Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems.

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Fractional Calculus

Background:

• Fractional partial differential equations (PDEs) are crucial for modeling complex phenomena.
• Nonlinear time fractional hyperbolic and pseudo hyperbolic PDEs with nonlocal conditions present significant analytical challenges.
• Existing numerical methods may lack accuracy or efficiency for these types of problems.

Purpose of the Study:

• To present and solve two specific problems involving nonlinear time fractional hyperbolic PDEs and time fractional pseudo hyperbolic PDEs.
• To introduce and validate the Collocation Technique for Shifted Chebyshev of the Second Kind with Residual Power Series Algorithm (CTSCSK-RPSA).
• To provide a detailed error analysis for the proposed numerical method.

Main Methods:

• The primary method employed is the Collocation Technique for Shifted Chebyshev of the Second Kind with Residual Power Series Algorithm (CTSCSK-RPSA).
• This technique combines spectral methods with a series expansion approach for efficient computation.
• The method is applied to solve nonlinear time fractional hyperbolic and pseudo hyperbolic PDEs with nonlocal conditions.

Main Results:

• Numerical solutions were obtained for the presented fractional PDEs.
• The accuracy and efficiency of the CTSCSK-RPSA method were demonstrated through detailed error analysis.
• The numerical results obtained using CTSCSK-RPSA were compared with existing techniques, showing competitive performance.

Conclusions:

• The CTSCSK-RPSA is an accurate, simple, and convenient method for solving linear and nonlinear fractional PDEs.
• This method offers a reliable approach for tackling complex problems in physics and engineering.
• The detailed error analysis supports the robustness of the CTSCSK-RPSA for fractional differential equations.