The authors derived the analytical solution to diffusion equations. The solution requires linearization of diffusion equations, as well as developing the obtained expression into a series. In particular, the result of the first procedure is highly de...
We investigate an intermittent process obtained from the combination of a nonlinear diffusion equation and pauses. We consider the porous media equation with reaction terms related to the rate of switching the particles from the diffusive mode to the...
It is well known that meshless methods are effective for solving fractional differential equations on both regular and irregular domains. However, many commonly used basis functions such as Legendre wavelets and B-splines are naturally defined on rec...
We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our cu...
In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elli...
This paper delves into a rapid and accurate numerical solution for the inverse problem of the nonlinear diffusion equation in the context of multiphase porous media flow. For the realization of this, the combination of the multigrid method with const...
We investigate a nonlinear convection—diffusion equation involving a non-polynomial square-root flux. By applying a travelling-wave reduction and introducing a structurally motivated square-root transformation, we show that the resulting ordina...
This study presents a higher-order uniformly convergent numerical method for a singularly perturbed nonlinear reaction–diffusion equation. The quasilinearization technique is used to transform a nonlinear term into a linear boundary value probl...
The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors. Mathematically, the problem is reduced to the calculation of the “energy” of the ground state in the Schrödinger equati...
In the article, we propose a combination method based on the multigrid method and constraint data to solve the inverse problem in the context of the nonlinear convection–diffusion equation in the multiphase porous media flow. The inverse proble...
In this paper, we study the nonlinear Riesz space-fractional convection–diffusion equation over a finite domain in two dimensions with a reaction term. The Crank–Nicolson difference method for the temporal and the weighted–shifted G...
We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum princ...
The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous...
The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in v...
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematic...
The sign-invariant theory is used to study the radially symmetric nonlinear diffusion equations with gradient-dependent diffusivities. The first-order non-stationary sign-invariants and the first-order non-autonomous sign-invariants admitted by the g...
The radially symmetric nonlinear reaction–diffusion equation with gradient-dependent diffusivity is investigated. We obtain conditions under which the equations admit second-order conditional Lie–Bäcklund symmetries and first-order H...
This review is devoted to search for Lie and Q-conditional (nonclassical) symmetries and exact solutions of a class of reaction-diffusion-convection equations with exponential nonlinearities. A complete Lie symmetry classification of the class is der...
A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmet...
In this paper, the nonlinear time-fractional convection-diffusion equations are solved by the Legendre spectral method. The Caputo time-fractional derivative is discretized by the
The paper studies a degenerate nonlinear parabolic equation containing a convective term and a source (reaction) term. It considers the construction of approximate solutions to this equation with a specified law of diffusion wave motion, the existenc...
In this paper, we prove the lower bounds of the blow-up time of solutions for certain Caputo time fractional diffusion equations and systems with nonlinear memory terms under homogeneous Dirichlet boundary conditions. The proofs of our results rely o...
The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on...
In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the invers...
Symmetry properties of a nonlinear two-dimensional space-fractional diffusion equation with the Riesz potential of the order
In this article, some high-order time discrete schemes with an
The approximate analytical solution of fractional order, nonlinear, reaction differential equations, namely the nonlinear diffusion equations, with a given initial condition, is obtained by using the homotopy analysis method. As a demonstration of a...
In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central differ...
It is generally known that classical point and potential Lie symmetries of differential equations (the latter calculated as point symmetries of an equivalent system) can be different. We question whether this is true when the symmetries are extended...
Diffusion equations play a crucial role in various scientific and technological domains, including mathematical biology, physics, electrical engineering, and mathematics. This article presents a new formulation of the diffusion equation in the contex...
Although one-dimensional non-linear diffusion equations are commonly used to model flow dynamics in aquifers and fissures, they disregard multiple effects of real-life flows. Similarity analysis may allow further analytical reduction of these equatio...
We present a finite-difference integration algorithm for solution of a system of differential equations containing a diffusion equation with nonlinear terms. The approach is based on Crank–Nicolson method with predictor–corrector algorith...
This study presents a novel fifth-order iterative method for solving nonlinear systems derived from a modified combination of Jarratt and Newton schemes, incorporating a frozen derivative of the Jacobian. The method is applied to approximate solution...
In this paper, we consider the inverse problem for identifying the initial value problem of the time–space fractional nonlinear diffusion equation. The uniqueness of the solution is proved by taking the fixed point theorem of Banach compression...
The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position...
This paper presents a novel B-spline wavelet-based scheme for solving multi-term time–space variable-order fractional nonlinear diffusion-wave equations. By combining semi-orthogonal B-spline wavelets with a collocation approach and a quasiline...
This study is devoted to reaction–diffusion equations with spatially anisotropic time delay. Reaction–diffusion PDEs with either constant or variable transfer coefficients are considered. Nonlinear equations of a fairly general form conta...
In this work, we aim to explore new exact traveling wave solutions for the reaction–diffusion equation, which describes complex nonlinear phenomena such as cell growth and chemical reactions in nature. Obtaining exact solutions to this equation...
We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown
This paper proposes phenomenological equations that describe various aspects of aftershock evolution: elementary master equation, logistic equation, stochastic equation, and nonlinear diffusion equation. The elementary master equation is a first-orde...
In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is...
The paper studies the Lie symmetries of the nonlinear Fokker-Planck equation in one dimension, which are associated to the weighted Kaniadakis entropy. In particular, the Lie symmetries of the nonlinear diffusive equation, associated to the weighted...
Four explicit numerical schemes are collected, which are stable and efficient for the diffusion equation. Using these diffusion solvers, several new methods are constructed for the nonlinear Huxley’s equation. Then, based on many successive num...
The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a po...
Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological...
This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (...
In this paper, two-dimensional Genocchi polynomials and the Ritz–Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein–Gordon Equation (FKGE). A satisfier function that satisf...
In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new suffici...
This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection–diffusion–reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive id...
The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study d...
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