Talaat El Danaf

Al-Azhar University

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Mohamed Ramadan

Al-Azhar University

Kamal Raslan

Al-Azhar University

Mohamed A. Abd Elsalam

Al-Azhar University

Khaled S. Mohamed

Higher Institute of Engineering

Khalid K. Ali

Al-Azhar University

Mohamed Abd

Northern Technical University

El Fatih Abdullahi Abdel Salam

Al-Azhar University

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Al-Azhar University

Cairo University

Menoufia University

Assiut University

National Research Centre

South Valley University

Aswan University

King Faisal Specialist Hospital & Research Centre

Taif University

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Solving systems of ordinary differential equations in unbounded domains by exponential Chebyshev collocation method

DOAJ (DOAJ: Directory of Open Access Journals) (2016)

Mohamed Ramadan Kamal Raslan Talaat El Danaf Mohamed A. Abd Elsalam Mohamed Ramadan

Collocation (remote sensing)

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A new exponential Chebyshev operational matrix of derivatives for solving high-order ordinary differential equations in unbounded domains

Journal of Modern Methods in Numerical Mathematics (2016)

Mohamed Ramadan Kamal Raslan Talaat El Danaf Mohamed A. Abd Elsalam

The purpose of this paper is to investigate a new exponential Chebyshev (EC) operational matrix of derivatives. The new operational matrix of derivatives of the EC functions is derived and introduced for solving high-order linear ordinary differential equations with variable coefficients in unbounded domain using the collocation method. This method transforms the given differential equation and conditions to matrix equation with unknown EC coefficients. These matrices together with the collocation method are utilized to reduce the solution of high-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of EC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive with good accuracy.

Matrix Exponential

Orthogonal collocation

Matrix (chemical analysis)

Exponential integrator

Algebraic equation

10.20454/jmmnm.2016.1068

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Citations (6)

Solving systems of high-order linear ordinary differential equations with variable coefficients by means of exponential Chebyshev collocation method

Journal of Modern Methods in Numerical Mathematics (2017)

Mohamed Ramadan Kamal Raslan Talaat El Danaf Mohamed A. Abd Elsalam

The purpose of this paper is to investigate the use of exponential Chebyshev (EC) collocation method for solving systems of high-order linear ordinary differential equations with variable coefficients with new scheme, using the EC collocation method in unbounded domains. The EC functions approach deals directly with infinite boundaries without singularities. The method transforms the system of differential equations and the given conditions to block matrix equations with unknown EC coefficients. By means of the obtained matrix equations, a new system of equations which corresponds to the system of linear algebraic equations is gained. Numerical examples are given to illustrative the validity and applicability of the method.

Coefficient matrix

Orthogonal collocation

Matrix Exponential

Chebyshev iteration

Exponential integrator

Algebraic equation

Reduction of order

Independent equation

Collocation (remote sensing)

10.20454/jmmnm.2017.1131

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Citations (3)

Collocation Method with Quintic B-Spline Method for Solving the Hirota equation

DOAJ (DOAJ: Directory of Open Access Journals) (2016)

Khalid K. Ali Kamal Raslan Talaat El Danaf

Quintic function

Collocation (remote sensing)

B-spline

Spline (mechanical)

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MODIFIED EXPONENTIAL CHEBYSHEV OPERATIONAL MATRICES OF DERIVATIVES FOR SOLVING HIGH-ORDER PARTIAL DIFFERENTIAL EQUATIONS IN UNBOUNDED DOMAINS Vol. 5(2) July 2017, No. 22, pp. 229-241 M. A. RAMADAN, K. R. MOHAMED, T. S. EL DANAF, M. A. ABD EL SALAM

Electronic Journal of Mathematical Analysis and Applications (2017)

Mohamed Ramadan Khaled S. Mohamed Talaat El Danaf Mohamed Abd El Fatih Abdullahi Abdel Salam

In this paper, a modified type of exponential Chebyshev operational matrices of derivatives is presented.The introduced operational matrices were employed for solving high-order linear partial differential equations (PDEs) with variable coefficients under general form of conditions by collocation method.The method is based on the approximation by the truncated double exponential Chebyshev (EC) series.The PDEs and conditions are transformed into block matrix equations, which correspond to a system of linear algebraic equations with the unknown EC coefficients, by using EC collocation points.Combining these matrix equations and then solving the system yields the EC coefficients of the solution function.Numerical examples are included to demonstrate the validity and applicability of the method.

10.21608/ejmaa.2017.310988

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