Applications of nonstandard analysis to partial differential equations—I. the diffusion equation
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Heat equation
FTCS scheme
Finite difference
Universal differential equation
Matrix difference equation
Central differencing scheme
The paper is devoted to an implicit finite difference method (FDM) for solving initial-boundary value problems (IBVP) for one-dimensional wave equation. The second-order derivatives in the wave equation have been approximated at the four intermediate points, as a consequence, an implicit nine-point difference scheme has been obtained. Von Neumann stability analysis has been conducted and we have demonstrated, that the presented difference scheme is unconditionally stable.
Central differencing scheme
Finite difference
Finite difference scheme
Matrix difference equation
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In this paper, a finite difference scheme of the generalized Rose- nau equation is proposed. Existence and uniqueness of numerical solution are proved. The convergent in the order of is showed. Numerical simulations show the method is efficient.
Central differencing scheme
Finite difference scheme
Finite difference
Matrix difference equation
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In this paper,a finite difference method for an initial-boundary value problem of generalized symmetric regularized-long-wave equation was considered.A energy conservative finite difference of three levels was proposed.This scheme simulates the conservation properties of the problem well.It is proved that the finite difference scheme is convergent in second-order and stable.The numerical examples show this scheme is feasible.
Finite difference scheme
Finite difference
Central differencing scheme
Conservation law
Matrix difference equation
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In this paper,the nonlinear partial equation with dissipation term-Burgers equation is adopted as model equation.The problems of the difference schemes matching of left item and right item of the equation and the influence of the stability of high-order difference schemes are discussed by adjusting right item of equation and analyzed the characteristics of the equation.The results improve the method of choosing the difference schemes.
Central differencing scheme
Matrix difference equation
FTCS scheme
Fisher's equation
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The numerical solution to the problem of generalized Rosenau equation with initial-boundary value is considered.A finite difference scheme of three levels is proposed.This scheme can be used to simulate the conservation properties of the problem well.It is proved that the finite difference scheme is convergent with order 2 and stable without condition by discrete functional analytical method.The numerical examples show this scheme is feasible.
Central differencing scheme
Finite difference scheme
Matrix difference equation
Finite difference
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The numerical solution for an initial-boundary value problem of Benjamin-Bona-Mahony equation was considered. A pseudo-compact finite difference of three levels was proposed. The existence and u-niqueness of solution are studied. It is proved that the finite difference scheme is convergent with the convergence order 2 and stable by the discrete functional analysis. And the results are demonstrated by the numerical examples.
Central differencing scheme
Compact finite difference
Finite difference scheme
Matrix difference equation
Finite difference
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The numerical solution for an initial-boundary value problem of Rosenau equation is considered.An implicit finite difference of three levels is proposed.This scheme simulates the conservation properties of the problem well.And the prior estimate of the solution is obtained.It is proved that the finite difference scheme is convergent with order 2and stable by the discrete functional analysis.Numerical examples demonstrate the theoretical results.
Finite difference scheme
Central differencing scheme
Finite difference
Matrix difference equation
Conservation law
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Matrix difference equation
Finite difference
Fisher's equation
Central differencing scheme
FTCS scheme
Universal differential equation
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In this paper,a finite difference method for a initial-boundary value problem of Benjamin-Bona-Mahony equation was considered.A pseudo-compact finite difference of two levels was proposed.Existence and uniqueness of numerical solutions are derived.It is proved that the finite difference scheme is convergent in order o(τ2+h2) and stable.Numerical experiments show that this method is efficient.
Finite difference
Compact finite difference
Finite difference scheme
Central differencing scheme
Matrix difference equation
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In this paper,a finite difference method is presented for the initial value problems of Rosenau-Kaw ahara Equation. A three level linear conservation finite difference scheme with one weighted coefficient is designed. The scheme has the advantages that it preserves two invariant properties of the original differential equation. It is proved that the finite difference scheme is convergent with second-order and unconditionally stable by discrete functional analysis method. Numerical identification also show s that appropriate adjustments to the one weighted parameter will significantly improve the computational accuracy.
Central differencing scheme
Finite difference scheme
Finite difference
Matrix difference equation
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