The design and implementation of accelerators capable of providing high-quality bunches require precise and efficient online modeling tools. Current comprehensive beam dynamics studies are prohibitively costly and challenging to use for interactive system design. A precise high-speed method for beam dynamics analysis in accelerator components is presented and compared to the results of the conventional particle-in-cell codes. Using powerful mathematical techniques, the suggested method evaluates the temporal evolution of a bunch shape in six-dimensional (6D) phase space along the accelerators. The moment equations that govern the evolution of the bunch envelope in 6D phase space are introduced. The three-dimensional space-charge, external, and emittance forces are calculated to be fully analytically insensitive to different beam envelopes. Substituting the obtained forces into the beam-envelope equations establishes a set of six modified equations describing the beam dynamics using simple algebraic expressions. The whole solution considers the energy spread inherent to an electron beam. The model accuracy is demonstrated by studying beam transport through various components of an accelerator. Applying this analytical approach not only forms a style of physical thinking by indicating the factors that affect the behavior of the charged particle bunches but also has an ultrafast computational speed that is at least 3 orders of magnitude faster than that of particle tracking codes for designing today's linear accelerators. Finally, the model's feasibility is benchmarked for successfully designing a photoinjector for the advanced proton driven plasma wakefield acceleration experiment.