Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation
oleh: McSylvester Ejighikeme Omaba, Eze R. Nwaeze
| Format: | Article |
|---|---|
| Diterbitkan: | MDPI AG 2019-04-01 |
Deskripsi
We study a class of conformable time-fractional stochastic equation <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>t</mi> </mrow> <mi>a</mi> </msubsup> <mi>u</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>σ</mi> <mrow> <mo>(</mo> <mi>u</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <msub> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> <mi>t</mi> </msub> <mo>,</mo> <mspace width="0.166667em"></mspace> <mspace width="0.166667em"></mspace> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mspace width="0.166667em"></mspace> <mi>t</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mi>a</mi> <mo>,</mo> <mi>T</mi> <mo>]</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt"></mspace> <mi>T</mi> <mo><</mo> <mo>∞</mo> <mo>,</mo> <mspace width="0.166667em"></mspace> <mspace width="0.166667em"></mspace> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> The initial condition <inline-formula> <math display="inline"> <semantics> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em"></mspace> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula> is a non-random function assumed to be non-negative and bounded, <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>T</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>t</mi> </mrow> <mi>a</mi> </msubsup> </semantics> </math> </inline-formula> is a conformable time-fractional derivative, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>:</mo> <mi mathvariant="double-struck">R</mi> <mo>→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula> is Lipschitz continuous and <inline-formula> <math display="inline"> <semantics> <msub> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> <mi>t</mi> </msub> </semantics> </math> </inline-formula> a generalized derivative of Wiener process. Some precise condition for the existence and uniqueness of a solution of the class of equation is given and we also give an upper bound estimate on the growth moment of the solution. Unlike the growth moment of stochastic fractional heat equation with Riemann–Liouville or Caputo–Dzhrbashyan fractional derivative which grows in time like <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>t</mi> <msub> <mi>c</mi> <mn>1</mn> </msub> </msup> <mo form="prefix">exp</mo> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em"></mspace> <mspace width="0.166667em"></mspace> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <mspace width="0.166667em"></mspace> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>; our result also shows that the energy of the solution (the second moment) grows exponentially in time for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mo>[</mo> <mi>a</mi> <mo>,</mo> <mi>T</mi> <mo>]</mo> <mo>,</mo> <mspace width="0.166667em"></mspace> <mspace width="0.166667em"></mspace> <mi>T</mi> <mo><</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula> but with at most <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo form="prefix">exp</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for some constants <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mspace width="0.166667em"></mspace> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> </inline-formula>.