We analyze the spherically symmetric complex diffusion and special type of the complex reaction–diffusion equations. These equations are form invariant to the free Schrödinger equations and to the Schrödinger equations with power-law space-dependent potentials. Our new type of solutions are important because we found a new realm of solutions which lie between the solutions of the classical regular diffusion equation and the usual quantum mechanical solutions of the Schrödinger equation. As the solution method, we applied the the self-similar Ansatz, which reduces the original partial differential equation (PDE) to an ordinary differential equation (ODE) which can be solved analytically. The self-similar Ansatz couples the spatial and temporal variables together instead of the usual separation which has to be used in ordinary quantum mechanics for time-independent Hamiltonian. For the complex diffusion equation—without any additional source term—the solutions are the Kummer’s M and Kummer’s U functions. For some parameter values we found L2 integrability, as in the Cartesian case. We interpret that this property can be a “quantum mechanical heritage” and can be a far relation to ordinary quantum mechanics. Therefore, in this sense, our solutions might have quantum mechanical interest in the future. For the complex reaction–diffusion-type equation we derived the Whittaker M and Whittaker W functions as solutions. These solutions have no L2 integrability at all. All derived solutions have complex quadratic arguments. These kind of analytic solutions are new and cannot be found in the existing scientific literature. Finally, the role of the complex angular momentum was investigated as well.

In this article the two dimensional heat diffusion equation is studied. We consider a system for infinite horizon, the heat can diffuse without spatial constraint. In case of polar coordinates one have a radial and an angular part of the spatial variation. With an appropriate self-similar transformation, we arrive to an ordinary differential equation. The equation admits a countable set of solutions which can be obtained by an algebraic method. These solutions decay in space and in time, the latter is a power law decay with different exponents for different solutions. The diffusion equation with a source term is also discussed by the self-similar method.

We investigate the partial differential equation system which describes the double-diffusion convection phenomena with the reduction formalism. Double-diffusion refers to when two scalar quantities with different diffusivity, such as heat and solute concentration, contribute to density gradients within a fluid under the influence of gravity. The time-dependent self-similar trial function is applied and analytic results are presented for the dynamical variables and analyzed in detail. Additionally, the entropy production was derived as well. In the second part of the study we investigate the role of an additional heat source.

In this paper, we derive and compare the solutions of the real and complex diffusion equations obtained by the self-similar solution to each other. Additionally, possible quantum mechanical aspects are analyzed as well. In the second part of the study, a complex reaction-diffusion equation is investigated and compared to the quantum mechanical solutions of the corresponding Schr & ouml;dinger equation with a power-law-type potential. We show that the same parameter ratio emerges in the quantum mechanical and the self-similar solutions as well, which is a remarkable property. For both investigated equations, the complex diffusion and complex reaction-diffusion equations, we found some solutions for which the absolute value squared has a convergent finite numerical integral. This is not a rigorous L2 integrability condition but a good hint for that, which is a key property in quantum mechanics.

To understand the complex problem of surface growth phenomena, we developed a model in which the regular diffusion equation is coupled to the Kardar–Parisi–Zhang (KPZ) equation. The fundamental or Gaussian solution of the regular diffusion equation is considered as an external noise or source term in the KPZ equation. The obtained system of partial differential equations is analytically solved by a self-similar Ansatz and expressed the solution as a combination of elementary and special functions. Using this solution, the effects of the physical parameters were explicitly investigated. Our recent explicit numerical method, the leapfrog-hopscotch algorithm, is also tested for this problem. It is shown that this method can be safely used with orders of magnitude larger time step sizes than the usual explicit (Euler) scheme as well if we are far from the cusp-like solutions. We pointed out that the cusps themselves cannot be properly simulated by any method that we know.

We present a surface growth model associated with diffusion on surface. The diffusion process id described by the diffusion equation. The growth phenomena is governed by the KPZ equation with an appropriate source term.

In this paper, we define and investigate a system of coupled regular diffusion equations in which each concentration acts as a driving term in the next diffusion equation. Such systems can be understood as a kind of cascade process which appear in different fields of physics like diffusion and reaction processes or turbulence. As a solution, we apply the time-dependent self-similar Ansatz method, the obtained solutions can be expressed as the product of a Gaussian and a Kummer’s function. This model physically means that the first diffusion works as a catalyst in the second diffusion system. The coupling of these diffusion systems is only one way. In the second part of the study we investigate mutually coupled diffusion equations which also have the self-similar trial function. The derived solutions show some similarities to the former one. To make our investigation more complete, different kinds of couplings were examined like the linear, the power-law, and the Lorentzian. Finally, a special coupling was investigated which is capable of describing isomerization with temporal decay.

We investigate diffusion equations which have concentration dependent diffusion coefficients with physically two relevant Ansatze, the self-similar and the traveling wave Ansatz. We found that for power-law concentration dependence some of the results can be expressed with a general analytic implicit formulas for both trial functions. For the self-similar case some of the solutions can be given with a formula containing the hypergeometric function. For the traveling wave case different analytic formulas are given for different exponents. For some physically reasonable parameter sets the direct solutions are given and analyzed in details.

In this study, five different time-dependent incompressible non-Newtonian boundary layer models in two dimensions are investigated with the self-similar Ansatz, including external magnetic field effects. The power-law, the Casson fluid, the Oldroyd-B model, the Walter fluid B model, and the Williamson fluid are analyzed. For the first two models, analytical results are given for the velocity and pressure distributions, which can be expressed by different types of hypergeometric functions. Depending on the parameters involved in the analytical solutions of the nonlinear ordinary differential equation obtained by the similarity transformation, a vast range of solution types is presented. It turned out that the last three models lack self-similar symmetry; therefore, no analytic solutions can be derived.

New analytical solutions of the heat conduction equation obtained by utilizing a self-similar Ansatz are presented in cylindrical and spherical coordinates. Then, these solutions are reproduced with high accuracy using recent explicit and unconditionally stable finite difference methods. After this, real experimental data from the literature regarding a heated cylinder are reproduced using the explicit numerical methods as well as using Finite Element Methods (FEM) ANSYS workbench. Convection and nonlinear radiation are also considered on the boundary of the cylinder. The verification results showed that the numerical methods have a high accuracy to deal with cylindrical and spherical bodies; also, the comparison of the temperatures for all approaches showed that the explicit methods are more accurate than the commercial software.

In the description of transport phenomena, diffusion represents an important aspect. In certain cases, the diffusion may appear together with convection. In this paper, we study the diffusion equation with the self-similar Ansatz. With an appropriate change of variables, we have found an original new type of solution of the diffusion equation for infinite horizon. We derive novel even solutions of diffusion equation for the boundary conditions presented. For completeness, the odd solutions are also mentioned as well, as part of the previous works. We have found a countable set of even and odd solutions, of which linear combinations also fulfill the diffusion equation. Finally, the diffusion equation with a constant source term is discussed, which also has even and odd solutions.

We investigate diffusion equations which have concentration dependent diffusion coefficients with physically two relevant Ans\"atze, the self-similar and the traveling wave Ansatz. We found that for power-law concentration dependence some of the results can be expressed with a general analytic implicit formulas for both trial functions. For the self-similar case some of the solutions can be given with a formula containing the hypergeometric function. For the traveling wave case different analytic formulas are given for different exponents. For some physically reasonable parameter sets the direct solutions are given and analyzed in details.

In our latest studies with the help of the self-similar Ansatz we derived new type of 1 solutions for the regular diffusion equation which are much more complex than the well-known 2 Gaussian and error functions. These solutions contain additional Kummer’s M and Kummer’s U 3 functions with quadratic argument. In the present treaties we perform an analogous analysis for the 4 regular diffusion equation which has a complex diffusion coefficient. Formally, it is equivalent to 5 the free Schrödinger equation however it is far from being trivial how the solutions can be given 6 quantum mechanical interpretation. We investigate both the one dimensional Cartesian and the 7 spherical symmetric equations. We find solutions which fulfill the L2 integrability criteria therefore 8 might have quantum mechanical relevance in the future.

We investigate a hydrodynamic equation system which—with some approximation—is capable of describing the tsunami propagation in the open ocean with the time-dependent self-similar Ansatz. We found analytic solutions of how the wave height and velocity behave in time and space for constant and linear seabed functions. First, we study waves on open water, where the seabed can be considered relatively constant, sufficiently far from the shore. We found original shape functions for the ocean waves. In the second part of the study, we also consider a seabed which is oblique. Most of the solutions can be expressed with special functions. Finally, we apply the most common traveling wave Ansatz and present relative simple, although instructive solutions as well.

We investigate the incompressible and compressible heat conducting boundary layer with applying the two-dimensional self-similar Ansatz. Analytic solutions can be found for the incompressible case which can be expressed with special functions. The parameter dependencies are studied and discussed in details. In the last part of our study we present the ordinary differential equation (ODE) system which is obtained for compressible boundary layers.

We examine the one-dimensional transient diffusion equation with a space-dependent diffusion coefficient. Such equations can be derived from the Fokker–Planck equation and are essential for understanding the diffusion mechanisms, e.g., in carbon nanotubes. First, we construct new, nontrivial analytical solutions with the classical self-similar Ansatz in one space dimension. Then we apply 14 different explicit numerical time integration methods, most of which are recently introduced unconditionally stable schemes, to reproduce the analytical solution. The test results show that the best algorithms, especially the leapfrog-hopscotch, are very efficient and severely outperform the conventional Runge–Kutta methods. Our results may attract attention in the community who develops multi-physics engineering software.

Transport phenomena plays an important role in science and technology. In the wide variety of applications both advection and diffusion may appear. Regarding diffusion, for long times, different type of decay rates are possible for different non-equilibrium systems. After summarizing the existing solutions of the regular diffusion equation, we present not so well known solution derived from three different trial functions, as a key point we present a family of solutions for the case of infinite horizon. By this we tried to make a step toward understanding the different long time decays for different diffusive systems.