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Articles de revues sur le sujet « Degenerate elliptic equation »

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Auteur : Grafiati

Publié le 11 mars 2023

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1

Trudinger, Neil S. « On degenerate fully nonlinear elliptic equations in balls ». Bulletin of the Australian Mathematical Society 35, n^o 2 (avril 1987) : 299–307. http://dx.doi.org/10.1017/s0004972700013253.

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We establish derivative estimates and existence theorems for the Dirichlet and Neumann problems for nonlinear, degenerate elliptic equations of the form F (D2u) = g in balls. The degeneracy arises through the possible vanishing of the function g and the degenerate Monge-Ampère equation is covered as a special case.

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2

Igisinov, S. Zh, L. D. Zhumaliyeva, A. O. Suleimbekova et Ye N. Bayandiyev. « Estimates of singular numbers (s-numbers) for a class of degenerate elliptic operators ». BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 107, n^o 3 (30 septembre 2022) : 51–58. http://dx.doi.org/10.31489/2022m3/51-58.

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In this paper we study a class of degenerate elliptic equations with an arbitrary power degeneracy on the line. Based on the research carried out in the course of the work, the authors propose methods to overcome various difficulties associated with the behavior of functions from the definition domain for a differential operator with piecewise continuous coefficients in a bounded domain, which affect the spectral characteristics of boundary value problems for degenerate elliptic equations. It is shown the conditions imposed on the coefficients at the lowest terms of the equation, which ensure the existence and uniqueness of the solution. The existence, uniqueness, and smoothness of a solution are proved, and estimates are found for singular numbers (s-numbers) and eigenvalues of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration.

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3

Le, Nam Q. « On the Harnack inequality for degenerate and singular elliptic equations with unbounded lower order terms via sliding paraboloids ». Communications in Contemporary Mathematics 20, n^o 01 (23 octobre 2017) : 1750012. http://dx.doi.org/10.1142/s0219199717500122.

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We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge–Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.

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4

Tanirbergen, Aisulu K. « A MIXED PROBLEM FOR A DEGENERATE MULTIDIMENSIONAL ELLIPTIC EQUATION ». UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, n^o 3 (211) (30 septembre 2021) : 37–41. http://dx.doi.org/10.18522/1026-2237-2021-3-37-41.

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This article shows the unique solvability and obtains an explicit form of the classical solution of the mixed prob-lem in a cylindrical domain for a model degenerate multidimensional elliptic equation. The correctness of boundary value problems in the plane for elliptic equations by the method of the theory of ana-lytic functions of a complex variable has been well studied. The first boundary value problem or the Dirichlet problem for multidimensional elliptic equations with degeneration on the boundary has been sufficiently analyzed. However, as we know, the mixed problem for the indicated equations has been studied very little.

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5

Andreu, F., V. Caselles et J. M. Mazón. « A strongly degenerate quasilinear elliptic equation ». Nonlinear Analysis : Theory, Methods & ; Applications 61, n^o 4 (mai 2005) : 637–69. http://dx.doi.org/10.1016/j.na.2004.11.020.

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6

Krasovitskii, T. I. « Degenerate elliptic equations and nonuniqueness of solutions to the Kolmogorov equation ». Доклады Академии наук 487, n^o 4 (27 août 2019) : 361–64. http://dx.doi.org/10.31857/s0869-56524874361-364.

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In this paper we propose a new method of constructing examples of nonuniqueness of probability solutions by reducing the stationary Fokker-Planck-Kolmogorov equation to a degenerate elliptic equation on a bounded domain.

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7

Rocca, Elisabetta, et Riccarda Rossi. « A degenerating PDE system for phase transitions and damage ». Mathematical Models and Methods in Applied Sciences 24, n^o 07 (14 avril 2014) : 1265–341. http://dx.doi.org/10.1142/s021820251450002x.

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In this paper, we analyze a PDE system arising in the modeling of phase transition and damage phenomena in thermoviscoelastic materials. The resulting evolution equations in the unknowns ϑ (absolute temperature), u (displacement), and χ (phase/damage parameter) are strongly nonlinearly coupled. Moreover, the momentum equation for u contains χ-dependent elliptic operators, which degenerate at the pure phases (corresponding to the values χ = 0 and χ = 1), making the whole system degenerate. That is why, we have to resort to a suitable weak solvability notion for the analysis of the problem: it consists of the weak formulations of the heat and momentum equation, and, for the phase/damage parameter χ, of a generalization of the principle of virtual powers, partially mutuated from the theory of rate-independent damage processes. To prove an existence result for this weak formulation, an approximating problem is introduced, where the elliptic degeneracy of the displacement equation is ruled out: in the framework of damage models, this corresponds to allowing for partial damage only. For such an approximate system, global-in-time existence and well-posedness results are established in various cases. Then, the passage to the limit to the degenerate system is performed via suitable variational techniques.

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8

Gutiérrez, Cristian E., et Federico Tournier. « Harnack Inequality for a Degenerate Elliptic Equation ». Communications in Partial Differential Equations 36, n^o 12 (décembre 2011) : 2103–16. http://dx.doi.org/10.1080/03605302.2011.618210.

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9

Horiuchi, Toshio. « Quasilinear degenerate elliptic equation with absorption term ». Nonlinear Analysis : Theory, Methods & ; Applications 47, n^o 3 (août 2001) : 1649–57. http://dx.doi.org/10.1016/s0362-546x(01)00298-x.

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10

Amano, Kazuo. « The Dirichlet problem for degenerate elliptic 2-dimensional Monge-Ampère equation ». Bulletin of the Australian Mathematical Society 37, n^o 3 (juin 1988) : 389–410. http://dx.doi.org/10.1017/s0004972700027015.

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We study the following Dirichlet problem for the degenerate elliptic Monge-Ampère equation: Given , f ≥ 0 and , find a solution , t ≥ 2, satisfying in Ω and u = g on ∂Ω. Since f is nonnegative, we cannot apply any standard elliptic methods. In this paper, we use an iteration scheme of Nash-Moser type and a priori estimates for degenerate elliptic operators, and solve the Dirichlet problem for a certain class of f and g.

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11

Djida, Jean-Daniel, et Arran Fernandez. « Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions ». Axioms 7, n^o 3 (1 septembre 2018) : 65. http://dx.doi.org/10.3390/axioms7030065.

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The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.

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12

Ding, Da-Jiang, Di-Qing Jin et Chao-Qing Dai. « Analytical solutions of differential-difference sine-Gordon equation ». Thermal Science 21, n^o 4 (2017) : 1701–5. http://dx.doi.org/10.2298/tsci160809056d.

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In modern textile engineering, non-linear differential-difference equations are often used to describe some phenomena arising in heat/electron conduction and flow in carbon nanotubes. In this paper, we extend the variable coefficient Jacobian elliptic function method to solve non-linear differential-difference sine-Gordon equation by introducing a negative power and some variable coefficients in the ansatz, and derive two series of Jacobian elliptic function solutions. When the modulus of Jacobian elliptic function approaches to 1, some solutions can degenerate into some known solutions in the literature.

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13

Albano, Paolo. « On the Eikonal equation for degenerate elliptic operators ». Proceedings of the American Mathematical Society 140, n^o 5 (1 mai 2012) : 1739–47. http://dx.doi.org/10.1090/s0002-9939-2011-11132-8.

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14

Kallel-Jallouli, Saoussen. « The Dirichlet Problem for Degenerate Elliptic Darboux Equation ». Communications in Partial Differential Equations 29, n^o 7-8 (11 janvier 2004) : 1097–125. http://dx.doi.org/10.1081/pde-200033756.

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15

Pao, C. V. « Eigenvalue Problems of a Degenerate Quasilinear Elliptic Equation ». Rocky Mountain Journal of Mathematics 40, n^o 1 (février 2010) : 305–11. http://dx.doi.org/10.1216/rmj-2010-40-1-305.

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16

Delgado, M., J. A. Montero et A. Suárez. « Optimal Control for the Degenerate Elliptic Logistic Equation ». Applied Mathematics and Optimization 45, n^o 3 (1 janvier 2002) : 325–45. http://dx.doi.org/10.1007/s00245-001-0039-1.

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17

Borsuk, Mikhail. « Boundary value problems for singular p- and p(x)- Laplacian equations in a domain with conical point on the boundary ». Ukrainian Mathematical Bulletin 17, n^o 4 (13 décembre 2020) : 455–83. http://dx.doi.org/10.37069/1810-3200-2020-17-4-1.

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This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

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18

Krasovitskii, T. I. « Degenerate Elliptic Equations and Nonuniqueness of Solutions to the Kolmogorov Equation ». Doklady Mathematics 100, n^o 1 (juillet 2019) : 354–57. http://dx.doi.org/10.1134/s1064562419040112.

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19

Benci, Vieri, et Donato Fortunato. « A strongly degenerate elliptic equation arising from the semilinear Maxwell equations ». Comptes Rendus Mathematique 339, n^o 12 (décembre 2004) : 839–42. http://dx.doi.org/10.1016/j.crma.2004.07.029.

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20

SHAKHMUROV, VELI B., et AIDA SAHMUROVA. « Mixed problems for degenerate abstract parabolic equations and applications ». Carpathian Journal of Mathematics 34, n^o 2 (2018) : 247–54. http://dx.doi.org/10.37193/cjm.2018.02.13.

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Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

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21

JOSEPH, ANCEMMA, et K. PORSEZIAN. « PERIODIC WAVE SOLUTIONS TO MODIFIED NONLINEAR SCHRÖDINGER EQUATION PERTAINING TO NEGATIVE INDEX MATERIALS ». Journal of Nonlinear Optical Physics & ; Materials 19, n^o 01 (mars 2010) : 177–87. http://dx.doi.org/10.1142/s0218863510005005.

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In this paper, we intend to determine periodic wave solutions for the modified nonlinear Schrödinger equation pertaining to negative index materials. We have treated the propagation equation possessing higher order linear and nonlinear dispersion terms with Jacobian elliptic function expansion method and arrived at the Jacobian elliptic periodic wave solutions. When the module of the Jacobian elliptic function m → 1, these solutions degenerate to the solitary wave solutions of the governing system.

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22

Langlais, Michel. « On the Continuous Solutions of a Degenerate Elliptic Equation ». Proceedings of the London Mathematical Society s3-50, n^o 2 (mars 1985) : 282–98. http://dx.doi.org/10.1112/plms/s3-50.2.282.

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23

Kvedaras, B. « On regular solutions of a strongly degenerate elliptic equation ». Lithuanian Mathematical Journal 35, n^o 2 (avril 1995) : 168–82. http://dx.doi.org/10.1007/bf02341496.

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24

French, Donald A. « The Finite Element Method for a Degenerate Elliptic Equation ». SIAM Journal on Numerical Analysis 24, n^o 4 (août 1987) : 788–815. http://dx.doi.org/10.1137/0724051.

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25

Wong, M. W. « Weyl transforms and a degenerate elliptic partial differential equation ». Proceedings of the Royal Society A : Mathematical, Physical and Engineering Sciences 461, n^o 2064 (26 septembre 2005) : 3863–70. http://dx.doi.org/10.1098/rspa.2005.1560.

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We give a formula for the inverse of a degenerate elliptic partial differential operator P on related to the Heisenberg group. The formula is in terms of pseudo-differential operators of the Weyl type, i.e. Weyl transforms. The technique is to use the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for . Using the formula for the inverse, we give an estimate for the L p norm of the solution u of the partial differential equation Pu = f on in terms of the L 2 norm of f , 2≤ p ≤∞.

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26

Dong, Wei, et Jian Tao Chen. « Existence and Multiplicity Results for a Degenerate Elliptic Equation ». Acta Mathematica Sinica, English Series 22, n^o 3 (14 mars 2006) : 665–70. http://dx.doi.org/10.1007/s10114-005-0696-0.

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27

Ji, Xinhua, et Tao Qian. « Properties of Poisson kernel for a degenerate elliptic equation ». Mathematical Methods in the Applied Sciences 23, n^o 1 (10 janvier 2000) : 71–80. http://dx.doi.org/10.1002/(sici)1099-1476(20000110)23:1<71 ::aid-mma104>3.0.co;2-2.

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28

Aliev, S., T. Gajiev, Ya Rustamov et T. Maharramova. « Forcing the system by a drift ». Matematychni Studii 55, n^o 2 (22 juin 2021) : 201–5. http://dx.doi.org/10.30970/ms.55.2.201-205.

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29

Kozhanov, Aleksandr I., et Oksana I. Bzheumikhova. « Elliptic and Parabolic Equations with Involution and Degeneration at Higher Derivatives ». Mathematics 10, n^o 18 (14 septembre 2022) : 3325. http://dx.doi.org/10.3390/math10183325.

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We study the solvability in Sobolev spaces of boundary value problems for elliptic and parabolic equations with variable coefficients in the presence of an involution (involutive deviation) at higher derivatives, both in the nondegenerate and degenerate cases. For the problems under study, we prove the existence theorems as well as the uniqueness of regular solutions, i.e., those that have all weak derivatives in the equation.

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30

Abashkin, A. A. « ONE-VALUED SOLVABILITY OF A NONLOCAL PROBLEM FOR THE AXISYMMETRIC HELMHOLTZ EQUATION ». Vestnik of Samara University. Natural Science Series 17, n^o 2 (16 juin 2017) : 5–14. http://dx.doi.org/10.18287/2541-7525-2011-17-2-5-14.

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A nonlocal boundary value problem for degenerate elliptic equation is considered. Boundary value of this problem considerably depend on low derivativecoefficient changes. Existence and uniqueness of a solution are proved.

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31

Ammar, Kaouther, et Petra Wittbold. « Existence of renormalized solutions of degenerate elliptic-parabolic problems ». Proceedings of the Royal Society of Edinburgh : Section A Mathematics 133, n^o 3 (juin 2003) : 477–96. http://dx.doi.org/10.1017/s0308210500002493.

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We consider a general class of degenerate elliptic-parabolic problems associated with the equation b(υ)t = div a(υ, Dυ) + f. Existence of renormalized solutions is established for general L1 data. Uniqueness of renormalized solutions has already been shown in a previous work.

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32

Huang, Genggeng. « A Liouville theorem of degenerate elliptic equation and its application ». Discrete & ; Continuous Dynamical Systems - A 33, n^o 10 (2013) : 4549–66. http://dx.doi.org/10.3934/dcds.2013.33.4549.

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33

HAYASIDA, Kazuya, et Yasuhiko KAWAI. « On a Degenerate Quasilinear Elliptic Equation with Mixed Boundary Conditions ». Tokyo Journal of Mathematics 10, n^o 2 (décembre 1987) : 437–70. http://dx.doi.org/10.3836/tjm/1270134525.

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34

Arbogast, Todd, et Abraham L. Taicher. « A Linear Degenerate Elliptic Equation Arising from Two-Phase Mixtures ». SIAM Journal on Numerical Analysis 54, n^o 5 (janvier 2016) : 3105–22. http://dx.doi.org/10.1137/16m1067846.

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35

Shen, Yao-tian, et Zhi-hui Chen. « Nonlinear degenerate elliptic equation with Hardy potential and critical parameter ». Nonlinear Analysis : Theory, Methods & ; Applications 69, n^o 4 (août 2008) : 1462–77. http://dx.doi.org/10.1016/j.na.2007.06.046.

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36

Petean, Jimmy. « Degenerate solutions of a nonlinear elliptic equation on the sphere ». Nonlinear Analysis : Theory, Methods & ; Applications 100 (mai 2014) : 23–29. http://dx.doi.org/10.1016/j.na.2013.12.024.

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37

Santambrogio, Filippo, et Vincenzo Vespri. « Continuity in two dimensions for a very degenerate elliptic equation ». Nonlinear Analysis : Theory, Methods & ; Applications 73, n^o 12 (décembre 2010) : 3832–41. http://dx.doi.org/10.1016/j.na.2010.08.008.

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38

Citti, Giovanna. « Positive solutions for a quasilinear degenerate elliptic equation inR n ». Rendiconti del Circolo Matematico di Palermo 35, n^o 3 (septembre 1986) : 364–75. http://dx.doi.org/10.1007/bf02843904.

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39

Takeuchi, Shingo. « Positive solutions of a degenerate elliptic equation with logistic reaction ». Proceedings of the American Mathematical Society 129, n^o 2 (29 août 2000) : 433–41. http://dx.doi.org/10.1090/s0002-9939-00-05723-3.

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40

Amattat, M. « Multiplicity results and global bifurcations for a degenerate elliptic equation ». Nonlinear Analysis and Differential Equations 2 (2014) : 1–44. http://dx.doi.org/10.12988/nade.2014.3615.

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41

Chen, Guanwei. « Nonlinear elliptic equation with lower order term and degenerate coercivity ». Mathematical Notes 93, n^o 1-2 (janvier 2013) : 224–37. http://dx.doi.org/10.1134/s0001434613010240.

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42

Bao, Jiguang. « The Dirichlet Problem for the Degenerate Elliptic Monge–Ampère Equation ». Journal of Mathematical Analysis and Applications 238, n^o 1 (octobre 1999) : 166–78. http://dx.doi.org/10.1006/jmaa.1999.6519.

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43

Takeuchi, Shingo. « Multiplicity Result for a Degenerate Elliptic Equation with Logistic Reaction ». Journal of Differential Equations 173, n^o 1 (juin 2001) : 138–44. http://dx.doi.org/10.1006/jdeq.2000.3914.

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44

Garain, Prashanta. « Properties of solutions to some weighted p-Laplacian equation ». Opuscula Mathematica 40, n^o 4 (2020) : 483–94. http://dx.doi.org/10.7494/opmath.2020.40.4.483.

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In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,\] on smooth domain and for varying nonlinearity \(f\).

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45

Al Taki, Bilal, et Christophe Lacave. « Degenerate lake equations : classical solutions and vanishing viscosity limit ». Nonlinearity 36, n^o 1 (14 décembre 2022) : 653–78. http://dx.doi.org/10.1088/1361-6544/aca865.

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Abstract The objective of this paper is twofold. First, we show the existence of global classical solutions to the degenerate inviscid lake equations. This result is achieved after revising the elliptic regularity for a degenerate equation on the associated stream-function, and adapting the method used for construction of classical solutions to the incompressible Euler equations. Second, we show that the weak solutions of the viscous lake equations converge to classical solutions of the inviscid lake equations when the viscosity coefficient goes to zero, which constitutes an important physical validation of these models. The later result is achieved by the use of energy method as in the proofs of Kato-type theorems. This method also allows us to expose a convergence rate.

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46

Mohammed, Ahmed. « Hölder continuity of solutions of some degenerate elliptic differential equations ». Bulletin of the Australian Mathematical Society 62, n^o 3 (décembre 2000) : 369–77. http://dx.doi.org/10.1017/s0004972700018888.

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47

Turov, M. M., V. E. Fedorov et B. T. Kien. « Linear Inverse Problems for Multi-term Equations with Riemann — Liouville Derivatives ». Bulletin of Irkutsk State University. Series Mathematics 38 (2021) : 36–53. http://dx.doi.org/10.26516/1997-7670.2021.38.36.

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The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann – Liouville derivatives in time.

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48

Salas, Alvaro H., Castillo H. Jairo E et M. R. Alharthi. « On the Approximate Solutions of the Constant Forced (Un)Damping Helmholtz Equation for Arbitrary Initial Conditions ». Mathematical Problems in Engineering 2021 (31 janvier 2021) : 1–10. http://dx.doi.org/10.1155/2021/8887566.

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This paper presents some novel solutions to the family of the Helmholtz equations (including the constant forced undamping Helmholtz equation (equation (1)) and the constant forced damping Helmholtz equation (equation (2))) which have been reported. In the beginning, equation (1) is solved analytically using two different techniques (direct and indirect solutions): in the first technique (direct solution), a new assumption is introduced to find the analytical solution of equation (1) in the form of the Weierstrass elliptic function with arbitrary initial conditions. In the second case (indirect solution), the solution of the undamping (standard) Duffing equation is devoted to determine the analytical solution to equation (1) in the form of Jacobian elliptic function with arbitrary initial conditions. Moreover, equation (2) is solved using a new ansatz and with the help of equation (1) solutions. Also, the evolution equations (equations (1) and (2)) are solved numerically via the Adomian decomposition method (ADM). Furthermore, a comparison between the approximate analytical solution and approximate numerical solutions using the fourth-order Runge–Kutta method (RK4) and ADM is reported. Furthermore, the maximum distance error for the obtained solutions is estimated. As a practical application, the Helmholtz-type equation will be derived from the fluid governing equations of quantum plasma particles with(out) taking the ionic kinematic viscosity into account for investigating the characteristics of (un)damping oscillations in a degenerate quantum plasma model.

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49

Neveu, André. « A Bäcklund Transformation for Elliptic Four-Point Conformal Blocks ». Reviews in Mathematical Physics 30, n^o 07 (25 juillet 2018) : 1840012. http://dx.doi.org/10.1142/s0129055x18400123.

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We apply an integral transformation to solutions of a partial differential equation for the five-point correlation functions in Liouville theory on a sphere with one degenerate field [Formula: see text]. By repeating this transformation, we can reach a whole lattice of values for the conformal dimensions of the four other operators. Factorizing out the degenerate field leads to integral representations of the corresponding four-point conformal blocks. We illustrate this procedure on the elliptic conformal blocks discovered in a previous publication.

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50

Cimpoiasu, Rodica. « Multiple explicit solutions of the 2D variable coefficients Chafee–Infante model via a generalized expansion method ». Modern Physics Letters B 35, n^o 19 (9 avril 2021) : 2150312. http://dx.doi.org/10.1142/s0217984921503127.

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In this work, we do apply a generalized expansion method to the realistic two-dimensional (2D) Chafee–Infante model with time-variable coefficients which is encountered in physical sciences.The key ideas of this method consist in: (i) to choose a nonlinear wave variable in respect to time-variable into the general finite series solution of the governing model; (ii) to take a full advantage from the general elliptic equation introduced as an auxiliary equation which can degenerate into sub-equations such as Riccati equation, the Jacobian elliptic equations, the generalized Riccati equation. Based upon this powerful technique, we successfully construct for the first time several types of non-autonomous solitary waves as well as some non-autonomous triangular solutions, rational or doubly periodic type ones. We investigate the propagation of non-autonomous solitons and we emphasize as well upon the influence of the variable coefficients. We are providing and analyzing a few graphical representations of some specific solutions. The results of this paper will be valuable in the study of nonlinear physical phenomena. The above- mentioned method could be employed to solve other partial differential equations with variable coefficients which describe various complicated natural phenomena.

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