A Suitable Algorithm to Solve a Nonlinear Fractional Integro-Differential Equation with Extended Singular Kernel in (2+1) Dimensions

In this paper, the authors consider a problem with comprehensive properties in terms of form and content in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi mathvariant="script">L</mi></mrow><mrow><mn>2</mn></mrow></msub><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></mfenced><mo>×</mo><mfenced separators="|"><mrow><mi mathvariant="normal">c</mi><mo>,</mo><mi mathvariant="normal">d</mi></mrow></mfenced></mrow></mfenced><mo>×</mo><mi>C</mi><mfenced open="[" close="]" separators="|"><mrow><mn>0</mn><mo>,</mo><mi>T</mi></mrow></mfenced><mo>,</mo><mi>T</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In terms of time form, we assume that the time phase delay is implicitly contained in a nonlinear differential integral equation. The positional part is considered in two dimensions, and the position’s kernel is a general singular kernel, many different forms of which will be derived. In terms of content, all of the previously established numerical techniques are only appropriate for studying special cases of the kernel separately but are not suitable for studying the general kernel. This led to the use of the Toeplitz matrix method, which deals with the kernel in its extended nonlinear form and the special kernels will be studied as applications of the method. Moreover, this method has the advantage of converting all single integrals into regular integrals that can be easily solved. Additionally, the researchers examine the solution’s existence, uniqueness, and convergence in this paper. The error and its stability are also studied. At the end of the research, the authors studied some numerical applications of some of the singular kernels derived from the general kernel, examining the approximation error in each application separately.