Tricomi Equation Research Articles

Some numerical solutions of local fractional tricomi equation in fractal transonic flow

We investigate numerical solutions of fractional-order Tricomi equation (LFTE) in fractal transonic flow media by employing the method of local fractional q-homotopy transform (LFq-HATM). This method is a combination of methods of homotopy analysis and q-Laplace transform. We express the solutions in terms of rapidly convergent power series where the basics functions are easily computable by Mathematica software. We present uniqueness and convergence analysis of the model via Banach‘s fixed point theory (BFPT). Reliability analysis of the numerical method is provided by figures and physical works. Obtained results indicate the strength and efficiency of the LEq-HATM.

Two step Adams Bashforth method for time fractional Tricomi equation with non-local and non-singular Kernel

Abstract Recently, Atangana and Baleanu (AB) introduced a new fractional differentiation concept using non-local and non-singular kernel. Later on, theoretical applications, more practical applications and new numerical methods was established for solving partial differential equations in the meaning of AB derivative. In this study, the new numerical scheme was formulated by Owolabi and Atangana [A. Atangana, K.M.Owolabi, New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena 13.1 (2018) 1–19.] is considered for solving fractional Tricomi equation which involves the Mittag- Leffler kernel. A novel two-step Adams-Bashforth scheme is applied for the approximation of the AB fractional derivative. Stability of the numerical scheme is examined with the help of von Neumann stability analysis and induction principle. To test the applicability and suitability of the proposed method, two notable examples are considered with numerical results presented for some fractional order values.

Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method

Purpose The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this paper is to present results on the numerical simulation for time-fractional partial differential equations arising in transonic multiphase flows, which are described by the Tricomi and the Keldysh equations of Robin functions types. Design/methodology/approach Those resulting mathematical models are solved by using the reproducing kernel method, which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations and error bounds under some hypotheses, which provide the theoretical basis of the proposed method are also discussed. Findings The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the method are discussed through academic validations. Originality/value In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional PDEs such as those found in the transonic multiphase flows. The authors applied the reproducing kernel method systematically for the numerical solutions of time-fractional Tricomi and Keldysh equations subject to Robin functions types.

Existence and uniqueness for some equation of a mixed elliptic-hyperbolic type

In the paper, we prove results on existence and uniqueness of weak solutions for closed Dirichlet problem for a transonic flow model, and interior regularity results are also given in the important special case of the Tricomi equation. The method employed consists in variants of the a-b-c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function.

Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip

In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tricomi equation in a strip with a polynomial right-hand side. The Fourier transform method shows that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing this polynomial solution is given and examples are considered. If the strip lies in the ellipticity region of the equation, then this solution is unique in the class of functions of polynomial growth. If the strip lies in a mixed domain, then the solution of the Dirichlet problem is not unique in the class of functions of polynomial growth, but it is unique in the class of polynomials.

Boundary-Value Problem for Functional-Differential Advanced-Retarded Tricomi Equation

We investigate the boundary-value problem for Tricomi mixed-type equation with multiple functional retarding and advancing. We construct the general solution to the equation. The problem is uniquely solvable.

Constructing analytic solutions on the Tricomi equation

AbstractIn this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

Solutions of time‐fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space

The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this research article is to present results on the numerical simulation for time‐fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space that were found in the transonic flows. Those resulting mathematical models are solved using the reproducing kernel algorithm which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the algorithm are discussed through academic validations.

On semilinear Tricomi equations with critical exponents or in two space dimensions

This paper is a complement of our recent works on the semilinear Tricomi equations in [9] and [10]. For the semilinear Tricomi equation ∂t2u−tΔu=|u|p with initial data (u(0,⋅),∂tu(0,⋅))=(u0,u1), where t≥0, x∈Rn (n≥3), p>1, and ui∈C0∞(Rn) (i=0,1), we have shown in [9] and [10] that there exists a critical exponent pcrit(n)>1 such that the solution u, in general, blows up in finite time when 1<p<pcrit(n), and there is a global small solution for p>pcrit(n). In the present paper, firstly, we prove that the solution of ∂t2u−tΔu=|u|p will generally blow up for the critical exponent p=pcrit(n) and n≥2, secondly, we establish the global existence of small data solution to ∂t2u−tΔu=|u|p for p>pcrit(n) and n=2. Thus, we have given a systematic study on the blowup or global existence of small data solution u to the equation ∂t2u−tΔu=|u|p for n≥2.

Numerical simulation of sonic boom focusing and its application to mission performance

This paper describes the effort undertaken to include numerical boom focusing on a lossy Burgers equation based sonic boom propagation tool called sBOOM. Traditional ray acoustics may break down during acceleration or other aircraft maneuvers that cause the ray tube area to approach zero. The paper describes a way to numerically predict focused sonic boom signatures using both Gill-Seebass similitude and the solution to the non-linear Tricomi equation that models the physics of boom focusing. The paper would then use this capability to determine focusing and non-focusing climb trajectories and their impact on mission performance.

Conformal invariance and new exact solutions of the elastostatics equations

We fulfilled a group foliation of the system of n-dimensional (n ≥ 2) Lame equations of the classical static theory of elasticity with respect to the infinite subgroup contained in normal subgroup of main group of this system. It permitted us to move from the Lame equations to the equivalent unification of two first-order systems: automorphic and resolving. We obtained a general solution of the automorphic system. This solution is an n-dimensional analogue of the Kolosov-Muskhelishvili formula. We found the main Lie group of transformations of the resolving system of this group foliation. It turned out that in the two-dimensional and three-dimensional cases, which have a physical meaning, this system is conformally invariant, while the Lame equations admit only a group of similarities of the Euclidean space. This is a big success, since in the method of group foliation, resolving equations usually inherit Lie symmetries subgroup of the full symmetry group that was not used for the foliation. In the three-dimensional case for the solutions of the resolving system, we found the general form of the transformations similar to the Kelvin transformation. These transformations are the consequence of the conformal invariance of the resolving system. In the three-dimensional case with a help of the complex dependent and independent variables, the resolving system is written as a simple complex system. This allowed us to find non-trivial exact solutions of the Lame equations, which direct for the Lame equations practically impossible to obtain. For this complex system, all the essentially distinct invariant solutions of the maximal rank we have found in explicit form, or we reduced the finding of those solutions to the solving of the classical one-dimensional equations of the mathematical physics: the heat equation, the telegraph equation, the Tricomi equation, the generalized Darboux equation, and other equations. For the resolving system, we obtained double wave of a special type and double waves of the shear deformations in an elastic medium. By the obtained formulas for generating new solutions, these solutions give an infinite set of exact solutions of this complex system. By the three-dimensional analog of the Kolosov-Muskhelishvili formula, the found solutions produce an infinite-dimensional vector space of exact solutions of the Lame equations.

A Reliable Algorithm for a Local Fractional Tricomi Equation Arising in Fractal Transonic Flow

The pivotal proposal of this work is to present a reliable algorithm based on the local fractional homotopy perturbation Sumudu transform technique for solving a local fractional Tricomi equation occurring in fractal transonic flow. The proposed technique provides the results without any transformation of the equation into discrete counterparts or imposing restrictive assumptions and is completely free of round-off errors. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

Bifurcation curves for synchronous electrical machine

One of the well-known and the simplest mathematical models of electrical synchronous machine is a pendulum with magnetoelectrical extinguishers (Stoker in Nonlinear vibrations. Interscience Publisher, New York, p 273, 1950; Skubov and Khodzhaev in Nonlinear electromechanics. Springer, Berlin, p 393, 2008). In case when the machine has a single conductive contour, this model is described by Tricomi equation (Tricomi in Annali della Roma Scuola Normale Superiore de Piza Scienza Phys E Mat 2:1–20, 1933; Leonov and Kondratyeva in Analysis of stability of electrical machines of alternating current. SPbSU, St. Petersburg, 2009). In this article, the new method of obtaining the set of parameters corresponding to heteroclinic separatrix of Tricomi equation is described. This set is a bifurcation curve, which separates the field of absolute stability of synchronous regime and the field of existence of the stable rotation of pendulum corresponding to asynchronous operating mode of the machine.

Integral transform approach to generalized Tricomi equations

In this article, we introduce the integral transform that can be used to obtain solutions of the generalized Tricomi equation via solutions of a simpler equation. We employed a particular version of this transform in [13,14,41–46] to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in the Einstein–de Sitter spacetime, the wave and the Klein–Gordon equations in the de Sitter and anti-de Sitter spacetimes.

Diffraction of high-intensity field in focal region as dynamics of nonlinear system with low-frequency dispersion

The stationary profile in the focal region of a focused nonlinear acoustic wave is described. Three models following from the Khokhlov-Zabolotskaya (KZ) equation with three independent variables are used: (i) the simplified one-dimensional Ostrovsky-Vakhnenko equation, (ii) the system of equations for paraxial series expansion of the acoustic field in powers of transverse coordinates, and (iii) the KZ equation reduced to two independent variables. The structure of the last equation is analogous to the Westervelt equation. Linearization through the Legendre transformation and reduction to the well-studied Euler-Tricomi equation is shown. At high intensities the stationary profiles are periodic sequences of arc sections having singularities of derivative in their matching points. The occurrence of arc profiles was pointed out by Makov. These appear in different nonlinear systems with low-frequency dispersion. Profiles containing discontinuities (shock fronts) change their form while passing through the focal region and are non-stationary waves. The numerical estimations of maximum pressure and intensity in the focus agree with computer calculations and experimental measurements.

Regarding ‘Information Preservation and Weather Forecasting for Black Holes’ by S. W. Hawking

It is proposed that the ‘apparent horizons’ assumed by Hawking to resolve the black hole information paradox, are in reality the regions where in Lorentzian relativity the absolute velocity against a preferred reference system at rest with the zero point vacuum energy reaches the velocity of light, and where an elliptical differential equation holding matter in a stable equilibrium goes over a transluminal Euler-Tricomi equation into a hyperbolic differential equation where such an equilibrium is not more possible, with matter in approaching this region disintegrating into radiation. Hawking’s proposal depends on the anti-de Sitter/conformal field theory (AdS/CFT) conjecture which in turn depends on string/M theory which in the absence of super-symmetry will not work.

Local Fractional Derivative Boundary Value Problems for Tricomi Equation Arising in Fractal Transonic Flow

The local fractional decomposition method is applied to obtain the nondifferentiable numerical solutions for the local fractional Tricomi equation arising in fractal transonic flow with the local fractional derivative boundary value conditions.

The Nondifferentiable Solution for Local Fractional Tricomi Equation Arising in Fractal Transonic Flow by Local Fractional Variational Iteration Method

We present the nondifferentiable approximate solution for local fractional Tricomi equation arising in fractal transonic flow by local fractional variational iteration method. Some illustrative examples are shown and graphs are also given.

A note on initial value problem for the generalized Tricomi equation in a mixed-type domain

In this paper, we study the well-posedness of initial value problem for n-dimensional generalized Tricomi equation in the mixed-type domain {(t, x): t ∈ [−1,+∞), x ∈ ℝn} with the initial data given on the line t = −1 in Hadamard’s sense. By taking partial Fourier transformation, we obtain the explicit expression of the solution in terms of two integral operators and further establish the global estimate of such a solution for a class of initial data and source term. Finally, we establish the global solution in time direction for a semilinear problem used the estimate.

A mixed problem with Tricomi and Frankl conditions for the gellerstedt equation with a singular coefficient

We consider a generalized Tricomi equation with a singular coefficient. For this equation in a mixed domain we study the corresponding problem in the case, when a part of the boundary characteristic is free of boundary conditions; the deficient Tricomi condition is equivalently substituted by a nonlocal Frankl condition on a segment of the degeneration line. We prove that the stated problem is well-posed.