| Summary: | In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) equation by replacing the integer-order time derivative with the Caputo fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">η</mi><mo>=</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> with variable coefficients. We consider homogeneous boundary conditions to find the approximate solutions for the bounded space variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo><</mo><mi mathvariant="italic">χ</mi><mo><</mo><mi>L</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>,</mo><mi>L</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. To confirm the effectiveness of the proposed methods of non-integer order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">η</mi></semantics></math></inline-formula>, the computation of two test problems was presented. A comparison is made between the obtained results of the (VIM), (ADM), and (HAM) through tables and graphs. The numerical results demonstrate the effectiveness of the three numerical methods.
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