An Efficient Multiscale Runge-Kutta Galerkin Method for Generalized Burgers-Huxley Equation
|
|
- Rebecca Willis
- 5 years ago
- Views:
Transcription
1 Applied Mathematical Sciences, Vol. 11, 2017, no. 30, HIKARI Ltd, An Efficient Multiscale Runge-Kutta Galerkin Method for Generalized Burgers-Huxley Equation Jian Chen Department of Mathematics, Foshan University Foshan , P. R. China Copyright c 2017 Jian Chen. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, an efficient multiscale Runge-Kutta Galerkin method (MRKGM) efficient numerical solution for the generalized Burgers-Huxley equation is presented. Firstly, the spacial variable is discretized by multiscale Galerkin method with the multiscale orthonormal bases in H0 1 (0, 1). This method yields Burgers-Huxley equation to a system of ordinary differential equations. Secondly, the strong stability preserving third-order Runge-Kutta method is employed to solve the system of ordinary differential equations. The numerical results obtained in this way are compared with the exact solution to demonstrate the high accuracy of the method even in the case of a small number of bases. Mathematics Subject Classification: 65J15; 65M60 Keywords: multiscale Galerkin method, multiscale orthonormal bases, Runge-Kutta method, generalized Burgers-Huxley equations 1 Introduction Mathematical modeling of many physical systems leads to nonlinear partial differential equations in various fields of science and engineering. Generalized Burgers-Huxley equation u t + αu δ u x u xx = βu(1 u δ )(u δ γ), a x b, t 0, (1.1)
2 1468 Jian Chen where α, β, γ and δ are constants with β 0, δ > 0, γ (0, 1), being a nonlinear partial differential equation is of great importance for describing different mechanisms. This equation investigated by Satsuma shows a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transport [1]. In recent years, many researchers have used various numerical methods in the literatures to solve the Burgers-Huxley equation (1.1). For instance, Adomian decomposition method was studied in [2], spectral collocation method, pseudospectral method and spectral domain decomposition method were studied in a series of paper [3, 4, 5, 6, 7], wavelet method was introduced [8]. More recently, a three-step Talyor-Galerkin method was presented for the singularly perturbed generalized Burgers-Huxley equation [9]. And the local discontinuous Galerkin method has been used to solve the Burgers-Huxley and Burgers-Fisher equations [10]. Multiscale or multilevel numerical methods received much attention recently. They have considerable advantages and become the standard approaches in solving integral equations([11, 12, 13, 14, 15]). The methods lead to a numerically sparse matrix presentation of the integral operator. A proper truncation of such a sparse matrix will result in a fast numerical algorithm for solving the equation. What s more, the methods allow us to design multilevel augmentation method (MAM) for fast solving the system resulting from discretization of the integral equation because of the multiscale structure of the coefficient matrix [13, 14, 15]. This method was then applied to solve differential equations [16, 17, 18]. The main advantage of multiscale methods for solving differential equations is the stability of the multiscale bases. It can reduce the condition number of the resulting discrete systems largely and obtain the uniform boundedness of the condition number under some conditions[16]. In this paper, the generalized Burgers-Huxley equation with Dirichlet boundary condition is solved by combination of the multiscale Galerkin method and the strong stability preserving Runge-Kutta method. After discreting the spacial variables by Galerkin method with the multiscale orthornomal bases constructed in [16], we apply the strong stability preserving three-oder Runge- Kutta method [19] to solve the resulting ordinary differential equations. The computed results show that the method is effective for obtaining numerical approximations of Burgers-Huxley equation. Furthermore, the use of considerably small number of bases and strong stability make the method advantageous. The remainder of this paper is organized as follows: In section 2 we introduce the multiscale orthornomal bases in Sobolev spaces. We describe the model problem and the proposed algorithm in detail in section 3 and section 4, respectively. In section 5 numerical results of several test experiments are reported, which show that the algorithm is stable and accurate to solve
3 Multiscale Runge-Kutta Galerkin method 1469 Burgers-Huxley equations. At last a short conclusion is drawn in section 6. 2 Multiscale orthonormal bases for H 1 0(0, 1) In this section, we introduce the multiscale orthonormal bases constructed in [16] for Sobolev spaces H0(E), 1 E := [0, 1]. Readers can refer to [16] for the details. Let X := H0(E) 1 denote the Sobolev space of elements u vanishing at both 0 and 1, the inner product and norm of H0(E) 1 are equipped by u, v 1 := u (x)v (x)dx, u 1 := u, u 1, u, v H0(E). 1 E For k 2, we denote by X n the finite dimensional subspace of X whose elements are the piecewise polynomials of order less than k with knots j/µ n, j 1 Z µ n 1, where µ > 1 and the notation Z n := {0, 1, 2,, n 1}. It is easily seen that the subspaces X n are nested, i.e., X n 1 X n, n N 0 := {0, 1, 2, }. By the definition of X n, the dimension of the space X n is dim X n = (k 1)µ n 1. Because of the nestedness property, X n can be expressed as an orthogonal direct sum of X n 1 and W n. It follows that for n N 0, X n = X 0 W 1 W n, where the notation S 1 S 2 stands for the orthogonal direct sum of spaces S 1 and S 2. It can be computed that the dimension of the subspace W n is obtained by ω(n) := dim W n = dim X n dim X n 1 = (k 1)µ n 1. When subspace W 1 has been given, W n can be recursively constructed via the linear operators T e : L 2 (E) L 2 (E), e Z µ, defined by: { µ (T e u)(x) := 1/2 u(φ 1 e (x)), x φ e (E), (2.1) 0, x / φ e (E), where φ e (x) := x+e are affine mappings which define the partitions of interval µ E. Specially, we introduce the construction of linear multiscale orthonormal bases which will be used to discretize the spatial variables of Burgers-Huxley equation. Choose k = 2, µ = 2. It is easily seen that X 0 =, dim W i = 2 i 1, i > 1. The desired basis of W 1 is given by { x, x [0, 1 ω 1,0 = ), 2 1 x, x [ 1, 1]. 2 Let {ω i,j (x), j Z ω(i) } be the bases of W i, then the operators (2.1) for bases of W i+1 have concrete forms: { 2 1 ω i,j (2x), x [0, 1 2 ω i+1,j = ), 0, x [ 1, 1], j = 0, 1, 2,, 2 i 1 1, 2
4 1470 Jian Chen and { 0, x [0, 1 ω i+1,2 i 1 2 +j = ), 1 2 ω i,j (2x 1), x [ 1, 1], j = 0, 1, 2,, 2 i Therefor, functions {ω i,j (x) : j Z ω(i) } form an orthonormal basis for W i, and X n = {ω i,j (x) : (i, j) J n }, J n := {(i, j) : i Z n+1, j Z ω(i) }. The graph of bases for X 3 are shown in Fig ω i,j (x) x Fig. 1: H 1 0 linear multiscale orthogonal bases for X 3. For an arbitrary interval [a, b], the multiscale orthonormal bases of Sobolev space H 1 0(a, b) can also be constructed just by a simple affine transformation. 3 The model problem As mentioned above, the generalized Burgers-Huxley equation represents behaviors of many physical systems encountered in models of reaction mechanisms, convection effects and diffusion transport. The governing initial boundary value generalized Burgers-Huxley equation under consideration is: u t + αu δ u x u xx = βu(1 u δ )(u δ γ), a x b, 0 < t T, u(x, 0) = u 0 (x) = [ γ + γ tanh(θ 2 2 1x))] 1/δ, a x b, u(a, t) = u a (t) = [ γ + γ tanh(θ 2 2 1(a θ 2 t))] 1/δ, 0 < t T, u(b, t) = u b (t) = [ γ + γ tanh(θ 2 2 1(b θ 2 t))] 1/δ, 0 < t T. (3.1)
5 Multiscale Runge-Kutta Galerkin method 1471 The exact solution of the problem (3.1) is given by ([20]) where u(x, t) = [ γ 2 + γ 2 tanh(θ 1(x θ 2 t))] 1/δ, θ 1 := αδ + δ α 2 + 4β(1 + δ) 4(1 + δ) θ 2 := αγ 1 + δ, (1 + δ γ)( α + α2 + 4β(1 + δ). 2(1 + δ) The aim of this paper is to numerically solve this equation by multiscale Galerkin method. The computed results are compared with the exact solutions, showing that the presented method is capable of achieving high accuracy and stable solution for problem (3.1). 4 The multiscale Galerkin method for Burgers- Huxley equation In this section, we combine the multiscale Galerkin method and the strong stability preserving third-order Runge-Kutta method to solve the generalized Burgers-Huxley equation. First we convert (3.1) to a zero boundary problem. To this end, we let ũ(x, t) = u(x, t) h(x, t) = u(x, t) b x b a u a(t) x a b a u b(t). Then ũ(a, t) = ũ(b, t) = 0, and the problem (3.1) can be rewritten as an initial-boundary value problem of functions ũ(x, t) ũ t + h t + α(ũ + h) δ (ũ x + h x ) ũ xx = f(ũ), a < x < b, 0 < t T, ũ(x, 0) = u 0 (x) h(x, 0), a x b, ũ(a, t) = ũ(b, t) = 0, 0 < t T, (4.1) where f(ũ) = β[1 (ũ + h)][1 (ũ + h) δ ][(ũ + h) δ γ]. Let X n be a finite dimensional subspace of H 1 0(E). The multiscale orthonormal basis functions {ω i,j (x) : (i, j) J n } introduced in section 2 are chosen as bases for X n. Then the approximate solution ũ n in space X n can be written as: ũ n (x, t) = (i,j) J n ũ i,j (t)ω i,j (x). (4.2)
6 1472 Jian Chen Then, the multiscale Galerkin method of problem (4.1) can be posed as: Find ũ n = ũ n (, t) X n such that (ũ n,t, ω) + (h t, ω) + (g n, ω) = (f n, ω), ω X n, ũ n (x, 0) = ũ i,j (0)ω i,j (x), (i,j) J n (4.3) where (u, v) := b a u(x)v(x)dx, g n = g(ũ n ) := α(ũ n + h) δ (ũ n,x + h x ) ũ n,xx and f n = f(ũ n ) := β[1 (ũ n + h)][1 (ũ n + h) δ ][(ũ n + h) δ γ]. The coefficients ũ i,j (0) can be computed by ũ i,j (0) = ũ(x, 0), ω i,j (x) 1, (i, j) J n. By substituting the above representation of ũ n (4.2) into the multiscale Galerkin scheme (4.3), we obtain an initial value problem of ordinary differential equation system d = L(Ũn, t), dtũn Ũ n (0) = [ũ i,j (0) : (i, j) J n ]. (4.4) Choose τ as the time step and apply the SSP-RK3 scheme to solve ordinary differential equation system (4.4), i.e., for m = 1, 2,, T/τ, compute K 1 = Ũ(m 1) n K 2 = 3 4Ũ(m 1) n Ũ (m) n = 1 3Ũ(m 1) + τl(ũ(m 1) n, (m 1)τ), + 1K τL(K 4 1, mτ), (4.5) n + 2K τL(K 3 2, (m 1)τ). 2 Thus for m = 1, 2,, T/τ, we obtain the coefficients Ũ m n = [ũ i,j (mτ) : (i, j) J n ]. Therefor, the approximate solution of the equation(1.1) can be expressed as: u n (x, mτ) = ũ n (x, mτ) + h(x, mτ) = ũ i,j (mτ)w i,j (x) + b x b a u a(mτ) + x a b a u b(mτ). (i,j) J n
7 Multiscale Runge-Kutta Galerkin method Numerical experiments In this section we obtain numerical solutions of generalized Burgers-Huxley equation in the form (1.1) by MRKGM. To verify the efficiency of the proposed method for the current problem in comparison with the exact solution, errors for various values of α, β, δ and γ are reported in the following tables. For all of the experiments, n and N := N(n) = 2 n 1 stand for the level and the corresponding dimension of X n. And the maximum error norm L at time level t = mτ, m = 0, 1, 2, is defined by L = u n (x, t) u(x, t) = max 1 j N u n(x j, t) u(x j, t), where u n (x, t) and u(x, t) denote the approximate solution and theoretical solution respectively. For the computational work, we select the following examples from [5, 8]. In all of the examples, n is taken as 3 and τ is taken as Example 1. Consider Burgers-Huxley equation (1.1) in the domain 0 x 1 with α = β = 1 and γ = The maximum error obtained by our method and domain decomposition algorithm in [5] for various δ and t are given in Table 1. Notice that the number of basis functions in [5] and in our method are (N + 1) M = 5 2 = 10 and = 7, respectively. But the time step in [5] is much less than this in present method. Furthermore, as indicated in this table, the present method is more accurate. To show the solitary wave evolution with time, we expand the computation domain to [ 10, 20] and plot the numerical solution and exact solution in Fig.2 for the values α = β = 1, γ = 2, δ = 1 at times t = 0, 5, 10. Table 1: Maximum error with α = β = 1, γ = for Example 1. t δ Error in [5] MRKGM t = t = δ = e e-8 δ = e e-5 δ = e e δ = e e-8 δ = e e-5 δ = e e-5
8 1474 Jian Chen t=0 t=5 t=10 u(x,t) x Fig.2: Numerical solution( o ) and exact solution( - ) at time t = 0, 5, 10 of Example 1 for α = β = δ = 1, γ = 2. Example 2. The proposed method is applied to equation (1.1) with α = 0.1, β = 0.001, γ = in [0, 1] at t = 0.2, 1.0. In Table 2, we show maximum error for various values of δ and compare with the corresponding results in [5]. Table 2: Maximum error with α = 0.1, β = 0.001, γ = for Example 2. t δ Error in [5] MRKGM t = t = δ = e e-13 δ = e e-10 δ = e e δ = e e-13 δ = e e-10 δ = e e-9 Example 3. Table 3 shows the absolute errors for various values of δ and x with α = 0.1, β = 0.1 and γ = at t = 0.9. The results are compared with [8] and it is found that these are better than the results presented in [8].
9 Multiscale Runge-Kutta Galerkin method 1475 Table 3: Absolute errors for various values of δ and x at t = 0.9 for Example 3. x δ = 1 δ = 2 δ = 8 Error in [8] MRKGM Error in [8] MRKGM Error in [8] MRKGM t = t = t = t = t = t = e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-7 Example 4. In Table 4 we present the absolute errors for various values of β and x with α = 1, δ = 1 and γ = at t = 0.9. The results are compared with [8]. For t = 0.9 and β = 50, graphical presentation of the absolute error between u n and u(x, t) is depicted in Fig. 3. Table 4: Absolute errors for various values of β and x at t = 0.9 for Example 4. x β = 1 β = 10 β = 50 Error in [8] MRKGM Error in [8] MRKGM Error in [8] MRKGM t = t = t = t = t = t = e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-7
10 1476 Jian Chen 3 x u n (x,t) u(x,t) x Fig.3: The error curve of u n (x, t) u(x, t) with α = δ = 1, β = 50, γ = at t = 0.9. Example 5. Table 5 gives the absolute errors for various values of γ and x with α = 5, β = 10 and δ = 2 at t = 0.9. Also, the results are compared with [8] and it is found that these are better than the results presented in [8]. Table 5: Absolute errors for various values of γ and x at t = 0.9 for Example 5. x γ = 10 2 γ = 10 3 γ = 10 4 Error in [8] MRKGM Error in [8] MRKGM Error in [8] MRKGM t = t = t = t = t = t = e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-8
11 Multiscale Runge-Kutta Galerkin method Conclusion In this paper, we combine the multiscale Galerkin method with multiscale orthonormal bases and third-order Runge-Kutta method to solve generalized Burgers-Huxley equation. Equation (1.1) is computed for various paraments by the present method, and the numerical examples demonstrate that the presented method gives highly accurate results even in the case of a small number of grid points. Compared with analytical solution and other numerical scheme, the proposed method can give more accurate results. Acknowledgements. This research was supported in part by the Natural Science Foundation of China under grant and , the Natural Science Foundation of Guangdong Province of China under grant 2016A and the Training Programme Foundation for Excellent Young Scholar of Guangdong Province under grant YQ References [1] J. Satsuma, M. Ablowitz, B. Fuchssteiner and M. Kruskal (Eds.), Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, [2] H.N.A. Ismail, K. Raslan and A.A.A. Rabboh, Adomian decomposition method for Burger s-huxley and Burger s-fisher equations, Appl. Math. Comput., 159 (2004), [3] M. Javidi, Spectral collocation method for the solution of the generalized Burger-Fisher equation, Appl. Math. Comput., 174 (2006), [4] M. Javidi, A numerical solution of the generalized Burger s-huxley equation by pseudospectral method and Darvishi s preconditioning, Appl. Math. Comput., 175 (2006), [5] M. Javidi and A. Golbabai, A new domain decomposition algorithm for generalized Burger s-huxley equation based on Chebyshev polynomials and preconditioning, Chaos Solitons Fract., 39 (2009), [6] M. Javidi, A numerical solution of the generalized Burger s-huxley equation by spectral collocation method, Appl. Math. Comput., 178 (2006),
12 1478 Jian Chen [7] A. Golbabai and M. Javidi, A spectral domain decomposition approach for the generalized Burger s-fisher equation, Chaos Solitons Fract., 39 (2009), [8] I. Celik, Haar wavelet method for solving generalized Burgers-Huxley equation, Arab J. Math. Sci., 18 (2012), [9] B.V. Rathish Kumar., Vivek Sangwan, S.V.S.S.N.V.G.K. Murthy and Mohit Nigam, A numerical study of singularly perturbed generalized Burgers- Huxley equation using three-step Taylor-Galerkin method, Comput. Math. Appl., 62 (2011), [10] R. Zhang, X. Yu and G. Zhao, The local discontinuous Galerkin method for Burger s-huxley and Burger s-fisher equations, Appl. Math. Comput., 218 (2012), [11] Z. Chen, C. A. Micchelli and Y. Xu, A multilevel method for solving operator equations, J. Math. Anal. Appl., 262 (2001), [12] Z. Chen, C. A. Micchelli and Y. Xu, Fast collocation methods for second kind integral equations, SIAM J. Numer. Anal., 40 (2002), [13] Z. Chen, B. Wu and Y. Xu, Multilevel augmentation methods for solving operator equations, Numer. Math. J. Chinese Univ., 14 (2005), [14] Z. Chen, B. Wu and Y. Xu, Fast multilevel augmentation methods for solving Hammerstein equations, SIAM J. Numer. Anal., 47 (2009), [15] J. Chen, Fast multilevel augmentation method for nonlinear integral equations, Inte. Jour. Comput. Math., 89 (2012), no. 1, [16] Z. Chen, B. Wu and Y. Xu, Multilevel augmentation methods for differential equations, Adv. Comput. Math., 24 (2006), [17] J. Chen, Fast multilevel augmentation method for nonlinear boundary value problems, Comput. Math. Appl., 61 (2011), [18] J. Chen, Z. Chen and S. Cheng, Fast multilevel augmentation methods for solving the Sine-Gordon equation, J. Math. Anal. Appl., 375 (2011),
13 Multiscale Runge-Kutta Galerkin method 1479 [19] S. Gottlieb, C. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), [20] X. Wang, Z. Zhu and Y. Lu, Solitary wave solutions of the generalised Burgers-Huxley equation, J. Phys. A Math. General., 23 (1990), no. 3, Received: February 14, 2017; Published: May 23, 2017
A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning
Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 22, 1097-1106 A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning M. T. Darvishi a,, S.
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationThe Modified Adomian Decomposition Method for. Solving Nonlinear Coupled Burger s Equations
Nonlinear Analysis and Differential Equations, Vol. 3, 015, no. 3, 111-1 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/nade.015.416 The Modified Adomian Decomposition Method for Solving Nonlinear
More informationPeriodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation
Contemporary Engineering Sciences Vol. 11 2018 no. 16 785-791 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ces.2018.8267 Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type
More informationA Study on Linear and Nonlinear Stiff Problems. Using Single-Term Haar Wavelet Series Technique
Int. Journal of Math. Analysis, Vol. 7, 3, no. 53, 65-636 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ijma.3.3894 A Study on Linear and Nonlinear Stiff Problems Using Single-Term Haar Wavelet Series
More informationDIFFERENTIAL QUADRATURE SOLUTIONS OF THE GENERALIZED BURGERS FISHER EQUATION WITH A STRONG STABILITY PRESERVING HIGH-ORDER TIME INTEGRATION
Mathematical and Computational Applications, Vol. 16, o. 2, pp. 477-486, 2011. Association for Scientific Research DIFFERETIAL QUADRATURE SOLUTIOS OF THE GEERALIZED BURGERS FISHER EQUATIO WITH A STROG
More informationThe Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 549-560 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.718 The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive
More informationNumerical solution of General Rosenau-RLW Equation using Quintic B-splines Collocation Method
Available online at www.ispacs.com/cna Volume 2012, Year 2012 Article ID cna-00129, 16 pages doi:10.5899/2012/cna-00129 Research Article Numerical solution of General Rosenau-RLW Equation using Quintic
More informationNumerical Solution of Heat Equation by Spectral Method
Applied Mathematical Sciences, Vol 8, 2014, no 8, 397-404 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams201439502 Numerical Solution of Heat Equation by Spectral Method Narayan Thapa Department
More informationSpectral Collocation Method for Parabolic Partial Differential Equations with Neumann Boundary Conditions
Applied Mathematical Sciences, Vol. 1, 2007, no. 5, 211-218 Spectral Collocation Method for Parabolic Partial Differential Equations with Neumann Boundary Conditions M. Javidi a and A. Golbabai b a Department
More informationVariational Theory of Solitons for a Higher Order Generalized Camassa-Holm Equation
International Journal of Mathematical Analysis Vol. 11, 2017, no. 21, 1007-1018 HIKAI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.710141 Variational Theory of Solitons for a Higher Order Generalized
More informationCHAPTER III HAAR WAVELET METHOD FOR SOLVING FISHER S EQUATION
CHAPTER III HAAR WAVELET METHOD FOR SOLVING FISHER S EQUATION A version of this chapter has been published as Haar Wavelet Method for solving Fisher s equation, Appl. Math.Comput.,(ELSEVIER) 211 (2009)
More informationSuperconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients
Superconvergence of discontinuous Galerkin methods for -D linear hyperbolic equations with degenerate variable coefficients Waixiang Cao Chi-Wang Shu Zhimin Zhang Abstract In this paper, we study the superconvergence
More informationResearch Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations
Abstract and Applied Analysis Volume 212, Article ID 391918, 11 pages doi:1.1155/212/391918 Research Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations Chuanjun Chen
More informationA Recursion Scheme for the Fisher Equation
Applied Mathematical Sciences, Vol. 8, 204, no. 08, 536-5368 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/ams.204.47570 A Recursion Scheme for the Fisher Equation P. Sitompul, H. Gunawan,Y. Soeharyadi
More informationRemarks on Fuglede-Putnam Theorem for Normal Operators Modulo the Hilbert-Schmidt Class
International Mathematical Forum, Vol. 9, 2014, no. 29, 1389-1396 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47141 Remarks on Fuglede-Putnam Theorem for Normal Operators Modulo the
More informationSeries Solution of Weakly-Singular Kernel Volterra Integro-Differential Equations by the Combined Laplace-Adomian Method
Series Solution of Weakly-Singular Kernel Volterra Integro-Differential Equations by the Combined Laplace-Adomian Method By: Mohsen Soori University: Amirkabir University of Technology (Tehran Polytechnic),
More informationNumerical Investigation of the Time Invariant Optimal Control of Singular Systems Using Adomian Decomposition Method
Applied Mathematical Sciences, Vol. 8, 24, no. 2, 6-68 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ams.24.4863 Numerical Investigation of the Time Invariant Optimal Control of Singular Systems
More informationAdomian Decomposition Method with Laguerre Polynomials for Solving Ordinary Differential Equation
J. Basic. Appl. Sci. Res., 2(12)12236-12241, 2012 2012, TextRoad Publication ISSN 2090-4304 Journal of Basic and Applied Scientific Research www.textroad.com Adomian Decomposition Method with Laguerre
More informationHigh order, finite volume method, flux conservation, finite element method
FLUX-CONSERVING FINITE ELEMENT METHODS SHANGYOU ZHANG, ZHIMIN ZHANG, AND QINGSONG ZOU Abstract. We analyze the flux conservation property of the finite element method. It is shown that the finite element
More informationRational Chebyshev pseudospectral method for long-short wave equations
Journal of Physics: Conference Series PAPER OPE ACCESS Rational Chebyshev pseudospectral method for long-short wave equations To cite this article: Zeting Liu and Shujuan Lv 07 J. Phys.: Conf. Ser. 84
More informationRiesz Representation Theorem on Generalized n-inner Product Spaces
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 18, 873-882 HIKARI Ltd, www.m-hikari.com Riesz Representation Theorem on Generalized n-inner Product Spaces Pudji Astuti Faculty of Mathematics and Natural
More informationNumerical Simulation of the Generalized Hirota-Satsuma Coupled KdV Equations by Variational Iteration Method
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.7(29) No.1,pp.67-74 Numerical Simulation of the Generalized Hirota-Satsuma Coupled KdV Equations by Variational
More informationHomotopy Perturbation Method for the Fisher s Equation and Its Generalized
ISSN 749-889 (print), 749-897 (online) International Journal of Nonlinear Science Vol.8(2009) No.4,pp.448-455 Homotopy Perturbation Method for the Fisher s Equation and Its Generalized M. Matinfar,M. Ghanbari
More informationNumerical Solution of the Generalized Burgers-Huxley Equation by Exponential Time Differencing Scheme
International Journal of Bioinformatics and Biomedical Engineering Vol., No., 5, pp. 3-5 http://www.aiscience.org/journal/ijbbe Numerical Solution of the Generalized Burgers-Huxley Equation by Exponential
More informationNumerical Solution of One-dimensional Telegraph Equation using Cubic B-spline Collocation Method
2014 (2014) 1-8 Available online at www.ispacs.com/iasc Volume 2014, Year 2014 Article ID iasc-00042, 8 Pages doi:10.5899/2014/iasc-00042 Research Article Numerical Solution of One-dimensional Telegraph
More informationRELIABLE TREATMENT FOR SOLVING BOUNDARY VALUE PROBLEMS OF PANTOGRAPH DELAY DIFFERENTIAL EQUATION
(c) 216 217 Rom. Rep. Phys. (for accepted papers only) RELIABLE TREATMENT FOR SOLVING BOUNDARY VALUE PROBLEMS OF PANTOGRAPH DELAY DIFFERENTIAL EQUATION ABDUL-MAJID WAZWAZ 1,a, MUHAMMAD ASIF ZAHOOR RAJA
More informationSymmetric Properties for Carlitz s Type (h, q)-twisted Tangent Polynomials Using Twisted (h, q)-tangent Zeta Function
International Journal of Algebra, Vol 11, 2017, no 6, 255-263 HIKARI Ltd, wwwm-hiaricom https://doiorg/1012988/ija20177728 Symmetric Properties for Carlitz s Type h, -Twisted Tangent Polynomials Using
More informationThe New Exact Solutions of the New Coupled Konno-Oono Equation By Using Extended Simplest Equation Method
Applied Mathematical Sciences, Vol. 12, 2018, no. 6, 293-301 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8118 The New Exact Solutions of the New Coupled Konno-Oono Equation By Using
More informationof a Two-Operator Product 1
Applied Mathematical Sciences, Vol. 7, 2013, no. 130, 6465-6474 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.39501 Reverse Order Law for {1, 3}-Inverse of a Two-Operator Product 1 XUE
More informationNumerical Solution of Two-Dimensional Volterra Integral Equations by Spectral Galerkin Method
Journal of Applied Mathematics & Bioinformatics, vol.1, no.2, 2011, 159-174 ISSN: 1792-6602 (print), 1792-6939 (online) International Scientific Press, 2011 Numerical Solution of Two-Dimensional Volterra
More informationON THE NUMERICAL SOLUTION FOR THE FRACTIONAL WAVE EQUATION USING LEGENDRE PSEUDOSPECTRAL METHOD
International Journal of Pure and Applied Mathematics Volume 84 No. 4 2013, 307-319 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v84i4.1
More informationResearch Article Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation
Applied Mathematics Volume 22, Article ID 39876, 9 pages doi:.55/22/39876 Research Article Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation Xiuming Li
More informationAn efficient algorithm on timefractional. equations with variable coefficients. Research Article OPEN ACCESS. Jamshad Ahmad*, Syed Tauseef Mohyud-Din
OPEN ACCESS Research Article An efficient algorithm on timefractional partial differential equations with variable coefficients Jamshad Ahmad*, Syed Tauseef Mohyud-Din Department of Mathematics, Faculty
More informationMathematical Models Based on Boussinesq Love equation
Applied Mathematical Sciences, Vol. 8, 2014, no. 110, 5477-5483 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.47546 Mathematical Models Based on Boussinesq Love equation A. A. Zamyshlyaeva
More informationAn improved collocation method based on deviation of the error for solving BBMB equation
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 6, No. 2, 2018, pp. 238-247 An improved collocation method based on deviation of the error for solving BBMB equation Reza
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationSolution of Differential Equations of Lane-Emden Type by Combining Integral Transform and Variational Iteration Method
Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 3, 143-150 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2016.613 Solution of Differential Equations of Lane-Emden Type by
More informationImprovements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method
International Journal of Mathematical Analysis Vol. 9, 2015, no. 39, 1919-1928 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.54124 Improvements in Newton-Rapshon Method for Nonlinear
More informationDouble Total Domination in Circulant Graphs 1
Applied Mathematical Sciences, Vol. 12, 2018, no. 32, 1623-1633 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.811172 Double Total Domination in Circulant Graphs 1 Qin Zhang and Chengye
More informationA highly accurate method to solve Fisher s equation
PRAMANA c Indian Academy of Sciences Vol. 78, No. 3 journal of March 2012 physics pp. 335 346 A highly accurate method to solve Fisher s equation MEHDI BASTANI 1 and DAVOD KHOJASTEH SALKUYEH 2, 1 Department
More informationThe Variants of Energy Integral Induced by the Vibrating String
Applied Mathematical Sciences, Vol. 9, 15, no. 34, 1655-1661 HIKARI td, www.m-hikari.com http://d.doi.org/1.1988/ams.15.5168 The Variants of Energy Integral Induced by the Vibrating String Hwajoon Kim
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationSolution of Non Linear Singular Perturbation Equation. Using Hermite Collocation Method
Applied Mathematical Sciences, Vol. 7, 03, no. 09, 5397-5408 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ams.03.37409 Solution of Non Linear Singular Perturbation Equation Using Hermite Collocation
More informationOne-Step Hybrid Block Method with One Generalized Off-Step Points for Direct Solution of Second Order Ordinary Differential Equations
Applied Mathematical Sciences, Vol. 10, 2016, no. 29, 142-142 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.6127 One-Step Hybrid Block Method with One Generalized Off-Step Points for
More informationSolution for a non-homogeneous Klein-Gordon Equation with 5th Degree Polynomial Forcing Function
Advanced Studies in Theoretical Physics Vol., 207, no. 2, 679-685 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/astp.207.7052 Solution for a non-homogeneous Klein-Gordon Equation with 5th Degree
More informationNumerical Solution of Nonlocal Parabolic Partial Differential Equation via Bernstein Polynomial Method
Punjab University Journal of Mathematics (ISSN 116-2526) Vol.48(1)(216) pp. 47-53 Numerical Solution of Nonlocal Parabolic Partial Differential Equation via Bernstein Polynomial Method Kobra Karimi Department
More informationAnti-synchronization of a new hyperchaotic system via small-gain theorem
Anti-synchronization of a new hyperchaotic system via small-gain theorem Xiao Jian( ) College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China (Received 8 February 2010; revised
More informationNumerical Solution of Space-Time Fractional Convection-Diffusion Equations with Variable Coefficients Using Haar Wavelets
Copyright 22 Tech Science Press CMES, vol.89, no.6, pp.48-495, 22 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations with Variable Coefficients Using Haar Wavelets Jinxia Wei, Yiming
More informationA Present Position-Dependent Conditional Fourier-Feynman Transform and Convolution Product over Continuous Paths
International Journal of Mathematical Analysis Vol. 9, 05, no. 48, 387-406 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.589 A Present Position-Dependent Conditional Fourier-Feynman Transform
More informationA Numerical-Computational Technique for Solving. Transformed Cauchy-Euler Equidimensional. Equations of Homogeneous Type
Advanced Studies in Theoretical Physics Vol. 9, 015, no., 85-9 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/astp.015.41160 A Numerical-Computational Technique for Solving Transformed Cauchy-Euler
More informationWeak Solutions to Nonlinear Parabolic Problems with Variable Exponent
International Journal of Mathematical Analysis Vol. 1, 216, no. 12, 553-564 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.216.6223 Weak Solutions to Nonlinear Parabolic Problems with Variable
More informationQuintic B-Spline Galerkin Method for Numerical Solutions of the Burgers Equation
5 1 July 24, Antalya, Turkey Dynamical Systems and Applications, Proceedings, pp. 295 39 Quintic B-Spline Galerkin Method for Numerical Solutions of the Burgers Equation İdris Dağ 1,BülentSaka 2 and Ahmet
More informationSome Reviews on Ranks of Upper Triangular Block Matrices over a Skew Field
International Mathematical Forum, Vol 13, 2018, no 7, 323-335 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/imf20188528 Some Reviews on Ranks of Upper Triangular lock Matrices over a Skew Field Netsai
More informationSolution of Nonlinear Fractional Differential. Equations Using the Homotopy Perturbation. Sumudu Transform Method
Applied Mathematical Sciences, Vol. 8, 2014, no. 44, 2195-2210 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4285 Solution of Nonlinear Fractional Differential Equations Using the Homotopy
More informationProlongation structure for nonlinear integrable couplings of a KdV soliton hierarchy
Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy Yu Fa-Jun School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China Received
More informationMorera s Theorem for Functions of a Hyperbolic Variable
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 32, 1595-1600 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.212354 Morera s Theorem for Functions of a Hyperbolic Variable Kristin
More informationNew Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation
International Conference on Computer Technology and Science (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Singapore DOI:.776/IPCSIT..V47.59 New Analytical Solutions For () Dimensional Kaup-Kupershmidt Equation
More informationDynamical Behavior for Optimal Cubic-Order Multiple Solver
Applied Mathematical Sciences, Vol., 7, no., 5 - HIKARI Ltd, www.m-hikari.com https://doi.org/.988/ams.7.6946 Dynamical Behavior for Optimal Cubic-Order Multiple Solver Young Hee Geum Department of Applied
More informationPotential Symmetries and Differential Forms. for Wave Dissipation Equation
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 42, 2061-2066 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.36163 Potential Symmetries and Differential Forms for Wave Dissipation
More informationEFFICIENT SPECTRAL COLLOCATION METHOD FOR SOLVING MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS BASED ON THE GENERALIZED LAGUERRE POLYNOMIALS
Journal of Fractional Calculus and Applications, Vol. 3. July 212, No.13, pp. 1-14. ISSN: 29-5858. http://www.fcaj.webs.com/ EFFICIENT SPECTRAL COLLOCATION METHOD FOR SOLVING MULTI-TERM FRACTIONAL DIFFERENTIAL
More informationResearch Article Constrained Solutions of a System of Matrix Equations
Journal of Applied Mathematics Volume 2012, Article ID 471573, 19 pages doi:10.1155/2012/471573 Research Article Constrained Solutions of a System of Matrix Equations Qing-Wen Wang 1 and Juan Yu 1, 2 1
More informationPositive Solution of a Nonlinear Four-Point Boundary-Value Problem
Nonlinear Analysis and Differential Equations, Vol. 5, 27, no. 8, 299-38 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/nade.27.78 Positive Solution of a Nonlinear Four-Point Boundary-Value Problem
More informationAbstract. 1. Introduction
Journal of Computational Mathematics Vol.28, No.2, 2010, 273 288. http://www.global-sci.org/jcm doi:10.4208/jcm.2009.10-m2870 UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS
More informationThe Shifted Data Problems by Using Transform of Derivatives
Applied Mathematical Sciences, Vol. 8, 2014, no. 151, 7529-7534 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49784 The Shifted Data Problems by Using Transform of Derivatives Hwajoon
More informationStrong Stability Preserving Properties of Runge Kutta Time Discretization Methods for Linear Constant Coefficient Operators
Journal of Scientific Computing, Vol. 8, No., February 3 ( 3) Strong Stability Preserving Properties of Runge Kutta Time Discretization Methods for Linear Constant Coefficient Operators Sigal Gottlieb
More informationOn Nonlinear Methods for Stiff and Singular First Order Initial Value Problems
Nonlinear Analysis and Differential Equations, Vol. 6, 08, no., 5-64 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/nade.08.8 On Nonlinear Methods for Stiff and Singular First Order Initial Value Problems
More informationThe Representation of Energy Equation by Laplace Transform
Int. Journal of Math. Analysis, Vol. 8, 24, no. 22, 93-97 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ijma.24.442 The Representation of Energy Equation by Laplace Transform Taehee Lee and Hwajoon
More informationBernstein operational matrices for solving multiterm variable order fractional differential equations
International Journal of Current Engineering and Technology E-ISSN 2277 4106 P-ISSN 2347 5161 2017 INPRESSCO All Rights Reserved Available at http://inpressco.com/category/ijcet Research Article Bernstein
More informationAn Efficient Numerical Method for Solving. the Fractional Diffusion Equation
Journal of Applied Mathematics & Bioinformatics, vol.1, no.2, 2011, 1-12 ISSN: 1792-6602 (print), 1792-6939 (online) International Scientific Press, 2011 An Efficient Numerical Method for Solving the Fractional
More informationHomotopy perturbation method for the Wu-Zhang equation in fluid dynamics
Journal of Physics: Conference Series Homotopy perturbation method for the Wu-Zhang equation in fluid dynamics To cite this article: Z Y Ma 008 J. Phys.: Conf. Ser. 96 08 View the article online for updates
More informationVariational iteration method for solving multispecies Lotka Volterra equations
Computers and Mathematics with Applications 54 27 93 99 www.elsevier.com/locate/camwa Variational iteration method for solving multispecies Lotka Volterra equations B. Batiha, M.S.M. Noorani, I. Hashim
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationProperties of BPFs for Approximating the Solution of Nonlinear Fredholm Integro Differential Equation
Applied Mathematical Sciences, Vol. 6, 212, no. 32, 1563-1569 Properties of BPFs for Approximating the Solution of Nonlinear Fredholm Integro Differential Equation Ahmad Shahsavaran 1 and Abar Shahsavaran
More informationA Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating Polynomials
Applied Mathematical Sciences, Vol. 8, 2014, no. 35, 1723-1730 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4127 A Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating
More informationResearch Article A Legendre Wavelet Spectral Collocation Method for Solving Oscillatory Initial Value Problems
Applied Mathematics Volume 013, Article ID 591636, 5 pages http://dx.doi.org/10.1155/013/591636 Research Article A Legendre Wavelet Spectral Collocation Method for Solving Oscillatory Initial Value Problems
More informationLocating Chromatic Number of Banana Tree
International Mathematical Forum, Vol. 12, 2017, no. 1, 39-45 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.610138 Locating Chromatic Number of Banana Tree Asmiati Department of Mathematics
More informationA New Analytical Method for Solutions of Nonlinear Impulsive Volterra-Fredholm Integral Equations
Nonlinear Analysis and Differential Equations, Vol. 6, 218, no. 4, 121-13 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/nade.218.8112 A New Analytical Method for Solutions of Nonlinear Impulsive
More informationZ. Omar. Department of Mathematics School of Quantitative Sciences College of Art and Sciences Univeristi Utara Malaysia, Malaysia. Ra ft.
International Journal of Mathematical Analysis Vol. 9, 015, no. 46, 57-7 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.015.57181 Developing a Single Step Hybrid Block Method with Generalized
More informationSharp Bounds for Seiffert Mean in Terms of Arithmetic and Geometric Means 1
Int. Journal of Math. Analysis, Vol. 7, 01, no. 6, 1765-177 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.01.49 Sharp Bounds for Seiffert Mean in Terms of Arithmetic and Geometric Means 1
More information(Received 05 August 2013, accepted 15 July 2014)
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.18(2014) No.1,pp.71-77 Spectral Collocation Method for the Numerical Solution of the Gardner and Huxley Equations
More informationA Numerical Solution of Classical Van der Pol-Duffing Oscillator by He s Parameter-Expansion Method
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 15, 709-71 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2013.355 A Numerical Solution of Classical Van der Pol-Duffing Oscillator by
More informationA Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations
Copyright 013 Tech Science Press CMES, vol.94, no.3, pp.5-38, 013 A Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations Xiaojing Liu 1, Jizeng Wang 1,, Youhe Zhou 1, Abstract:
More informationResearch Article Hermite Wavelet Method for Fractional Delay Differential Equations
Difference Equations, Article ID 359093, 8 pages http://dx.doi.org/0.55/04/359093 Research Article Hermite Wavelet Method for Fractional Delay Differential Equations Umer Saeed and Mujeeb ur Rehman School
More informationDifferential transformation method for solving one-space-dimensional telegraph equation
Volume 3, N 3, pp 639 653, 2 Copyright 2 SBMAC ISSN -825 wwwscielobr/cam Differential transformation method for solving one-space-dimensional telegraph equation B SOLTANALIZADEH Young Researchers Club,
More informationA Legendre Computational Matrix Method for Solving High-Order Fractional Differential Equations
Mathematics A Legendre Computational Matrix Method for Solving High-Order Fractional Differential Equations Mohamed Meabed KHADER * and Ahmed Saied HENDY Department of Mathematics, Faculty of Science,
More informationNumerical study of time-fractional hyperbolic partial differential equations
Available online at wwwisr-publicationscom/jmcs J Math Computer Sci, 7 7, 53 65 Research Article Journal Homepage: wwwtjmcscom - wwwisr-publicationscom/jmcs Numerical study of time-fractional hyperbolic
More informationResearch Article Soliton Solutions for the Wick-Type Stochastic KP Equation
Abstract and Applied Analysis Volume 212, Article ID 327682, 9 pages doi:1.1155/212/327682 Research Article Soliton Solutions for the Wick-Type Stochastic KP Equation Y. F. Guo, 1, 2 L. M. Ling, 2 and
More informationNew Integrable Decomposition of Super AKNS Equation
Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 803 808 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 5, November 15, 2010 New Integrable Decomposition of Super AKNS Equation JI Jie
More informationDouble Total Domination on Generalized Petersen Graphs 1
Applied Mathematical Sciences, Vol. 11, 2017, no. 19, 905-912 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.7114 Double Total Domination on Generalized Petersen Graphs 1 Chengye Zhao 2
More informationA New Characterization of A 11
International Journal of Algebra, Vol. 8, 2014, no. 6, 253-266 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4211 A New Characterization of A 11 Yong Yang, Shitian Liu and Yanhua Huang
More informationA New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources
Commun. Theor. Phys. Beijing, China 54 21 pp. 1 6 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 1, July 15, 21 A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent
More informationContra θ-c-continuous Functions
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 1, 43-50 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.714 Contra θ-c-continuous Functions C. W. Baker
More informationSolution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, 307-315 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7418 Solution of the Hirota Equation Using Lattice-Boltzmann and the
More informationAn Analytic Study of the (2 + 1)-Dimensional Potential Kadomtsev-Petviashvili Equation
Adv. Theor. Appl. Mech., Vol. 3, 21, no. 11, 513-52 An Analytic Study of the (2 + 1)-Dimensional Potential Kadomtsev-Petviashvili Equation B. Batiha and K. Batiha Department of Mathematics, Faculty of
More informationOn Numerical Solutions of Systems of. Ordinary Differential Equations. by Numerical-Analytical Method
Applied Mathematical Sciences, Vol. 8, 2014, no. 164, 8199-8207 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ams.2014.410807 On Numerical Solutions of Systems of Ordinary Differential Equations
More information256 Summary. D n f(x j ) = f j+n f j n 2n x. j n=1. α m n = 2( 1) n (m!) 2 (m n)!(m + n)!. PPW = 2π k x 2 N + 1. i=0?d i,j. N/2} N + 1-dim.
56 Summary High order FD Finite-order finite differences: Points per Wavelength: Number of passes: D n f(x j ) = f j+n f j n n x df xj = m α m dx n D n f j j n= α m n = ( ) n (m!) (m n)!(m + n)!. PPW =
More informationApproximation to the Dissipative Klein-Gordon Equation
International Journal of Mathematical Analysis Vol. 9, 215, no. 22, 159-163 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.5236 Approximation to the Dissipative Klein-Gordon Equation Edilber
More informationOn a Certain Representation in the Pairs of Normed Spaces
Applied Mathematical Sciences, Vol. 12, 2018, no. 3, 115-119 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712362 On a Certain Representation in the Pairs of ormed Spaces Ahiro Hoshida
More informationResearch Article Solution of the Porous Media Equation by a Compact Finite Difference Method
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2009, Article ID 9254, 3 pages doi:0.55/2009/9254 Research Article Solution of the Porous Media Equation by a Compact Finite Difference
More information