Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation

Koroche, Kedir Aliyi

doi:10.11648/j.mcs.20200504.12

Cited by 8 publications

(8 citation statements)

References 9 publications

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“…(9) , Wick, T. (10) , Salsa, S., & Verzini, G. (11) , Yang, W. Y. (12) , Koroche, K. A (13) , S. Sathyapriya (14) , Sharma, T., Pathak, D. S., Trivedi, G. J. & Sanghvi, R. Yang (15) , Omowo B. J.…”

Section: Comparative Analysis Through Previous Workmentioning

confidence: 99%

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Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

Sharma,

Pathak,

Trivedi

2024

IJST

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Objectives: This paper aims to address the limitations of the Crank-Nicolson Finite Difference method and propose an improved version called the modified Crank-Nicolson method. Methods: Utilized implicit discretization in time and space, with parameters k = 0.001, h = 0.1, and γ = 0.1. Conducted extensive testing on various partial differential equations. Findings: Results, displayed in Table 1, showcase the method's stability and accuracy. Comparative analysis in Table 2 demonstrates the Modified Crank-Nicolson method consistently outperforming the traditional approach, reaffirming its superiority in accuracy. Novelty: The modified Crank-Nicolson method offers a significant enhancement to the traditional Crank-Nicolson finite difference method, making it a valuable tool for effectively solving partial differential equations. Keywords: CrankNicolson Method, Modified CrankNicolson Method, Finite Difference, Partial Differential Equations, Parabolic Equations, Python Software

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Section: Comparative Analysis Through Previous Workmentioning

confidence: 99%

“…To derive the Crank Nicolson finite difference method (13,14) , we used the finite difference approximations, LHS of equation ( 1) is replace by…”

Section: Mathematical Foundationmentioning

confidence: 99%

Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

Sharma,

Pathak,

Trivedi

2024

IJST

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“…The Fourier analysis (Von-Neumann) stability [32] analysis technique is applied to investigate the stability of the proposed method. Such an approach has been used by many researchers like [23][24][25][26]32]. Now consider that the make nonlinearity in the difference scheme is linear by taking ] = max ( such that Thus this shows that the scheme is stable.…”

Section: Stability and Convergent Analysismentioning

confidence: 99%

Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation

Koroche¹

2021

IJAMTP

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In this paper, the piecewise parabolic method is presented for solving the one-dimensional advection-diffusion type equation and its application to the burger equation. First, the given solution domain is discretized by using a uniform Discretization grid point. Next by applying the integration in terms of spatial variable, we discretized the given advectiondiffusion type equation and converting it into the system of first-order ordinary differential equation in terms of temporarily variable. Next, by using Taylor series expansion we discretized the obtained system of ordinary differential equation and obtain the central finite difference equation. Then using this difference equation, the given advection-diffusion type equation is solved by using the parabolic method at each specified grid point. To validate the applicability of the proposed method, four model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm L 2 and maximum absolute error norm L ∞ and the local behavior of the solution is captured exactly. Numerical, exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root means square error norm L 2 and maximum absolute error norm L ∞ presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving advection-diffusion type equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve the advection-diffusion type equation.

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“…( 12) is stable if for which the eigenvalues of the coefficient matrix of the system of the differential equation are satisfied ��𝜆𝜆 � � � 0. Proof: See reference [15] Since from the principal part of the local truncation error, the derived local truncation error for the proposed scheme is…”

Section: Stability and Convergent Analysismentioning

confidence: 99%

Lax-Wendroff Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected To Integral Condition

Koroche

2021

ijmph

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In this paper, the Lax-Wendroff difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion and the Lax-Wendroff difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L � and L � norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

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Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation

Cited by 8 publications

References 9 publications

Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation

Lax-Wendroff Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected To Integral Condition

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