. The problems of interest are elliptic PDEs in thin domains. The goal is to obtain estimates that take into account both the thickness of the domain and the order of the model. The techniques involved do not require the models to be energy minimizers, and are based on asymptotic expansions for the exact and model solutions. We obtain estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).

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Bezerra Júnior, Elzon C., João Vitor da Silva, and Gleydson C. Ricarte. "Geometric estimates for doubly nonlinear parabolic PDEs." Nonlinearity 35, no. 5 (April 21, 2022): 2334–62. http://dx.doi.org/10.1088/1361-6544/ac636e.

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Abstract In this manuscript, we establish C loc α , α θ regularity estimates for bounded weak solutions of a certain class of doubly degenerate evolution PDEs, whose simplest model case is given by ∂ u ∂ t − d i v ( m | u | m − 1 | ∇ u | p − 2 ∇ u ) = f ( x , t ) in Ω T ≔ Ω × ( 0 , T ) , where m ⩾ 1, p ⩾ 2 and f belongs to a suitable anisotropic Lebesgue space. Employing intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In this scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence of our findings and approach, we address a Liouville type result for entire weak solutions of a related homogeneous problem with frozen coefficients and asymptotic estimates under a certain approximating regime, which may have their own mathematical interest. We also present examples of degenerate PDEs where our results can be applied.

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Zhang, Zhao. "Numerical Analysis and Comparison of Gridless Partial Differential Equations." International Journal of Circuits, Systems and Signal Processing 15 (August 31, 2021): 1223–31. http://dx.doi.org/10.46300/9106.2021.15.133.

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In the field of science and engineering, partial differential equations play an important role in the process of transforming physical phenomena into mathematical models. Therefore, it is very important to get a numerical solution with high accuracy. In solving linear partial differential equations, meshless solution is a very important method. Based on this, we propose the numerical solution analysis and comparison of meshless partial differential equations (PDEs). It is found that the interaction between the numerical solutions of gridless PDEs is better, and the absolute error and relative error are lower, which proves the superiority of the numerical solutions of gridless PDEs

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Brio, M., J. G. Caputo, and H. Kravitz. "Spectral solutions of PDEs on networks." Applied Numerical Mathematics 172 (February 2022): 99–117. http://dx.doi.org/10.1016/j.apnum.2021.09.021.

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Polyanin, Andrei D., and Vsevolod G. Sorokin. "Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay." Communications in Nonlinear Science and Numerical Simulation 95 (April 2021): 105634. http://dx.doi.org/10.1016/j.cnsns.2020.105634.

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28

Calogero, F., M. Euler, and N. Euler. "New evolution PDEs with many isochronous solutions." Journal of Mathematical Analysis and Applications 353, no. 2 (May 2009): 481–88. http://dx.doi.org/10.1016/j.jmaa.2008.12.038.

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29

Moradi, F., N. Moradi, M. Addam, and S. El Habib. "Existence of solutions for 4p-order PDES." Moroccan Journal of Pure and Applied Analysis 8, no. 1 (January 1, 2022): 179–90. http://dx.doi.org/10.2478/mjpaa-2022-0013.

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Abstract In this paper, we study the following nonlinear eigenvalue problem: { Δ 2 p u = λ m ( x ) u i n Ω , u = Δ u = … Δ 2 p − 1 u = 0 o n ∂ Ω . \left\{ {\matrix{ {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\Omega ,} \cr {u = \Delta u = \ldots {\Delta ^{2p - 1}}u = 0\,\,\,\,on\,\,\partial \Omega .} \cr } } \right. Where Ω is a bounded domain in ℝ N with smooth boundary ∂Ω, N ≥1, p ∈ ℕ*, m ∈ L ∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2 pu := Δ (Δ...(Δu)), 2p times the operator Δ. Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.

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30

Bhrawy, A. H., M. A. Alghamdi, and Eman S. Alaidarous. "An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions." Abstract and Applied Analysis 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/295936.

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One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

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Sarwar, S., M. A. Zahid, and S. Iqbal. "Mathematical study of fractional-order biological population model using optimal homotopy asymptotic method." International Journal of Biomathematics 09, no. 06 (August 2, 2016): 1650081. http://dx.doi.org/10.1142/s1793524516500819.

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In this paper, we study the fractional-order biological population models (FBPMs) with Malthusian, Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method (OHAM) for partial differential equations (PDEs) is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional-order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.

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Benci, Vieri, and Lorenzo Luperi Baglini. "Generalized solutions in PDEs and the Burgers' equation." Journal of Differential Equations 263, no. 10 (November 2017): 6916–52. http://dx.doi.org/10.1016/j.jde.2017.07.034.

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Malek, Stéphane. "On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity." Abstract and Applied Analysis 2017 (2017): 1–32. http://dx.doi.org/10.1155/2017/9405298.

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We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ϵ. The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ϵ of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in ϵ as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

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Liu, Hanze. "Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/156965.

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The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis and the generalized power series method. All of the geometric vector fields of the equations are obtained; the symmetry reductions and exact solutions to the equations are presented, including the exponentiated solutions and the similarity solutions. Furthermore, the exact analytic solutions are provided by the transformation technique and generalized power series method, which has shown that the combination of Lie symmetry analysis and the generalized power series method is a feasible approach to dealing with exact solutions to the variable-coefficients PDEs.

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35

Polyanin, Andrei D. "Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions." Mathematics 7, no. 5 (April 28, 2019): 386. http://dx.doi.org/10.3390/math7050386.

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The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

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Malek, Stephane. "On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities." Mathematics 8, no. 2 (February 4, 2020): 189. http://dx.doi.org/10.3390/math8020189.

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We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.

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Neilan, Michael, Abner J. Salgado, and Wujun Zhang. "Numerical analysis of strongly nonlinear PDEs." Acta Numerica 26 (May 1, 2017): 137–303. http://dx.doi.org/10.1017/s0962492917000071.

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We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

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Asmolov, Evgeny S., Tatiana V. Nizkaya, and Olga I. Vinogradova. "Accurate Solutions to Non-Linear PDEs Underlying a Propulsion of Catalytic Microswimmers." Mathematics 10, no. 9 (May 1, 2022): 1503. http://dx.doi.org/10.3390/math10091503.

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Catalytic swimmers self-propel in electrolyte solutions thanks to an inhomogeneous ion release from their surface. Here, we consider the experimentally relevant limit of thin electrostatic diffuse layers, where the method of matched asymptotic expansions can be employed. While the analytical solution for ion concentration and electric potential in the inner region is known, the electrostatic problem in the outer region was previously solved but only for a linear case. Additionally, only main geometries such as a sphere or cylinder have been favoured. Here, we derive a non-linear outer solution for the electric field and concentrations for swimmers of any shape with given ion surface fluxes that then allow us to find the velocity of particle self-propulsion. The power of our formalism is to include the complicated effects of the anisotropy and inhomogeneity of surface ion fluxes under relevant boundary conditions. This is demonstrated by exact solutions for electric potential profiles in some particular cases with the consequent calculations of self-propulsion velocities.

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Anzellotti, G. "BV solutions of quasilinear PDEs in divergence form." Communications in Partial Differential Equations 12, no. 1 (January 1987): 77–122. http://dx.doi.org/10.1080/03605308708820485.

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Jakobsen, Espen R., and Kenneth H. Karlsen. "Continuous dependence estimates for viscosity solutions of integro-PDEs." Journal of Differential Equations 212, no. 2 (May 2005): 278–318. http://dx.doi.org/10.1016/j.jde.2004.06.021.

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Schwab, Christoph, and Claude Jeffrey Gittelson. "Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs." Acta Numerica 20 (April 28, 2011): 291–467. http://dx.doi.org/10.1017/s0962492911000055.

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Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.

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Nteumagne, Bienvenue Feugang, and Raseelo J. Moitsheki. "OPTIMAL SYSTEMS AND GROUP INVARIANT SOLUTIONS FOR A MODEL ARISING IN FINANCIAL MATHEMATICS." Mathematical Modelling and Analysis 14, no. 4 (December 31, 2009): 495–502. http://dx.doi.org/10.3846/1392-6292.2009.14.495-502.

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We consider a bond‐pricing model described in terms of partial differential equations (PDEs). Classical Lie point symmetry analysis of the considered PDEs resulted in a number of point symmetries being admitted. The one‐dimensional optimal system of subalgebras is constructed. Following the symmetry reductions, we determine the group‐invariant solutions.

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van der Walt, Jan Harm. "Solutions of Smooth Nonlinear Partial Differential Equations." Abstract and Applied Analysis 2011 (2011): 1–37. http://dx.doi.org/10.1155/2011/658936.

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The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension.

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Arioli, Gianni, and Hans Koch. "Families of Periodic Solutions for Some Hamiltonian PDEs." SIAM Journal on Applied Dynamical Systems 16, no. 1 (January 2017): 1–15. http://dx.doi.org/10.1137/16m1070177.

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45

Barrasso, Adrien, and Francesco Russo. "Decoupled Mild Solutions of Path-Dependent PDEs and Integro PDEs Represented by BSDEs Driven by Cadlag Martingales." Potential Analysis 53, no. 2 (March 20, 2019): 449–81. http://dx.doi.org/10.1007/s11118-019-09775-x.

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Negri, Matteo. "A unilateral L2L^{2}-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement." Advances in Calculus of Variations 12, no. 1 (January 1, 2019): 1–29. http://dx.doi.org/10.1515/acv-2016-0028.

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AbstractWe consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic {L^{2}}-penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework of {W^{1,2}(0,T;L^{2})}.

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Abdulaziz, O., I. Hashim, and A. Saif. "Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method." Differential Equations and Nonlinear Mechanics 2008 (2008): 1–16. http://dx.doi.org/10.1155/2008/686512.

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The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.

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Salani *, Paolo. "Starshapedness of level sets of solutions to elliptic PDEs." Applicable Analysis 84, no. 12 (December 2005): 1185–97. http://dx.doi.org/10.1080/00036810412331297262.

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Gripenberg, G. "Boundary regularity for viscosity solutions to degenerate elliptic PDEs." Journal of Mathematical Analysis and Applications 352, no. 1 (April 2009): 175–83. http://dx.doi.org/10.1016/j.jmaa.2008.06.036.

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Cheng, Hongyu, and Rafael de la Llave. "Stable manifolds to bounded solutions in possibly ill-posed PDEs." Journal of Differential Equations 268, no. 8 (April 2020): 4830–99. http://dx.doi.org/10.1016/j.jde.2019.10.042.

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