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Journal articles on the topic 'Diffusion partial differential equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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1

Benedetti, I., N. V. Loi, L. Malaguti, and V. Taddei. "Nonlocal diffusion second order partial differential equations." Journal of Differential Equations 262, no. 3 (February 2017): 1499–523. http://dx.doi.org/10.1016/j.jde.2016.10.019.

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2

Bertrand, Nicolas, Jocelyn Sabatier, Olivier Briat, and Jean-Michel Vinassa. "An Implementation Solution for Fractional Partial Differential Equations." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/795651.

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The link between fractional differentiation and diffusion equation is used in this paper to propose a solution for the implementation of fractional diffusion equations. These equations permit us to take into account species anomalous diffusion at electrochemical interfaces, thus permitting an accurate modeling of batteries, ultracapacitors, and fuel cells. However, fractional diffusion equations are not addressed in most commercial software dedicated to partial differential equations simulation. The proposed solution is evaluated in an example.

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3

Bonito, Andrea, Albert Cohen, Ronald DeVore, Guergana Petrova, and Gerrit Welper. "Diffusion Coefficients Estimation for Elliptic Partial Differential Equations." SIAM Journal on Mathematical Analysis 49, no. 2 (January 2017): 1570–92. http://dx.doi.org/10.1137/16m1094476.

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4

Goyal, Kavita, and Mani Mehra. "Fast diffusion wavelet method for partial differential equations." Applied Mathematical Modelling 40, no. 7-8 (April 2016): 5000–5025. http://dx.doi.org/10.1016/j.apm.2015.10.054.

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5

Carvalho, Alexandre N., and Luiz Augusto F. Oliveira. "Delay-partial differential equations with some large diffusion." Nonlinear Analysis: Theory, Methods & Applications 22, no. 9 (May 1994): 1057–95. http://dx.doi.org/10.1016/0362-546x(94)90228-3.

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6

Aziz, Imran, and Imran Khan. "Numerical Solution of Diffusion and Reaction–Diffusion Partial Integro-Differential Equations." International Journal of Computational Methods 15, no. 06 (September 2018): 1850047. http://dx.doi.org/10.1142/s0219876218500470.

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In this paper, a collocation method based on Haar wavelet is developed for numerical solution of diffusion and reaction–diffusion partial integro-differential equations. The equations are parabolic partial integro-differential equations and we consider both one-dimensional and two-dimensional cases. Such equations have applications in several practical problems including population dynamics. An important advantage of the proposed method is that it can be applied to both linear as well as nonlinear problems with slide modification. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency and robustness of the proposed method.

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Lin, Xiaodong, Joseph W. H. So, and Jianhong Wu. "Centre manifolds for partial differential equations with delays." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 122, no. 3-4 (1992): 237–54. http://dx.doi.org/10.1017/s0308210500021090.

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SynopsisA centre manifold theory for reaction-diffusion equations with temporal delays is developed. Besides an existence proof, we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order. As an illustration, this reduction procedure is applied to the Hutchinson equation with diffusion.

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8

Zhao, Weng Cang, and Fan Wang. "Face Image Denoising Method Based on Fourth-Order Partial Differential Equations." Advanced Engineering Forum 6-7 (September 2012): 700–703. http://dx.doi.org/10.4028/www.scientific.net/aef.6-7.700.

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In order to improve the effect of face image denoising, this paper put forward several face image denoising methods based on partial differential equations, including P-M non-linear diffusion equations and fourth-order partial differential equations. We use those two methods by establishing non-linear diffusion equations and fourth-order anisotropic diffusion partial differential equation. The P-M non-linear diffusion denoising method can remove noise in intra-regions sufficiently but noise at edges can not be eliminated successfully and line-like structures can not be held very well.While the fourth-order partial differential equations denoising can retain the local detail characteristics of the original face image. Finally, through the experimental results we can see the effect of the fourth-order partial differential equations denoising is better, which makes the later face image processing more accurate and promotes the development of face image processing.

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Aljedhi, Reem Abdullah, and Adem Kılıçman. "Fractional Partial Differential Equations Associated with Lêvy Stable Process." Mathematics 8, no. 4 (April 2, 2020): 508. http://dx.doi.org/10.3390/math8040508.

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In this study, we first present a time-fractional L e ^ vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L e ^ vy-time fractional diffusion equation of European-style options. Further, we introduce a more general model based on the L e ^ vy-time fractional diffusion equation and review some recent findings associated with risk-neutral free European option pricing.

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10

Abdel-Gawad, H. I. "Solution of a class of partial differential equations. Some diffusion equations." Mathematical Methods in the Applied Sciences 12, no. 3 (March 1990): 221–28. http://dx.doi.org/10.1002/mma.1670120305.

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11

Su, Qiuyi, and Shigui Ruan. "Periodic solutions of partial functional differential equations." Proceedings of the American Mathematical Society, Series B 8, no. 13 (May 24, 2021): 145–57. http://dx.doi.org/10.1090/bproc/63.

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In this paper we study the existence of periodic solutions to the partial functional differential equation { d y ( t ) d t = B y ( t ) + L ^ ( y t ) + f ( t , y t ) , ∀ t ≥ 0 , y 0 = φ ∈ C B . \begin{equation*} \left \{ \begin {array}{l} \frac {dy(t)}{dt}=By(t)+\hat {L}(y_{t})+f(t,y_{t}), \;\forall t\geq 0,\\ y_{0}=\varphi \in C_{B}. \end{array} \right . \end{equation*} where B : Y → Y B: Y \rightarrow Y is a Hille-Yosida operator on a Banach space Y Y . For C B ≔ { φ ∈ C ( [ − r , 0 ] ; Y ) : φ ( 0 ) ∈ D ( B ) ¯ } C_{B}≔\{\varphi \in C([-r,0];Y): \varphi (0)\in \overline {D(B)}\} , y t ∈ C B y_{t}\in C_{B} is defined by y t ( θ ) = y ( t + θ ) y_{t}(\theta )=y(t+\theta ) , θ ∈ [ − r , 0 ] \theta \in [-r,0] , L ^ : C B → Y \hat {L}: C_{B}\rightarrow Y is a bounded linear operator, and f : R × C B → Y f:\mathbb {R}\times C_{B}\rightarrow Y is a continuous map and is T T -periodic in the time variable t t . Sufficient conditions on B B , L ^ \hat {L} and f ( t , y t ) f(t,y_{t}) are given to ensure the existence of T T -periodic solutions. The results then are applied to establish the existence of periodic solutions in a reaction-diffusion equation with time delay and the diffusive Nicholson’s blowflies equation.

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12

Li, Biao, and Yong Chen. "Nonlinear Partial Differential Equations Solved by Projective Riccati Equations Ansatz." Zeitschrift für Naturforschung A 58, no. 9-10 (October 1, 2003): 511–19. http://dx.doi.org/10.1515/zna-2003-9-1007.

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Based on the general projective Riccati equations method and symbolic computation, some new exact travelling wave solutions are obtained for a nonlinear reaction-diffusion equation, the highorder modified Boussinesq equation and the variant Boussinesq equation. The obtained solutions contain solitary waves, singular solitary waves, periodic and rational solutions. From our results, we can not only recover the known solitary wave solutions of these equations found by existing various tanh methods and other sophisticated methods, but also obtain some new and more general travelling wave solutions.

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13

Liu, Fu Xiang. "Manufacturing Image in Painting Based on Partial Differential Equations." Applied Mechanics and Materials 484-485 (January 2014): 949–52. http://dx.doi.org/10.4028/www.scientific.net/amm.484-485.949.

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For TV model, repair algorithm only along the gradient perpendicular to the direction of diffusion that will cause staircase effect, iterative low efficiency and generate false edge of the shortcomings. The article improved K and P parameters of manufacturing image restoration algorithm. The algorithm contains both isotropic and anisotropic diffusion, taking regional frequency difference realized in different areas using different iterative equation. The experimental results show that the algorithm is compared with the TV model algorithm has the same restorative effects, avoid the staircase effect and better than the TV model repair speed.

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14

Nadler, Boaz, Zeev Schuss, Amit Singer, and R. S. Eisenberg. "Ionic diffusion through confined geometries: from Langevin equations to partial differential equations." Journal of Physics: Condensed Matter 16, no. 22 (May 22, 2004): S2153—S2165. http://dx.doi.org/10.1088/0953-8984/16/22/015.

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15

Huang, Pan, Jun Yue, and Mao Lin Wang. "Coupled Partial Differential Equations Method for InSAS Interferogram Filtering." Advanced Materials Research 487 (March 2012): 103–6. http://dx.doi.org/10.4028/www.scientific.net/amr.487.103.

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In this paper, a coupled nonlinear diffusion partial differential equations (PDE) method for Interferometric Synthetic Aperture Sonar(InSAS) interferogram filtering was introduced. Many previous PDE methods in this area usually use Gauss pre-filtering. The choice of variance in Gauss function plays a very important role in the quality of the image obtained. Manually choice of the variance can hardly reach the self-adaptation aim. Using nonlinear diffusion equation to instead Gauss pre-filtering can overcome the disadvantage mentioned above. Numerical experiment results indicate that this coupled PDE method is able to effectively reduce the noise and preserve edge information. And it is important for InSAS real time processing.

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16

Blanco, Pablo, Paola Gervasio, and Alfio Quarteroni. "Extended Variational Formulation for Heterogeneous Partial Differential Equations." Computational Methods in Applied Mathematics 11, no. 2 (2011): 141–72. http://dx.doi.org/10.2478/cmam-2011-0008.

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AbstractWe address the coupling of an advection equation with a diffusion-advection equation, for solutions featuring boundary layers. We consider non-overlapping domain decompositions and we face up the heterogeneous problem using an extended variational formulation. We will prove the equivalence between the latter formulation and a treatment based on a singular perturbation theory. An exhaustive comparison in terms of solution and computational efficiency between these formulations is carried out.

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17

Lefebvre, Mario. "Similarity Solutions of Partial Differential Equations in Probability." Journal of Probability and Statistics 2011 (2011): 1–13. http://dx.doi.org/10.1155/2011/689427.

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Two-dimensional diffusion processes are considered between concentric circles and in angular sectors. The aim of the paper is to compute the probability that the process will hit a given part of the boundary of the stopping region first. The appropriate partial differential equations are solved explicitly by using the method of similarity solutions and the method of separation of variables. Some solutions are expressed as generalized Fourier series.

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18

Çenesiz, Y., and A. Kurt. "New Fractional Complex Transform for Conformable Fractional Partial Differential Equations." Journal of Applied Mathematics, Statistics and Informatics 12, no. 2 (December 1, 2016): 41–47. http://dx.doi.org/10.1515/jamsi-2016-0007.

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Abstract Conformable fractional complex transform is introduced in this paper for converting fractional partial differential equations to ordinary differential equations. Hence analytical methods in advanced calculus can be used to solve these equations. Conformable fractional complex transform is implemented to fractional partial differential equations such as space fractional advection diffusion equation and space fractional telegraph equation to obtain the exact solutions of these equations.

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19

Yamazaki, Kazuo. "Threshold dynamics of reaction–diffusion partial differential equations model of Ebola virus disease." International Journal of Biomathematics 11, no. 08 (November 2018): 1850108. http://dx.doi.org/10.1142/s1793524518501085.

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We study the reaction–diffusion Ebola PDE model that consists of equations that govern the evolution of susceptible, infected, recovered and deceased human individuals, as well as Ebola virus pathogens in the environment, with diffusive terms in all except the equation of the deceased human individuals. Under the setting of a spatial domain that is bounded, we prove the global well-posedness of the system; in contrast to the previous work on similar models such as cholera, avian influenza, malaria and dengue fever, diffusion coefficients may be different. Moreover, we derive its basic reproduction number, and under the condition that the diffusion coefficients of the susceptible and infected hosts are same, we prove the global stability of the disease-free-equilibrium, and uniform persistence in cases when the basic reproduction number lies beneath and above one, respectively. Again, we do not require that the diffusion coefficients of the recovered hosts be the same as the diffusion coefficients of the susceptible and infected hosts, in contrast to previous work on other models of infectious diseases. Another technical difficulty in our model is that the solution semiflow is not compact due to the lack of diffusion in the equation of the deceased human individuals; we overcome this difficulty using functional analysis techniques concerning Kuratowski measure of non-compactness.

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20

Kansal, Tarun. "Fundamental solution of the system of equations of pseudo oscillations in the theory of thermoelastic diffusion materials with double porosity." Multidiscipline Modeling in Materials and Structures 15, no. 2 (February 21, 2019): 317–36. http://dx.doi.org/10.1108/mmms-01-2018-0006.

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PurposeThe purpose of this paper to construct the fundamental solution of partial differential equations in the generalized theory of thermoelastic diffusion materials with double porosity.Design/methodology/approachThe paper deals with the study of pseudo oscillations in the generalized theory of thermoelastic diffusion materials with double porosity.FindingsThe paper finds the fundamental solution of partial differential equations in terms of elementary functions.Originality/valueAssuming the displacement vector, volume fraction fields, temperature change and chemical potential functions in terms of oscillation frequency in the governing equations, pseudo oscillations have been studied and finally the fundamental solution of partial differential equations in case of pseudo oscillations in terms of elementary functions has been constructed.

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21

Kpoumiè, Moussa El-Khalil, Khalil Ezzinbi, and David Békollè. "Nonautonomous partial functional differential equations; existence and regularity." Nonautonomous Dynamical Systems 4, no. 1 (November 27, 2017): 108–27. http://dx.doi.org/10.1515/msds-2017-0010.

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Abstract The aim of this work is to establish several results on the existence and regularity of solutions for some nondensely nonautonomous partial functional differential equations with finite delay in a Banach space. We assume that the linear part is not necessarily densely defined and generates an evolution family under the conditions introduced by N. Tanaka.We show the local existence of the mild solutions which may blow up at the finite time. Secondly,we give sufficient conditions ensuring the existence of the strict solutions. Finally, we consider a reaction diffusion equation with delay to illustrate the obtained results.

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22

Dogan, Elife, and Edward J. Allen. "Derivation of Stochastic Partial Differential Equations for Reaction-Diffusion Processes." Stochastic Analysis and Applications 29, no. 3 (April 20, 2011): 424–43. http://dx.doi.org/10.1080/07362994.2011.548987.

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23

Lotfi, El Mehdi, Mehdi Maziane, Khalid Hattaf, and Noura Yousfi. "Partial Differential Equations of an Epidemic Model with Spatial Diffusion." International Journal of Partial Differential Equations 2014 (February 10, 2014): 1–6. http://dx.doi.org/10.1155/2014/186437.

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The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

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24

Nie, Ningming, Jianfei Huang, Wenjia Wang, and Yifa Tang. "Solving spatial-fractional partial differential diffusion equations by spectral method." Journal of Statistical Computation and Simulation 84, no. 6 (May 30, 2013): 1173–89. http://dx.doi.org/10.1080/00949655.2013.803243.

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25

Kirr, E., and M. I. Weinstein. "Diffusion of Power in Randomly Perturbed Hamiltonian Partial Differential Equations." Communications in Mathematical Physics 255, no. 2 (February 4, 2005): 293–328. http://dx.doi.org/10.1007/s00220-004-1273-6.

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26

Ghanbari, Behzad, and Abdon Atangana. "An efficient numerical approach for fractional diffusion partial differential equations." Alexandria Engineering Journal 59, no. 4 (August 2020): 2171–80. http://dx.doi.org/10.1016/j.aej.2020.01.042.

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27

Deng, Shu Xian, and Ming Jun Wang. "Researches on a Class of Reaction-Diffusion Thermo-Plastic Material Equations." Advanced Materials Research 219-220 (March 2011): 1022–25. http://dx.doi.org/10.4028/www.scientific.net/amr.219-220.1022.

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This paper deals with a class of hyperbolic thermo-plastic material equation. The equation includes a reaction-diffusion-taxis partial differential equation, a reaction-diffusion partial differential equation. In the actual course of the discussion, we append a motility term in the equation. Then, the existence of unique global strong solution is proved using the theory of fractional powers of analytic semi group generators to new equation.

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Hill, James M., and Barry D. Hughes. "On the general random walk formulation for diffusion in media with Diffusivities." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 27, no. 1 (July 1985): 73–87. http://dx.doi.org/10.1017/s033427000000477x.

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AbstractA general discrete multi-dimensional and multi-state random walk model is proposed to describe the phenomena of diffusion in media with multiple diffusivities. The model is a generalization of a two-state one-dimensional discrete random walk model (Hill [8]) which gives rise to the partial differential equations of double diffusion. The same partial differential equations are shown to emerge as a special case of the continuous version of the present general model. For two states a particular generalization of the model given in [8] is presented which is not restricted to nearest neighbour transitions. Under appropriate circumstances this two-state model still yields the partial differential equations of double diffusion in the continuum limit, but an example of circumstances leading to a radically different continuum limit is presented.

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Choudhury, A. H. "Wavelet Method for Numerical Solution of Parabolic Equations." Journal of Computational Engineering 2014 (February 27, 2014): 1–12. http://dx.doi.org/10.1155/2014/346731.

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We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions.

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Abrashina-Zhadaeva, N., and N. Romanova. "Vector Additive Decomposition for 2D Fractional Diffusion Equation." Nonlinear Analysis: Modelling and Control 13, no. 2 (April 25, 2008): 137–43. http://dx.doi.org/10.15388/na.2008.13.2.14574.

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Such physical processes as the diffusion in the environments with fractal geometry and the particles’ subdiffusion lead to the initial value problems for the nonlocal fractional order partial differential equations. These equations are the generalization of the classical integer order differential equations. An analytical solution for fractional order differential equation with the constant coefficients is obtained in [1] by using Laplace-Fourier transform. However, nowadays many of the practical problems are described by the models with variable coefficients. In this paper we discuss the numerical vector decomposition model which is based on a shifted version of usual Gr¨unwald finite-difference approximation [2] for the non-local fractional order operators. We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions. Moreover, a numerical example using a finite difference algorithm for 2D fractional order partial differential equations is also presented and compared with the exact analytical solution.

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31

Zhang, Yang. "Analysis and Recognition Method of Internet Image Public Opinion Based on Partial Differential Equation." Advances in Mathematical Physics 2021 (September 14, 2021): 1–10. http://dx.doi.org/10.1155/2021/9759199.

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This article comprehensively and systematically expounds the development trends and basic theory of partial differential methods, analyzes the characteristics of sampling multiscale transformation in detail, and deeply studies the network image denoising and network image restoration methods that perform partial differential diffusion in the pixel domain and the transform domain. An adaptive diffusion method of partial differential equations is proposed. Among them, the key parameters can be adaptively changed according to the curvature and gradient of the local geometric information of the network image, and the diffusion direction and intensity of the diffusion can be controlled. First, using the principle of variation, we derive the Euler equation corresponding to the diffusion method of partial differential equations and analyze its diffusion ability using the local orthogonal coordinate system of the network image. Based on the theoretical analysis of public opinion, this article applies opinion mining technology to the online public opinion early warning system to achieve the purpose of grasping the opinions of netizens in time and guiding the trend of public opinion. Opinion mining is the use of natural language processing technology to automatically extract the emotional tendencies and evaluation objects contained in the subjective text. In the edge area of the network image, the diffusion along the edge direction should have a large diffusion coefficient, and the diffusion along the vertical edge direction should have a small diffusion coefficient; in the flat area of the network image, it diffuses to the surrounding with equal intensity, and the diffusion intensity value is relatively high. Secondly, based on the analysis of the adaptive partial differential equation diffusion method, using the half-point difference format, a numerical method for network image recognition is designed. Both theoretical analysis and experimental results show that the network image recognition model based on adaptive partial differential equation diffusion is more effective than the model based on partial differential equation recognition; at the same time, experiments show that the network image recognition model based on adaptive partial differential equation diffusion is more effective than the network image recognition model based on ordinary diffusion. The network image recognition model based on constant partial differential equation diffusion is more effective in improving the quality of network image recognition.

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Bin, Liu, Wang Bo, Li Zhuo, and Lv Yanfang. "The diffusion of CO2-brine storage based on stochastic partial differential equations." E3S Web of Conferences 206 (2020): 03031. http://dx.doi.org/10.1051/e3sconf/202020603031.

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The migration of CO2 is stochastic in heterogeneous porous media. This paper considers the CO2 diffusion with the case of steady flow in heterogeneous porous media. The partial differential equations of CO2 diffusion in random velocity field are established based on the mass conservation equations of CO2- brine two-phase flow with the change of time scale and spatial scale under the influence of heterogeneity such as permeability and porosity. The random travel process of CO2 is quantified by joint probability distributions and joint statistical moments (mean and variance), and the diffusion model of CO2 particle in random velocity field is established under the condition of non-linear and immiscibility in heterogeneous porous media. The micro mechanism of diffusion in heterogeneous porous media is revealed by numerical simulation. The general conclusion of steady state flow of CO2 diffusion in heterogeneous porous media was verified by simulating Sleipner CO2-brine storage in Norway.

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Gürbüz, Burcu. "A Computational Technique for Solving Singularly Perturbed Delay Partial Differential Equations." Foundations of Computing and Decision Sciences 46, no. 3 (September 1, 2021): 221–33. http://dx.doi.org/10.2478/fcds-2021-0015.

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Abstract In this work, a matrix method based on Laguerre series to solve singularly perturbed second order delay parabolic convection-diffusion and reaction-diffusion type problems involving boundary and initial conditions is introduced. The approximate solution of the problem is obtained by truncated Laguerre series. Moreover convergence analysis is introduced and stability is explained. Besides, a test case is given and the error analysis is considered by the different norms in order to show the applicability of the method.

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Hattaf, Khalid, and Noura Yousfi. "Global Stability for Fractional Diffusion Equations in Biological Systems." Complexity 2020 (August 5, 2020): 1–6. http://dx.doi.org/10.1155/2020/5476842.

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This paper proposes a new method of construction of Lyapunov functionals for the dynamical systems described by fractional differential equations and fractional partial differential equations. The proposed method is rigorously presented. Furthermore, the method is applied to establish the global stability of some fractional biological models with and without diffusion.

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Wu, G., Eric Wai Ming Lee, and Gao Li. "Numerical solutions of the reaction-diffusion equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 2 (March 2, 2015): 265–71. http://dx.doi.org/10.1108/hff-04-2014-0113.

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Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

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36

Adomian, G. "Application of decomposition to hyperbolic, parabolic, and elliptic partial differential equations." International Journal of Mathematics and Mathematical Sciences 12, no. 1 (1989): 137–43. http://dx.doi.org/10.1155/s0161171289000190.

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The decomposition method is applied to examples of hyperbolic, parabolic, and elliptic partial differential equations without use of linearizatlon techniques. We consider first a nonlinear dissipative wave equation; second, a nonlinear equation modeling convectlon-diffusion processes; and finally, an elliptic partial differential equation.

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Yang, Xiao-Jun, Zhi-Zhen Zhang, Tenreiro Machado, and Dumitru Baleanu. "On local fractional operators View of computational complexity: Diffusion and relaxation defined on cantor sets." Thermal Science 20, suppl. 3 (2016): 755–67. http://dx.doi.org/10.2298/tsci16s3755y.

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This paper treats the description of non-differentiable dynamics occurring in complex systems governed by local fractional partial differential equations. The exact solutions of diffusion and relaxation equations with Mittag-Leffler and exponential decay defined on Cantor sets are calculated. Comparative results with other versions of the local fractional derivatives are discussed.

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Barth, Andrea, and Andreas Stein. "A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients." SIAM/ASA Journal on Uncertainty Quantification 6, no. 4 (January 2018): 1707–43. http://dx.doi.org/10.1137/17m1148888.

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39

Reimers, Mark. "One dimensional stochastic partial differential equations and the branching measure diffusion." Probability Theory and Related Fields 81, no. 3 (April 1989): 319–40. http://dx.doi.org/10.1007/bf00340057.

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40

Teixeira, João. "Stable Schemes for Partial Differential Equations: The One-Dimensional Diffusion Equation." Journal of Computational Physics 153, no. 2 (August 1999): 403–17. http://dx.doi.org/10.1006/jcph.1999.6283.

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41

Bai, Yu-Shan, Jian-Ting Pei, and Wen-Xiu Ma. "λ-Symmetry and μ-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations." Mathematics 8, no. 7 (July 12, 2020): 1138. http://dx.doi.org/10.3390/math8071138.

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On one hand, we construct λ-symmetries and their corresponding integrating factors and invariant solutions for two kinds of ordinary differential equations. On the other hand, we present μ-symmetries for a (2+1)-dimensional diffusion equation and derive group-reductions of a first-order partial differential equation. A few specific group invariant solutions of those two partial differential equations are constructed.

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Babin, A. V., and M. I. Vishik. "Attractors of partial differential evolution equations in an unbounded domain." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 116, no. 3-4 (1990): 221–43. http://dx.doi.org/10.1017/s0308210500031498.

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SynopsisThere is a large number of papers in which attractors of parabolic reaction-diffusion equations in bounded domains are investigated. In this paper, these equations are considered in the whole unbounded space, and a theory of attractors of such equations is built. While investigating these equations, specific difficulties arise connected with the noncompactness of operators, with the continuity of their spectra, etc. Therefore some new conditions on nonlinear terms arise. In this paper weighted spaces are widely applied. An important feature of this problem is worth mentioning: namely, properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ.

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Lin, Yong Xing, Xiao Yan Xu, and Xian Dong Zhang. "A New Image Demising Method Based on Partial Differential Equations." Applied Mechanics and Materials 443 (October 2013): 22–26. http://dx.doi.org/10.4028/www.scientific.net/amm.443.22.

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In the paper, we discuss the image demising models, based on partial differential equations. It is through the use of the concept of variations in the calculus of the objective function minimization problem, defines the image processing tasks. The results show that the model expands 2d thermal diffusion equation. Therefore, it is easy to get solution is to use a simple iterative process.

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Velasco, María Pilar, David Usero, Salvador Jiménez, Luis Vázquez, José Luis Vázquez-Poletti, and Mina Mortazavi. "About Some Possible Implementations of the Fractional Calculus." Mathematics 8, no. 6 (June 2, 2020): 893. http://dx.doi.org/10.3390/math8060893.

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We present a partial panoramic view of possible contexts and applications of the fractional calculus. In this context, we show some different applications of fractional calculus to different models in ordinary differential equation (ODE) and partial differential equation (PDE) formulations ranging from the basic equations of mechanics to diffusion and Dirac equations.

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Stevens, A., and J. J. L. Velázquez. "Partial differential equations and non-diffusive structures." Nonlinearity 21, no. 12 (November 13, 2008): T283—T289. http://dx.doi.org/10.1088/0951-7715/21/12/t04.

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CRAUEL, H., P. E. KLOEDEN, and MEIHUA YANG. "RANDOM ATTRACTORS OF STOCHASTIC REACTION–DIFFUSION EQUATIONS ON VARIABLE DOMAINS." Stochastics and Dynamics 11, no. 02n03 (September 2011): 301–14. http://dx.doi.org/10.1142/s0219493711003292.

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It is shown that a stochastic partial differential equation of the reaction–diffusion type on time-varying domains obtained by a temporally continuous dependent spatially diffeomorphic transformation of a reference domain, which is bounded with a smooth boundary, generates a "partial-random" dynamical system, which has a pathwise nonautonomous pullback attractor.

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D’Acunto, Berardino, Luigi Frunzo, Vincenzo Luongo, and Maria Rosaria Mattei. "Modeling Heavy Metal Sorption and Interaction in a Multispecies Biofilm." Mathematics 7, no. 9 (August 24, 2019): 781. http://dx.doi.org/10.3390/math7090781.

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A mathematical model able to simulate the physical, chemical and biological interactions prevailing in multispecies biofilms in the presence of a toxic heavy metal is presented. The free boundary value problem related to biofilm growth and evolution is governed by a nonlinear ordinary differential equation. The problem requires the integration of a system of nonlinear hyperbolic partial differential equations describing the biofilm components evolution, and a systems of semilinear parabolic partial differential equations accounting for substrates diffusion and reaction within the biofilm. In addition, a semilinear parabolic partial differential equation is introduced to describe heavy metal diffusion and sorption. The biosoption process modeling is completed by the definition and integration of other two systems of nonlinear hyperbolic partial differential equations describing the free and occupied binding sites evolution, respectively. Numerical simulations of the heterotrophic-autotrophic interaction occurring in biofilm reactors devoted to wastewater treatment are presented. The high biosorption ability of bacteria living in a mature biofilm is highlighted, as well as the toxicity effect of heavy metals on autotrophic bacteria, whose growth directly affects the nitrification performance of bioreactors.

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Lugon Junior, Jader, João Flávio Vieira Vasconcellos, Diego Campos Knupp, Gisele Moraes Marinho, Luiz Bevilacqua, and Antônio José da Silva Neto. "Solution of Fourth Order Diffusion Equations and Analysis Using the Second Moment." Defect and Diffusion Forum 399 (February 2020): 10–20. http://dx.doi.org/10.4028/www.scientific.net/ddf.399.10.

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The classical concept of diffusion characterized by Fick’s law is well suited for describing a wide class of practical problems of interest. Nevertheless, it has been observed that it is not enough to properly represent other relevant applications of practical interest. When in a system of particles their spreading is slower or faster than predicted by the classical diffusion model, such a phenomenon is referred to as anomalous diffusion. Time fractional, space fractional and even space-time fractional equations are widely used to model phenomena such as solute transport in porous media, financial modelling and cancer tumor behavior. Considering the effects of partial and temporary retention in dispersion processes a new analytical formulation was derived to simulate anomalous diffusion. The new approach leads to a fourth-order partial differential equation (PDE) and assumes the existence of two concomitant fluxes. This work investigates the behavior of the bi-flux approach in one dimensional (1D) medium evaluating the mean square displacement for different cases in order to classify the diffusion process in normal, sub-diffusive or super-diffusive.

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Riza, Ion, and Constantin Gheorghe Opran. "Methods and Special Functions Used in Diffusion Modelling." Applied Mechanics and Materials 865 (June 2017): 64–72. http://dx.doi.org/10.4028/www.scientific.net/amm.865.64.

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Natural phenomena can be modelled adequately precise with differential equations. An example of such phenomena is diffusion. All differential equations with partial derivatives, of two variables and of the second order, which model the diffusion are parabolic with simple solutions. This simple solution is: erf(x), exponential functions or power series. Adding new differential terms requires the use of more complicated functions, some degenerated, modified or confluent, of real or imaginary variables. This functions are: Bessel, Kummer, Whittaker, Hermite, hyper-geometrical or Heun. The benefit of using these functions lies in greater clarity and precision in mathematical expression. Modelling methods by change-of-variable and function, adding new differential terms and transformations of special functions increase modelling possibilities of convective diffusion in different environments using differential equations. The paper presents a novel approach to diffusion modelling for dry corrosion.

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Yin, Hong. "Forward–backward stochastic partial differential equations with non-monotonic coefficients." Stochastics and Dynamics 16, no. 06 (November 6, 2016): 1650025. http://dx.doi.org/10.1142/s0219493716500258.

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In this paper we study the solvability of a class of fully-coupled forward–backward stochastic partial differential equations (FBSPDEs) with non-monotonic coefficients. These FBSPDEs cannot be put into the framework of stochastic evolution equations in general, and the usual decoupling methods for the Markovian forward–backward SDEs are difficult to apply. We prove the well-posedness of such FBSPDEs by using the method of continuation. Contrary to the common belief, we show that the usual monotonicity assumption can be removed by a change of the diffusion term.

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