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Temperature solutions in thin films using thermal wave Green's function solution equation

International Journal of Heat and Mass Transfer, ISSN 0017-9310, 2013, Volume 62, Issue 1, pp. 78 - 86

The thermal wave effect occurs during rapid heating of pure dielectric materials. It also serves as a limiting solution for rapid heating of other materials....

Conduction | Energy pulse | Hyperbolic heat conduction | Green's function | Thermal wave | MECHANICS | THERMODYNAMICS | HYPERBOLIC HEAT-CONDUCTION | FINITE MEDIUM | LA CHALEUR | PROPAGATION | ENGINEERING, MECHANICAL | Thin films | Dielectric films | Dielectrics

Conduction | Energy pulse | Hyperbolic heat conduction | Green's function | Thermal wave | MECHANICS | THERMODYNAMICS | HYPERBOLIC HEAT-CONDUCTION | FINITE MEDIUM | LA CHALEUR | PROPAGATION | ENGINEERING, MECHANICAL | Thin films | Dielectric films | Dielectrics

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 09/2013, Volume 220, pp. 226 - 234

Linear superposition principles of hyperbolic and trigonometric function solutions are analyzed for Hirota bilinear equations, with an aim to construct a...

Trigonometric function solution | Hyperbolic function solution | Linear subspace of solutions | Hirota bilinear form | SYSTEM | MATHEMATICS, APPLIED | BOUSSINESQ EQUATION | WAVE SOLUTIONS | COMPLEXITON SOLUTIONS | Algorithms

Trigonometric function solution | Hyperbolic function solution | Linear subspace of solutions | Hirota bilinear form | SYSTEM | MATHEMATICS, APPLIED | BOUSSINESQ EQUATION | WAVE SOLUTIONS | COMPLEXITON SOLUTIONS | Algorithms

Journal Article

Entropy, ISSN 1099-4300, 2015, Volume 17, Issue 6, pp. 4255 - 4270

In this study, we investigate some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation and the (2 +...

(2 + 1)-dimensional cubic Klein-Gordon equation | Rational function solutions | Trigonometric function solutions | Exponential function solution | (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation | Hyperbolic function solutions | The generalized Kudryashov method | Soliton solutions | EXISTENCE | PERIODIC-SOLUTIONS | PHYSICS, MULTIDISCIPLINARY | STABILITY | (2+1)-dimensional cubic Klein-Gordon equation | WAVE SOLUTIONS | trigonometric function solutions | UNIQUENESS | rational function solutions | the generalized Kudryashov method | (1+1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation | hyperbolic function solutions | exponential function solution | soliton solutions | ENTROPY | Mathematical analysis | Klein-Gordon equation | Rational functions | Nonlinearity | Entropy | Hyperbolic functions | Three dimensional | (2 + 1)-dimensional cubic Klein–Gordon equation | (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin–Bona–Mahony equation

(2 + 1)-dimensional cubic Klein-Gordon equation | Rational function solutions | Trigonometric function solutions | Exponential function solution | (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation | Hyperbolic function solutions | The generalized Kudryashov method | Soliton solutions | EXISTENCE | PERIODIC-SOLUTIONS | PHYSICS, MULTIDISCIPLINARY | STABILITY | (2+1)-dimensional cubic Klein-Gordon equation | WAVE SOLUTIONS | trigonometric function solutions | UNIQUENESS | rational function solutions | the generalized Kudryashov method | (1+1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation | hyperbolic function solutions | exponential function solution | soliton solutions | ENTROPY | Mathematical analysis | Klein-Gordon equation | Rational functions | Nonlinearity | Entropy | Hyperbolic functions | Three dimensional | (2 + 1)-dimensional cubic Klein–Gordon equation | (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin–Bona–Mahony equation

Journal Article

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, 06/2014, Volume 470, Issue 2166, p. 20140189

We derive deterministic cumulative distribution function (CDF) equations that govern the evolution of CDFs of state variables whose dynamics are described by...

Random coefficient | Uncertainty quantification | Stochastic modelling | Hyperbolic conservation law | uncertainty quantification | TRANSPORT | random coefficient | stochastic modelling | MULTIDISCIPLINARY SCIENCES | HETEROGENEOUS POROUS-MEDIA | DYNAMICS | hyperbolic conservation law | KINEMATIC WAVES | FLOW | 1008 | 119 | 120

Random coefficient | Uncertainty quantification | Stochastic modelling | Hyperbolic conservation law | uncertainty quantification | TRANSPORT | random coefficient | stochastic modelling | MULTIDISCIPLINARY SCIENCES | HETEROGENEOUS POROUS-MEDIA | DYNAMICS | hyperbolic conservation law | KINEMATIC WAVES | FLOW | 1008 | 119 | 120

Journal Article

5.
Full Text
Temperature solutions in thin films using thermal wave Green’s function solution equation

International Journal of Heat and Mass Transfer, ISSN 0017-9310, 07/2013, Volume 62, pp. 78 - 86

The thermal wave effect occurs during rapid heating of pure dielectric materials. It also serves as a limiting solution for rapid heating of other materials....

Conduction | Energy pulse | Hyperbolic heat conduction | Green’s function | Thermal wave

Conduction | Energy pulse | Hyperbolic heat conduction | Green’s function | Thermal wave

Journal Article

AIP Conference Proceedings, ISSN 0094-243X, 01/2017, Volume 1798, Issue 1

In this study, we have applied the improved Bernoulli sub-equation function method to the generalized double combined Sinh-Cosh-Gordon equation. This method...

Two dimensional analysis | Hyperbolic functions

Two dimensional analysis | Hyperbolic functions

Journal Article

Mathematical and Computational Applications, ISSN 1300-686X, 06/2016, Volume 21, Issue 2

Journal Article

Optik, ISSN 0030-4026, 05/2016, Volume 127, Issue 10, pp. 4222 - 4245

We analytically study the Schrödinger type nonlinear evolution equations by improved tan(Φ( )/2)-expansion method. Explicit solutions are derived, which...

Trigonometric function solution | Hyperbolic function solution | Improved tan(Φ(ξ)/2)-expansion method | Schrödinger equation | Exponential solution and rational solution | Improved tan(Pdbl(ξ)/2)-expansion method | TRAVELING-WAVE SOLUTIONS | Improved tan(Phi(xi)/2)-expansion method | Schrodinger equation | PARTIAL-DIFFERENTIAL-EQUATIONS | OPTICS | Searching | Mathematical analysis | Nonlinear differential equations | Solitons | Nonlinear evolution equations | Mathematical models | Schroedinger equation | Hyperbolic functions

Trigonometric function solution | Hyperbolic function solution | Improved tan(Φ(ξ)/2)-expansion method | Schrödinger equation | Exponential solution and rational solution | Improved tan(Pdbl(ξ)/2)-expansion method | TRAVELING-WAVE SOLUTIONS | Improved tan(Phi(xi)/2)-expansion method | Schrodinger equation | PARTIAL-DIFFERENTIAL-EQUATIONS | OPTICS | Searching | Mathematical analysis | Nonlinear differential equations | Solitons | Nonlinear evolution equations | Mathematical models | Schroedinger equation | Hyperbolic functions

Journal Article

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A generalized ( G ′ G ) -expansion method for the mKdV equation with variable coefficients

Physics Letters A, ISSN 0375-9601, 2008, Volume 372, Issue 13, pp. 2254 - 2257

In this Letter, a generalized -expansion method is proposed to seek exact solutions of nonlinear evolution equations. Being concise and straightforward, this...

Nonlinear evolution equations | Generalized [formula omitted]-expansion method | Trigonometric function solution | Hyperbolic function solution | Rational solution | Generalized (frac(G | G))-expansion method | PHYSICS, MULTIDISCIPLINARY | ELLIPTIC FUNCTION EXPANSION | KADOMSTEV-PETVIASHVILI EQUATION | hyperbolic function solution | F-EXPANSION METHOD | generalized (G '/G)-expansion method | HOMOTOPY PERTURBATION METHOD | BROER-KAUP EQUATIONS | VARIATIONAL ITERATION METHOD | TRAVELING-WAVE SOLUTIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | EXP-FUNCTION METHOD | trigonometric function solution | rational solution | nonlinear evolution equations | NONLINEAR EVOLUTION-EQUATIONS

Nonlinear evolution equations | Generalized [formula omitted]-expansion method | Trigonometric function solution | Hyperbolic function solution | Rational solution | Generalized (frac(G | G))-expansion method | PHYSICS, MULTIDISCIPLINARY | ELLIPTIC FUNCTION EXPANSION | KADOMSTEV-PETVIASHVILI EQUATION | hyperbolic function solution | F-EXPANSION METHOD | generalized (G '/G)-expansion method | HOMOTOPY PERTURBATION METHOD | BROER-KAUP EQUATIONS | VARIATIONAL ITERATION METHOD | TRAVELING-WAVE SOLUTIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | EXP-FUNCTION METHOD | trigonometric function solution | rational solution | nonlinear evolution equations | NONLINEAR EVOLUTION-EQUATIONS

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 09/2018, Volume 339, pp. 297 - 305

In this paper, a variable-coefficient nonlinear time fractional partial differential equation (PDE) with initial and boundary conditions is solved by using the...

Trigonometric function solution | Time fractional PDE | Variable separation method | Airy function solution | Hyperbolic function solution | Rational solution | GENERALIZED TRAVELING SOLUTIONS | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | EXP-FUNCTION METHOD | DIFFUSION EQUATION | COMPLEXITON SOLUTIONS | Methods | Differential equations

Trigonometric function solution | Time fractional PDE | Variable separation method | Airy function solution | Hyperbolic function solution | Rational solution | GENERALIZED TRAVELING SOLUTIONS | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | EXP-FUNCTION METHOD | DIFFUSION EQUATION | COMPLEXITON SOLUTIONS | Methods | Differential equations

Journal Article

Journal of Applied Mathematics, ISSN 1110-757X, 2012, Volume 2012, pp. 1 - 15

With the help of the generalized hyperbolic function, the subsidiary ordinary differential equation method is improved and proposed to construct exact...

SYSTEM | ORDER | MATHEMATICS, APPLIED | TRAVELING-WAVE SOLUTIONS | TANH METHOD | TERMS | SYMMETRICAL LUCAS FUNCTION | DIFFERENTIAL-EQUATIONS | SUB-ODE METHOD | KDV-MKDV EQUATION | Studies | Algebra | Derivatives | Partial differential equations | Fluid dynamics | Computer simulation | Mathematical analysis | Subsidiaries | Exact solutions | Differential equations | Nonlinearity | Mathematical models | Hyperbolic functions

SYSTEM | ORDER | MATHEMATICS, APPLIED | TRAVELING-WAVE SOLUTIONS | TANH METHOD | TERMS | SYMMETRICAL LUCAS FUNCTION | DIFFERENTIAL-EQUATIONS | SUB-ODE METHOD | KDV-MKDV EQUATION | Studies | Algebra | Derivatives | Partial differential equations | Fluid dynamics | Computer simulation | Mathematical analysis | Subsidiaries | Exact solutions | Differential equations | Nonlinearity | Mathematical models | Hyperbolic functions

Journal Article

Smart Materials and Structures, ISSN 0964-1726, 02/2016, Volume 25, Issue 3, pp. 35022 - 35029

In this study, we improve a new analytical method called the 'Modified exp(-Omega(xi)) expansion function method'. This method is based on the exp(-Omega(xi))...

hyperbolic function solution | modified exp(-Ω(ξ)) expansion function method | complex hyperbolic function solution | nonlinear longitudinal wave equation | (G'/G)-EXPANSION METHOD | INSTRUMENTS & INSTRUMENTATION | MATERIALS SCIENCE, MULTIDISCIPLINARY | modified exp(-Omega(xi)) expansion function method | NONLINEAR EVOLUTION-EQUATIONS | Functions (mathematics) | Mathematical analysis | Circularity | Longitudinal waves | Nonlinearity | Hyperbolic functions | Smart materials and structures

hyperbolic function solution | modified exp(-Ω(ξ)) expansion function method | complex hyperbolic function solution | nonlinear longitudinal wave equation | (G'/G)-EXPANSION METHOD | INSTRUMENTS & INSTRUMENTATION | MATERIALS SCIENCE, MULTIDISCIPLINARY | modified exp(-Omega(xi)) expansion function method | NONLINEAR EVOLUTION-EQUATIONS | Functions (mathematics) | Mathematical analysis | Circularity | Longitudinal waves | Nonlinearity | Hyperbolic functions | Smart materials and structures

Journal Article

Mathematical Problems in Engineering, ISSN 1024-123X, 2013, Volume 2013, pp. 1 - 6

Based on the F-expansion method with a new subequation, an improved F-expansion method is introduced. As illustrative examples, some new exact solutions...

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PERIODIC-WAVE SOLUTIONS | ENGINEERING, MULTIDISCIPLINARY | Usage | Functions | Research | Functional equations | Differential equations, Nonlinear | Mathematical research | Nonlinear equations | Partial differential equations | Exact solutions | Fractals | Hyperbolic functions | Physics | Studies | Engineering | Applied mathematics | Mathematical analysis | Nonlinear evolution equations | Elliptic functions | Trigonometric functions | Methods

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PERIODIC-WAVE SOLUTIONS | ENGINEERING, MULTIDISCIPLINARY | Usage | Functions | Research | Functional equations | Differential equations, Nonlinear | Mathematical research | Nonlinear equations | Partial differential equations | Exact solutions | Fractals | Hyperbolic functions | Physics | Studies | Engineering | Applied mathematics | Mathematical analysis | Nonlinear evolution equations | Elliptic functions | Trigonometric functions | Methods

Journal Article

AIP Conference Proceedings, ISSN 0094-243X, 2013, Volume 1558, Issue 1, pp. 1914 - 1918

The main goal of the present research, the Modified Kudryashov Method has been used to obtain the exact solutions of Generalized Fisher Equation with...

Fractional Generalized Fisher Equation | Modified Kudryashov Method | Symmetrical Hyberbolic Fibonacci Function | Mathematical analysis | Nonlinear differential equations | Exact solutions | Differential equations | Mathematical models | Hyperbolic functions | Trends | Elliptic functions

Fractional Generalized Fisher Equation | Modified Kudryashov Method | Symmetrical Hyberbolic Fibonacci Function | Mathematical analysis | Nonlinear differential equations | Exact solutions | Differential equations | Mathematical models | Hyperbolic functions | Trends | Elliptic functions

Conference Proceeding

Physics Letters A, ISSN 0375-9601, 2006, Volume 352, Issue 3, pp. 233 - 238

Variable-coefficient Sawada–Kotere equation is researched. By the means of modified mapping method, we establish a mapping relation between the known solutions...

Sawada–Kotere equation | Jacobian elliptic function | Mapping method | Variable-coefficient equation | Hyperbolic function | Sawada-Kotere equation | EXPANSION METHOD | variable-coefficient equation | PHYSICS, MULTIDISCIPLINARY | NONLINEAR-WAVE EQUATIONS | KDV EQUATION | mapping method | hyperbolic functions Sawada-Kotere equation

Sawada–Kotere equation | Jacobian elliptic function | Mapping method | Variable-coefficient equation | Hyperbolic function | Sawada-Kotere equation | EXPANSION METHOD | variable-coefficient equation | PHYSICS, MULTIDISCIPLINARY | NONLINEAR-WAVE EQUATIONS | KDV EQUATION | mapping method | hyperbolic functions Sawada-Kotere equation

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 2011, Volume 217, Issue 18, pp. 7377 - 7384

Using the differential transformation method and the homogeneous balance method, some new solutions of an auxiliary elliptic equation are obtained. These...

Nonlinear evolution equation | Auxiliary elliptic equation | Quasi-rational function solutions | EXPANSION METHOD | MATHEMATICS, APPLIED | ORDER PERIODIC-SOLUTIONS | WAVE EQUATIONS | Computation | Mathematical analysis | Differential equations | Exact solutions | Nonlinear evolution equations | Mathematical models | Transformations | Elliptic functions | Hyperbolic functions

Nonlinear evolution equation | Auxiliary elliptic equation | Quasi-rational function solutions | EXPANSION METHOD | MATHEMATICS, APPLIED | ORDER PERIODIC-SOLUTIONS | WAVE EQUATIONS | Computation | Mathematical analysis | Differential equations | Exact solutions | Nonlinear evolution equations | Mathematical models | Transformations | Elliptic functions | Hyperbolic functions

Journal Article

Journal of King Saud University - Science, ISSN 1018-3647, 07/2013, Volume 25, Issue 3, pp. 271 - 274

Formula solutions to the modified Korteweg–de Vries (mKdV) equation with constant coefficients are obtained via the Jacobi elliptic periodic function transform...

Trigonal solutions | Jacobi elliptic function | Modified Korteweg–de Vries equation | Hyperbolic solutions | Modified Korteweg-de Vries equation

Trigonal solutions | Jacobi elliptic function | Modified Korteweg–de Vries equation | Hyperbolic solutions | Modified Korteweg-de Vries equation

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 11/2018, Volume 41, Issue 16, pp. 6312 - 6325

In this paper, a system representing the coupling between the nonlinear Schrödinger equation and the inviscid burgers equation in modeling the interactions...

rational function solution | trigonometric function solution | complex function solution | hyperbolic function solution | exponential function solution | improved Bernoulli subequation function method | nonlinear Schrödinger equation | Schrödinger‐inviscid burgers system | improved tan(Φ(ξ)/2)‐expansion method | Schrödinger-inviscid burgers system | improved tan(Φ(ξ)/2)-expansion method | EXPANSION METHOD | MATHEMATICS, APPLIED | nonlinear Schrodinger equation | Schrodinger-inviscid burgers system | WHITHAM-BROER-KAUP | TRAVELING-WAVE SOLUTIONS | TANH-COTH METHOD | improved content style-type=mathematics tan(phi | LIE SYMMETRY ANALYSIS | 2)-content>-expansion method | RICCATI EQUATION | ZAKHAROV-KUZNETSOV | Nonlinear equations | Computational fluid dynamics | Rational functions | Nonlinear evolution equations | Schroedinger equation | Hyperbolic functions | Burgers equation | Trigonometric functions

rational function solution | trigonometric function solution | complex function solution | hyperbolic function solution | exponential function solution | improved Bernoulli subequation function method | nonlinear Schrödinger equation | Schrödinger‐inviscid burgers system | improved tan(Φ(ξ)/2)‐expansion method | Schrödinger-inviscid burgers system | improved tan(Φ(ξ)/2)-expansion method | EXPANSION METHOD | MATHEMATICS, APPLIED | nonlinear Schrodinger equation | Schrodinger-inviscid burgers system | WHITHAM-BROER-KAUP | TRAVELING-WAVE SOLUTIONS | TANH-COTH METHOD | improved content style-type=mathematics tan(phi | LIE SYMMETRY ANALYSIS | 2)-content>-expansion method | RICCATI EQUATION | ZAKHAROV-KUZNETSOV | Nonlinear equations | Computational fluid dynamics | Rational functions | Nonlinear evolution equations | Schroedinger equation | Hyperbolic functions | Burgers equation | Trigonometric functions

Journal Article

Ocean Engineering, ISSN 0029-8018, 07/2015, Volume 103, pp. 153 - 159

In this paper, we consider exact solutions of Maccari system. We tackle extended trial equation method and generalized Kudryashov method to find exact...

Maccari system | Generalized Kudryashov method | Extended trial equation method | Soliton solutions | Jacobi elliptic function solutions and hyperbolic function solutions | Rational function solutions | FIBONACCI FUNCTION SOLUTIONS | ENGINEERING, CIVIL | ENGINEERING, MARINE | ENGINEERING, OCEAN | OCEANOGRAPHY | EXP-FUNCTION METHOD | TRIAL EQUATION METHOD | Mathematical analysis | Exact solutions | Solitons | Ocean engineering | Hyperbolic functions | Elliptic functions | Solitary waves | Three dimensional

Maccari system | Generalized Kudryashov method | Extended trial equation method | Soliton solutions | Jacobi elliptic function solutions and hyperbolic function solutions | Rational function solutions | FIBONACCI FUNCTION SOLUTIONS | ENGINEERING, CIVIL | ENGINEERING, MARINE | ENGINEERING, OCEAN | OCEANOGRAPHY | EXP-FUNCTION METHOD | TRIAL EQUATION METHOD | Mathematical analysis | Exact solutions | Solitons | Ocean engineering | Hyperbolic functions | Elliptic functions | Solitary waves | Three dimensional

Journal Article

THERMAL SCIENCE, ISSN 0354-9836, 2019, Volume 23, Issue 4, pp. 2403 - 2411

In this paper, the (4+1)-dimensional Fokas equation is solved by the generalized F-expansion method, and new exact solutions with arbitrary functions are...

(4+1)-dimensional Fokas equation | THERMODYNAMICS | generalized F-expansion method | EXP-FUNCTION METHOD | trigonometric function solution | Jacobi elliptic function solution | hyperbolic function solution | F-EXPANSION METHOD | PERIODIC-WAVE SOLUTIONS | Elliptic functions | Hyperbolic functions | Partial differential equations | Trigonometric functions | Mathematical analysis | Exact solutions

(4+1)-dimensional Fokas equation | THERMODYNAMICS | generalized F-expansion method | EXP-FUNCTION METHOD | trigonometric function solution | Jacobi elliptic function solution | hyperbolic function solution | F-EXPANSION METHOD | PERIODIC-WAVE SOLUTIONS | Elliptic functions | Hyperbolic functions | Partial differential equations | Trigonometric functions | Mathematical analysis | Exact solutions

Journal Article

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