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Journal articles on the topic 'Semi linear equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Wang, Feizhi, and Yisheng Huang. "On a semi-linear Schrödinger equation in." Nonlinear Analysis: Theory, Methods & Applications 62, no. 5 (August 2005): 833–48. http://dx.doi.org/10.1016/j.na.2005.03.087.

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2

Krieger, J., and W. Schlag. "On the focusing critical semi-linear wave equation." American Journal of Mathematics 129, no. 3 (2007): 843–913. http://dx.doi.org/10.1353/ajm.2007.0021.

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3

Bokanowski, Olivier, Athena Picarelli, and Christoph Reisinger. "Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations." Numerische Mathematik 148, no. 1 (May 2021): 187–222. http://dx.doi.org/10.1007/s00211-021-01202-x.

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AbstractWe study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the $$L^2$$ L 2 norm for linear and semi-linear equations, and in the $$H^1$$ H 1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in $$L^2$$ L 2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms.

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4

Mahomed, F. M., A. Qadir, and A. Ramnarain. "Laplace-Type Semi-Invariants for a System of Two Linear Hyperbolic Equations by Complex Methods." Mathematical Problems in Engineering 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/202973.

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In 1773 Laplace obtained two fundamental semi-invariants, called Laplace invariants, for scalar linear hyperbolic partial differential equations (PDEs) in two independent variables. He utilized this in his integration theory for such equations. Recently, Tsaousi and Sophocleous studied semi-invariants for systems of two linear hyperbolic PDEs in two independent variables. Separately, by splitting a complex scalar ordinary differential equation (ODE) into its real and imaginary parts PDEs for two functions of two variables were obtained and their symmetry structure studied. In this work we revisit semi-invariants under equivalence transformations of the dependent variables for systems of two linear hyperbolic PDEs in two independent variables when such systems correspond to scalar complex linear hyperbolic equations in two independent variables, using the above-mentioned splitting procedure. The semi-invariants under linear changes of the dependent variables deduced for this class of hyperbolic linear systems correspond to the complex semi-invariants of the complex scalar linear hyperbolic equation. We show thatthe adjoint factorization corresponds precisely to the complex splitting. We also study the reductions and the inverse problem when such systems of two linear hyperbolic PDEs arise from a linear complex hyperbolic PDE. Examples are given to show the application of this approach.

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5

ANH, CUNG THE, and TANG QUOC BAO. "PULLBACK ATTRACTORS FOR A NON-AUTONOMOUS SEMI-LINEAR DEGENERATE PARABOLIC EQUATION." Glasgow Mathematical Journal 52, no. 3 (August 25, 2010): 537–54. http://dx.doi.org/10.1017/s0017089510000418.

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AbstractIn this paper, using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semi-linear degenerate parabolic equation in an arbitrary domain, without restriction on the growth order of the polynomial type non-linearity and with a suitable exponential growth of the external force. The obtained results improve some recent ones for the non-autonomous reaction–diffusion equations.

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6

Yin, Zhongqi. "Lipschitz stability for a semi-linear inverse stochastic transport problem." Journal of Inverse and Ill-posed Problems 28, no. 2 (April 1, 2020): 185–93. http://dx.doi.org/10.1515/jiip-2018-0115.

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AbstractThis paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations.

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7

Avdonin, S. A., B. P. Belinskiy, and John V. Matthews. "Inverse problem on the semi-axis: local approach." Tamkang Journal of Mathematics 42, no. 3 (August 24, 2011): 275–93. http://dx.doi.org/10.5556/j.tkjm.42.2011.916.

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We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

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8

Castro, Hernán. "Oscillations in a semi-linear singular Sturm–Liouville equation." Asymptotic Analysis 94, no. 3-4 (September 16, 2015): 363–73. http://dx.doi.org/10.3233/asy-151318.

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9

Oghre, E. O., and B. I. Olajuwon. "Fourier Transform Solution of the Semi-linear Parabolic Equation." Journal of Applied Sciences 5, no. 3 (February 15, 2005): 492–95. http://dx.doi.org/10.3923/jas.2005.492.495.

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10

V. Lê, Út. "Contraction-Galerkin method for a semi-linear wave equation." Communications on Pure & Applied Analysis 9, no. 1 (2010): 141–60. http://dx.doi.org/10.3934/cpaa.2010.9.141.

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11

Qi-guang, Wu, and Li Ji-chun. "Numerical solutions for singularly perturbed semi-linear parabolic equation." Applied Mathematics and Mechanics 14, no. 9 (September 1993): 793–801. http://dx.doi.org/10.1007/bf02457474.

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12

Liu, Can, Xinming Zhang, and Boying Wu. "Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations." Mathematics 8, no. 9 (September 10, 2020): 1549. http://dx.doi.org/10.3390/math8091549.

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In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral operator and the Caputo fractional differential operator. The quasilinearization method is utilized to convert the multi-term non-linear fractional order equation into a multi-term linear fractional order equation which, subsequently, is solved by means of semi-orthogonal B-spline wavelets. Herein, we investigate the operational matrix and the convergence of the proposed scheme. Several numerical results are delivered to confirm the accuracy and efficiency of our scheme.

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13

Talay Akyildiz, F., and K. Vajravelu. "Galerkin-Chebyshev Pseudo Spectral Method and a Split Step New Approach for a Class of Two dimensional Semi-linear Parabolic Equations of Second Order." Applied Mathematics and Nonlinear Sciences 3, no. 1 (June 8, 2018): 255–64. http://dx.doi.org/10.21042/amns.2018.1.00019.

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AbstractIn this paper, we use a time splitting method with higher-order accuracy for the solutions (in space variables) of a class of two-dimensional semi-linear parabolic equations. Galerkin-Chebyshev pseudo spectral method is used for discretization of the spatial derivatives, and implicit Euler method is used for temporal discretization. In addition, we use this novel method to solve the well-known semi-linear Poisson-Boltzmann (PB) model equation and obtain solutions with higher-order accuracy. Furthermore, we compare the results obtained by our method for the semi-linear parabolic equation with the available analytical results in the literature for some special cases, and found excellent agreement. Furthermore, our new technique is also applicable for three-dimensional problems.

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14

Blömker, Dirk, Giuseppe Cannizzaro, and Marco Romito. "Random initial conditions for semi-linear PDEs." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 3 (January 29, 2019): 1533–65. http://dx.doi.org/10.1017/prm.2018.157.

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AbstractWe analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.

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15

BUCKWAR, E., M. G. RIEDLER, and P. E. KLOEDEN. "THE NUMERICAL STABILITY OF STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE." Stochastics and Dynamics 11, no. 02n03 (September 2011): 265–81. http://dx.doi.org/10.1142/s0219493711003279.

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An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

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16

Pereira, D. C., and C. A. Raposo. "Asymptotic behavior for semi-linear wave equation with weak damping." International Journal of Mathematical Analysis 7 (2013): 713–18. http://dx.doi.org/10.12988/ijma.2013.13068.

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17

Tucsnak, Marius. "Semi-internal Stabilization for a Non-linear Bernoulli-Euler Equation." Mathematical Methods in the Applied Sciences 19, no. 11 (July 25, 1996): 897–907. http://dx.doi.org/10.1002/(sici)1099-1476(19960725)19:11<897::aid-mma801>3.0.co;2-#.

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18

Shen, Ruipeng. "A semi-linear energy critical wave equation with an application." Journal of Differential Equations 261, no. 11 (December 2016): 6437–84. http://dx.doi.org/10.1016/j.jde.2016.08.043.

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19

Castro, Rodrigo, and Patricio L. Felmer. "Semi-Classical Limit for Radial Non-Linear Schr�dinger Equation." Communications in Mathematical Physics 256, no. 2 (March 15, 2005): 411–35. http://dx.doi.org/10.1007/s00220-005-1320-y.

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20

Angenent, S. B. "The Morse-Smale property for a semi-linear parabolic equation." Journal of Differential Equations 62, no. 3 (May 1986): 427–42. http://dx.doi.org/10.1016/0022-0396(86)90093-8.

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21

D'AVENIA, PIETRO, EUGENIO MONTEFUSCO, and MARCO SQUASSINA. "ON THE LOGARITHMIC SCHRÖDINGER EQUATION." Communications in Contemporary Mathematics 16, no. 02 (April 2014): 1350032. http://dx.doi.org/10.1142/s0219199713500326.

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In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

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22

Zhang, Rongpei, Xijun Yu, Mingjun Li, and Zhen Wang. "A semi-implicit integration factor discontinuous Galerkin method for the non-linear heat equation." Thermal Science 23, no. 3 Part A (2019): 1623–28. http://dx.doi.org/10.2298/tsci180921232z.

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In this paper, a new discontinuous Galerkin method is employed to study the non-linear heat conduction equation with temperature dependent thermal conductivity. We present practical implementation of the new discontinuous Galerkin scheme with weighted flux averages. The second-order implicit integration factor for time discretization method is applied to the semi discrete form. We obtain the L2 stability of the discontinuous Galerkin scheme. Numerical examples show that the error estimates are of second order when linear element approximations are applied. The method is applied to the non-linear heat conduction equations with source term.

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23

Lissy, Pierre, Yannick Privat, and Yacouba Simporé. "Insensitizing control for linear and semi-linear heat equations with partially unknown domain." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 50. http://dx.doi.org/10.1051/cocv/2018035.

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We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝN (N ∈ ℕ*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.

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24

Huang, Chen. "A variant of Clark’s theorem and its applications for nonsmooth functionals without the global symmetric condition." Advances in Nonlinear Analysis 11, no. 1 (July 29, 2021): 285–303. http://dx.doi.org/10.1515/anona-2020-0197.

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Abstract We give a new non-smooth Clark’s theorem without the global symmetric condition. The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations. Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11].

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25

Filimonova, I. V., and T. S. Khachlaev. "ON ASYMPTOTIC PROPERTIES OF SOLUTIONS, DEFINED ON THE HALF OF AXIS OF ONE SEMILINEAR ODE." Vestnik of Samara University. Natural Science Series 21, no. 6 (May 17, 2017): 130–34. http://dx.doi.org/10.18287/2541-7525-2015-21-6-130-134.

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The paper deals with the solutions of ordinary diﬀerential semi-linear equa- tion, the coeﬃcients of which depend on several real parameters. If the coeﬃcient is chosen so that the equation does not contain the ﬁrst-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at inﬁnity is described in the book of Richard Bellman. The equations with the ﬁrst-order derivative, considered in this work, erase in some problems for elliptic partial diﬀerential equations in unbounded domains. The sign of the coeﬃcient in ﬁrst-order deriva- tive term essentially inﬂuences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential. The paper deals with the solutions of ordinary diﬀerential semi-linear equa- tion, the coeﬃcients of which depend on several real parameters. If the coeﬃcient is chosen so that the equation does not contain the ﬁrst-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at inﬁnity is described in the book of Richard Bellman. The equations with the ﬁrst-order derivative, considered in this work, erase in some problems for elliptic partial diﬀerential equations in unbounded domains. The sign of the coeﬃcient in ﬁrst-order deriva- tive term essentially inﬂuences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.

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26

Tumolo, Giovanni. "A mass conservative TR-BDF2 semi-implicit semi-Lagrangian DG discretization of the shallow water equations on general structured meshes of quadrilaterals." Communications in Applied and Industrial Mathematics 7, no. 3 (September 1, 2016): 165–90. http://dx.doi.org/10.1515/caim-2016-0026.

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Abstract As an extension of a previous work considering a fully advective formulation on Cartesian meshes, a mass conservative discretization approach is presented here for the shallow water equations, based on discontinuous finite elements on general structured meshes of quadrilaterals. A semi-implicit time integration is performed by employing the TR-BDF2 scheme and is combined with the semi-Lagrangian technique for the momentum equation only. Indeed, in order to simplify the derivation of the discrete linear Helmoltz equation to be solved at each time-step, a non-conservative formulation of the momentum equation is employed. The Eulerian flux form is considered instead for the continuity equation in order to ensure mass conservation. Numerical results show that on distorted meshes and for relatively high polynomial degrees, the proposed numerical method fully conserves mass and presents a higher level of accuracy than a standard off-centered Crank Nicolson approach. This is achieved without any significant imprinting of the mesh distortion on the solution.

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27

Dehghan, Mehdi. "Determination of an unknown parameter in a semi-linear parabolic equation." Mathematical Problems in Engineering 8, no. 2 (2002): 111–22. http://dx.doi.org/10.1080/10241230212906.

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Some finite difference approximations are developed for an inverse problem of determining an unknown parameterp(t)which is a coefficient of the solutionuin a semi-linear parabolic partial differential equation subject to a boundary integral overspecification. The accuracy and efficiency of the procedures are discussed. Some computational results using the newly proposed numerical techniques are presented. CPU times needed for this problem are reported.

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28

Kosmatov, Nickolai. "A coincidence problem for a second-order semi-linear differential equation." Electronic Journal of Qualitative Theory of Differential Equations, no. 82 (2020): 1–12. http://dx.doi.org/10.14232/ejqtde.2020.1.82.

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29

Nahas, J., and G. Ponce. "On the Persistent Properties of Solutions to Semi-Linear Schrödinger Equation." Communications in Partial Differential Equations 34, no. 10 (September 30, 2009): 1208–27. http://dx.doi.org/10.1080/03605300903129044.

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30

SUGITANI, YOUSUKE, and SHUICHI KAWASHIMA. "DECAY ESTIMATES OF SOLUTIONS TO A SEMI-LINEAR DISSIPATIVE PLATE EQUATION." Journal of Hyperbolic Differential Equations 07, no. 03 (September 2010): 471–501. http://dx.doi.org/10.1142/s0219891610002207.

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We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

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31

Pan, Xingbin. "Existence of singular solutions of semi-linear elliptic equation in Rn." Journal of Differential Equations 94, no. 1 (November 1991): 191–203. http://dx.doi.org/10.1016/0022-0396(91)90108-l.

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32

Du, Qiang, Lili Ju, Xiao Li, and Zhonghua Qiao. "Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation." Journal of Computational Physics 363 (June 2018): 39–54. http://dx.doi.org/10.1016/j.jcp.2018.02.023.

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33

Kosmatov, Nickolai. "A coincidence problem for a second-order semi-linear differential equation." Electronic Journal of Qualitative Theory of Differential Equations, no. 82 (2020): 1–12. http://dx.doi.org/10.14232/ejqtde.2020.1.82.

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34

Li, Yuan, and Jiang Qin. "Infinitely many Solutions of Semi-Linear Elliptic Equation with a Logarithmic Nonlinear Term." Applied Mechanics and Materials 496-500 (January 2014): 2216–19. http://dx.doi.org/10.4028/www.scientific.net/amm.496-500.2216.

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The semi-linear elliptic equation is an important model in Mathematic, Physics. In this paper, we study the Dirichlet problem of semi-linear elliptic equation with a logarithmic nonlinear term. By using the logarithmic Sobolev inequality, mountain pass theorem and perturbation theorem, we obtain infinitely many nontrivial weak solutions, and also the energy of the solution is positive.

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35

Zhang, Yinnan, and Weian Zheng. "Discretizing a backward stochastic differential equation." International Journal of Mathematics and Mathematical Sciences 32, no. 2 (2002): 103–16. http://dx.doi.org/10.1155/s0161171202110234.

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We show a simple method to discretize Pardoux-Peng's nonlinear backward stochastic differential equation. This discretization scheme also gives a numerical method to solve a class of semi-linear PDEs.

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36

Naito, Yūki. "An ODE approach to the multiplicity of self-similar solutions for semi-linear heat equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 4 (August 2006): 807–35. http://dx.doi.org/10.1017/s0308210500004741.

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The Cauchy problem for semi-linear heat equations with singular initial data is studied, where N > 2, p > (N + 2)/N, and l > 0 is a parameter. We establish the existence and multiplicity of positive self-similar solutions for the problem by applying the ordinary differential equation shooting method to the corresponding spatial profile problem.

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37

Zhou, Jun. "Numerical simulations of the energy-stable scheme for Swift-Hohenberg equation." Thermal Science 23, Suppl. 3 (2019): 669–76. http://dx.doi.org/10.2298/tsci180515080z.

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A collocation Fourier scheme for Swift-Hohenberg equation based on the convex splitting idea is implemented. To ensure an efficient numerical computation, we propose a general framework with linear iteration algorithm to solve the non-linear coupled equations which arise with the semi-implicit scheme. Following the contraction mapping theorem, we present a detailed convergence analysis for the linear iteration algorithm. Various numerical simulations, including verification of accuracy, dissipative property of discrete energy and pattern formation, are presented to demonstrate the efficiency and the robustness of proposed method.

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38

Matus, P. P., and H. T. K. Anh. "Compact difference schemes for Klein-Gordon equation." Doklady of the National Academy of Sciences of Belarus 64, no. 5 (November 5, 2020): 526–33. http://dx.doi.org/10.29235/1561-8323-2020-64-5-526-533.

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In this paper, we consider compact difference approximation of the fourth-order schemes for linear, semi-linear, and quasilinear Klein-Gordon equations. with respect to a small perturbation of initial conditions, right-hand side, and coefficients of the linear equations the strong stability of difference schemes is proved. The conducted numerical experiment shows how Runge rule is used to determine the orders of convergence of the difference scheme in the case of two independent variables.

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39

Diblík, Josef, Irada Dzhalladova, Mária Michalková, and Miroslava Růžičková. "Moment Equations in Modeling a Stable Foreign Currency Exchange Market in Conditions of Uncertainty." Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/172847.

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The paper develops a mathematical model of foreign currency exchange market in the form of a stochastic linear differential equation with coefficients depending on a semi-Markov process. The boundaries of the domain of its instability is determined by using moment equations.

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40

Xue, Chun Fang. "Method of Lines to Solve the Linear Temperature Field of LENS." Advanced Materials Research 1120-1121 (July 2015): 1441–45. http://dx.doi.org/10.4028/www.scientific.net/amr.1120-1121.1441.

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This article introduces a semi-analytical numerical method ——method of lines(MOLs) to solve steady temperature field of Laser Engineered Net shaping (LENS). The main idea of MOLs is to semi-discretized the governing equation of thermal transfer problem into a system of ordinary differential equations (ODEs) defined on discrete lines by means of the finite difference method. The steady linear temperature fields of functionally graded materials were obtained using MOLs and the regularities of different temperature functions were also found. The effects of thermal conductivity coefficient under different formal functions on thermal temperature fields were analys. Numerical results showed that different material thermal conductivity function had obvious different effect on the temperature field.

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41

Liang, Chuangchuang, and Pengchao Wang. "BLOW-UP RATE FOR THE SEMI-LINEAR WAVE EQUATION IN BOUNDED DOMAIN." Bulletin of the Korean Mathematical Society 52, no. 1 (January 31, 2015): 173–82. http://dx.doi.org/10.4134/bkms.2015.52.1.173.

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42

Liu, Yongqin. "The point-wise estimates of solutions for semi-linear dissipative wave equation." Communications on Pure and Applied Analysis 12, no. 1 (September 2012): 237–52. http://dx.doi.org/10.3934/cpaa.2013.12.237.

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43

Pokrovskii, A., and O. Rasskazov. "A symmetric solution of a semi-linear Duffing equation with Preisach nonlinearity." Physica B: Condensed Matter 372, no. 1-2 (February 2006): 30–32. http://dx.doi.org/10.1016/j.physb.2005.10.120.

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44

Fila, Marek, John R. King, Michael Winkler, and Eiji Yanagida. "Very slow grow-up of solutions of a semi-linear parabolic equation." Proceedings of the Edinburgh Mathematical Society 54, no. 2 (March 30, 2011): 381–400. http://dx.doi.org/10.1017/s0013091509001497.

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AbstractWe consider large-time behaviour of global solutions of the Cauchy problem for a parabolic equation with a supercritical nonlinearity. It is known that the solution is global and unbounded if the initial value is bounded by a singular steady state and decays slowly. In this paper we show that the grow-up of solutions can be arbitrarily slow if the initial value is chosen appropriately.

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Varedi, S. M., M. J. Hosseini, M. Rahimi, and D. D. Ganji. "He's variational iteration method for solving a semi-linear inverse parabolic equation." Physics Letters A 370, no. 3-4 (October 2007): 275–80. http://dx.doi.org/10.1016/j.physleta.2007.05.100.

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Mabrouk, Mongi. "A variational approach for a semi-linear parabolic equation with measure data." Annales de la faculté des sciences de Toulouse Mathématiques 9, no. 1 (2000): 91–112. http://dx.doi.org/10.5802/afst.955.

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Dao, Tuan Anh, and Ahmad Z. Fino. "Critical exponent for semi‐linear structurally damped wave equation of derivative type." Mathematical Methods in the Applied Sciences 43, no. 17 (June 25, 2020): 9766–75. http://dx.doi.org/10.1002/mma.6649.

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Feizmohammadi, Ali, and Lauri Oksanen. "An inverse problem for a semi-linear elliptic equation in Riemannian geometries." Journal of Differential Equations 269, no. 6 (September 2020): 4683–719. http://dx.doi.org/10.1016/j.jde.2020.03.037.

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49

Ganji, D. D. "A semi-Analytical technique for non-linear settling particle equation of Motion." Journal of Hydro-environment Research 6, no. 4 (December 2012): 323–27. http://dx.doi.org/10.1016/j.jher.2012.04.002.

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Ang, Dang Dinh, and A. Pham Ngoc Dinh. "Mixed problem for some semi-linear wave equation with a nonhomogeneous condition." Nonlinear Analysis: Theory, Methods & Applications 12, no. 6 (June 1988): 581–92. http://dx.doi.org/10.1016/0362-546x(88)90016-8.

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